{VERSION 6 0 "IBM INTEL NT" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 
0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 
0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 
0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 
102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times
" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 
261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 262 "" 
1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 264 "Times" 1 12 96 
52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 265 "" 1 18 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 260 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 
260 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 269 "" 1 
18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 270 "" 1 18 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 
0 0 }{CSTYLE "" 261 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "
" 258 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 274 "" 0 
1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 
0 0 0 0 0 0 0 }{CSTYLE "" 260 276 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
}{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 
278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 279 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" 260 281 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 
283 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 284 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" 260 286 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 289 "" 1 18 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
293 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 297 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 299 "" 1 18 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 300 "" 1 18 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 }{CSTYLE "" -1 301 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 260 302 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
303 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 306 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 258 307 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 
308 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 309 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 310 "" 0 1 0 0 0 0 0 0 1 0 0 
0 0 0 0 0 }{CSTYLE "" 261 311 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 312 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 
313 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 314 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 315 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 316 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 317 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
318 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 319 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 320 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 321 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 322 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
323 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 324 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 325 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" 260 326 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 327 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
328 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 329 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 330 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 331 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 332 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
333 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 334 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 335 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 336 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 337 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 
338 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 339 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 340 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" 260 341 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 342 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
343 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 344 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 345 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 346 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 347 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
348 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 349 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 350 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 351 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 352 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
353 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 354 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 355 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" 260 356 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 357 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
358 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 359 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 360 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 361 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 362 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
363 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 364 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 365 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 366 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 367 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
368 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 369 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 370 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 371 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 372 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
373 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 374 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 375 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 376 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 377 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
378 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 379 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 380 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 381 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 382 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 
383 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 384 "" 1 18 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 385 "" 0 1 0 0 0 0 0 0 1 0 0 
0 0 0 0 0 }{CSTYLE "" -1 386 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 387 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 
388 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 389 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 390 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 391 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 
{CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 
8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Time
s" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }
{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 
1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output
" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }
3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 
-1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 
0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 
2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot
" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }
1 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 84 "Inverse trigonometric functions a
nd the general solution of trigonometric equations " }}{PARA 0 "" 0 "
" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "
" {TEXT -1 18 "Version: 23.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Th
e inverse sine function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 684 204 204 
{PLOTDATA 2 "6/-%'CURVESG6$7S7$$!\"\"\"\"!$!3Z_t9*zYYA\"!#L7$$!3%pmm\"
HU,\"*)*!#=$!3!eJMu.?KU$!#>7$$!3=L$3FH'='z*F1$!3+<szX@i)R'F47$$!3SmmTg
Ba*o*F1$!3IEHfN$eyt*F47$$!3_mm\"H_\">#e*F1$!376(*>`#=)38F17$$!3!HL3_!4
Nv%*F1$!3(o,$*[,\"yS;F17$$!3Um;/wfHw$*F1$!3U+o7h)4p%>F17$$!3%***\\PM.t
t#*F1$!3S1OoW#)*=E#F17$$!3-n;/,oln\"*F1$!3'\\Bn!=d=&e#F17$$!3%)**\\(oW
B>1*F1$!3YY;AE+e/HF17$$!3sKL$epjJ&*)F1$!3)>\"y++%p(HKF17$$!3?nm\"z/ot&
))F1$!3i`2Z&)e38NF17$$!35++]P[_\\()F1$!31&)R!zT8#GQF17$$!3f++]7)Q7k)F1
$!3GB\"=b]8-9%F17$$!3u****\\i^)o`)F1$!3Jc2P:VNOWF17$$!3Cn;/^?7U%)F1$!3
&>g\"zvj:,ZF17$$!3oLL$eaR%H$)F1$!31YhK!3!f5]F17$$!30MLL$o#)RB)F1$!3)y(
\\-qS#yE&F17$$!3S+]PfO%H7)F1$!3#[')*p<E2hbF17$$!3&QLL$3`lC!)F1$!3vAC3]
G,:eF17$$!3*)**\\P4u\"o\"zF1$!3'y2'=t&Qs3'F17$$!3C+]7G-89yF1$!3=J6&p-t
*RjF17$$!3bLL3Fp)pq(F1$!31A.=Qxj'f'F17$$!3!GL3-$ff3wF1$!3_REO*Qgd#oF17
$$!3rmm\"z%zY-vF1$!3#*)[P_G$elqF17$$!33n;/,3B#R(F1$!3*oMuftsjI(F17$$!3
#3+D1/piH(F1$!3Rn/S!)o&)3vF17$$!3%pm;/rFE>(F1$!3E[F3a\\()>xF17$$!3;+++
vgb&3(F1$!3$z\\a!>0GHzF17$$!3_++vot!3)pF1$!3*)eN'HVja7)F17$$!3Q+]P4wXz
oF1$!3VD(R4x2pI)F17$$!3W++D\")f#pw'F1$!3eo>W6w\\)\\)F17$$!3oLLL38\"em'
F1$!3wf)y)=tfh')F17$$!3m++]7)\\yb'F1$!3uu<Xi)*3E))F17$$!3!pmT5l?+Y'F1$
!378nwU;Qm*)F17$$!33++]Pv1`jF1$!3V=A=!zi+6*F17$$!3UM$3F*HV_iF1$!3(eH:l
fneB*F17$$!3I+](o*3CZhF1$!3/4v%fq+vN*F17$$!3aLL$eR'RWgF1$!3]Rc\"eCTlY*
F17$$!3B+]PMxsOfF1$!3k*Q6\"[t5q&*F17$$!35nmm;+.LeF1$!3]0[.Z[]f'*F17$$!
3Onm\"H7%)ps&F1$!3fA7())[C.u*F17$$!3TL$3FM;=i&F1$!35A)QN9*z4)*F17$$!38
+++v!y^_&F1$!3eZ^y.2?k)*F17$$!3$RL$e9(>WT&F1$!3OVD&*GvO:**F17$$!3Tnmm;
hN:`F1$!3s\"*z!pxj4&**F17$$!3))**\\PfOt4_F1$!3u(=<=a+$y**F17$$!3*)**\\
i!\\R'3^F1$!3s\"H^eCwT***F17$$!3++++++++]F1F(-%'COLOURG6&%$RGBG$\"*+++
+\"!\")$F*F*F_[l-F$6$7S7$$\"3++++++++]F1$\"\"\"F*7$$\"35mmmT:(z@&F1$\"
3G;1:HJcw**F17$$\"3jLLe9ui2aF1$\"3fF1Fp_6=**F17$$\"3Anm;z_\"4i&F1$\"34
[=2I#[.\")*F17$$\"3$pmmT&phNeF1$\"3s_#=g&*)Rd'*F17$$\"35LLe*=)H\\gF1$
\"3OILBKvch%*F17$$\"3;nm\"z/3uC'F1$\"3oV%H&)R3>C*F17$$\"37++DJ$RDX'F1$
\"3'pWRhEjn(*)F17$$\"3'fm;zR'okmF1$\"3/S-igHOj')F17$$\"3I++D1J:woF1$\"
3f`[DuHo7$)F17$$\"3WLLL3En$4(F1$\"3TPYbCzr8zF17$$\"3qmm;/RE&G(F1$\"3DI
7E7bkJvF17$$\"3\")*****\\K]4](F1$\"3qzfR0k&*oqF17$$\"3$******\\PAvr(F1
$\"3#e1XdUE<d'F17$$\"3`+++v'Hi#zF1$\"3kb#)px?vjgF17$$\"3jmm\"z*ev:\")F
1$\"3S$zqguD)zbF17$$\"3kKLL347T$)F1$\"3_^uQ4uzy\\F17$$\"3,LLLLY.K&)F1$
\"3Q$4`M\"f+]WF17$$\"3?***\\7o7Tv)F1$\"34hP8oU*[\"QF17$$\"3IKLL$Q*o]*)
F1$\"3(=Ev16DrB$F17$$\"3A++D\"=lj;*F1$\"3A.Ji5`5*e#F17$$\"3]***\\PaR<P
*F1$\"33Ik].%[4'>F17$$\"3!HLLe9Ege*F1$\"3%*QIC5T(oH\"F17$$\"3GLLeR\"3G
y*F1$\"3B-h,9-*z\"oF47$$\"3cmm;/T1&***F1$\"3)p\"HX(yc1b\"!#?7$$\"3em;z
RQb@5!#<$!3iD&yR6lhw'F47$$\"3%)**\\(=>Y2/\"Fdcl$!3=$[n+L'ew7F17$$\"3im
m\"zXu91\"Fdcl$!3LA[LhQH>>F17$$\"3'******\\y))G3\"Fdcl$!35I9(*4zpuDF17
$$\"3!****\\i_QQ5\"Fdcl$!3W\"e&3+7j/KF17$$\"3#***\\7y%3T7\"Fdcl$!3Ac(z
HMV4!QF17$$\"3#****\\P![hY6Fdcl$!3jjpLIB*[W%F17$$\"3ELLLQx$o;\"Fdcl$!3
k\"*=)Gi`Y+&F17$$\"3')****\\P+V)=\"Fdcl$!3U'z9?Xs*zbF17$$\"3im;zpe*z?
\"Fdcl$!3My([F&**>zgF17$$\"3)*****\\#\\'QH7Fdcl$!3'Q_7#3)[')f'F17$$\"3
7L$e9S8&\\7Fdcl$!3KVA2_+DgqF17$$\"3%***\\i?=bq7Fdcl$!3!yihB*Qc7vF17$$
\"3GLL$3s?6H\"Fdcl$!3=RduJ13BzF17$$\"3&***\\7`Wl78Fdcl$!30<^(3$4R<$)F1
7$$\"3emmm'*RRL8Fdcl$!31zx(zx17m)F17$$\"3_mmTvJga8Fdcl$!3=')3qHUyu*)F1
7$$\"3KL$e9tOcP\"Fdcl$!3y**\\JZ=VY#*F17$$\"3'******\\Qk\\R\"Fdcl$!3#H3
6\\5\"\\g%*F17$$\"3@LL3dg6<9Fdcl$!3I*GTok-Hm*F17$$\"3_mmmw(GpV\"Fdcl$!
3F!*p'y,OV!)*F17$$\"3-+]7oK0e9Fdcl$!3kLecSjH8**F17$$\"3-+](=5s#y9Fdcl$
!3eSwim<rw**F17$$\"3++++++++:FdclF(Fhz-F$6%7SFez7$$!3MLLLe%G?y%F1$!3G;
1:HJcw**F17$$!3OmmT&esBf%F1$!3fF1Fp_6=**F17$$!3KLL$3s%3zVF1$!34[=2I#[.
\")*F17$$!33LL$e/$QkTF1$!3g^#=g&*)Rd'*F17$$!3!pm;/\"=q]RF1$!3OILBKvch%
*F17$$!3SLL3_>f_PF1$!3oV%H&)R3>C*F17$$!3))***\\(o1YZNF1$!3'eWRhEjn(*)F
17$$!3]LL3-OJNLF1$!3#*Q-igHOj')F17$$!3C++v$*o%Q7$F1$!3qa[DuHo7$)F17$$!
3ammm\"RFj!HF1$!3>NYbCzr8zF17$$!3JLL$e4OZr#F1$!3DI7E7bkJvF17$$!3=+++v'
\\!*\\#F1$!3!3)fR0k&*oqF17$$!33+++DwZ#G#F1$!3#e1XdUE<d'F17$$!3-+++D.xt
?F1$!3'yD)px?vjgF17$$!3OLL3-TC%)=F1$!3=\"zqguD)zbF17$$!3!omm;4z)e;F1$!
3kZuQ4uzy\\F17$$!3+nmmm`'zY\"F1$!3;\"4`M\"f+]WF17$$!3E++v=t)eC\"F1$!3?
dP8oU*[\"QF17$$!39nmm;1J\\5F1$!3wg_n5^7PKF17$$!3&y***\\(=[jL)F4$!3M/Ji
5`5*e#F17$$!3M****\\iXg#G'F4$!3kDk].%[4'>F17$$!3WlmmT&Q(RTF4$!3)p.V-6u
oH\"F17$$!3;nm;/'=><#F4$!3`ng,9-*z\"oF47$$!3vDMLLe*e$\\!#@$!3+qEX(yc1b
\"F`cl7$$\"3[em;zRQb@F4$\"3o=&yR6lhw'F47$$\"3'[***\\(=>Y2%F4$\"3G*[n+L
'ew7F17$$\"3Qhmm\"zXu9'F4$\"3\\@[LhQH>>F17$$\"3]'******\\y))G)F4$\"3KK
9(*4zpuDF17$$\"3'*)***\\i_QQ5F1$\"3A%e&3+7j/KF17$$\"3@***\\7y%3T7F1$\"
3We(zHMV4!QF17$$\"35****\\P![hY\"F1$\"3&e'pLIB*[W%F17$$\"3kKLL$Qx$o;F1
$\"3`!*=)Gi`Y+&F17$$\"3!)*****\\P+V)=F1$\"3w*z9?Xs*zbF17$$\"3?mm\"zpe*
z?F1$\"3My([F&**>zgF17$$\"3%)*****\\#\\'QH#F1$\"33ED@3)[')f'F17$$\"3GK
Le9S8&\\#F1$\"3wZA2_+DgqF17$$\"3R***\\i?=bq#F1$\"3\"*G;O#*Qc7vF17$$\"3
\"HLL$3s?6HF1$\"3STduJ13BzF17$$\"3a***\\7`Wl7$F1$\"3Q?^(3$4R<$)F17$$\"
3#pmmm'*RRL$F1$\"3Q#yxzx17m)F17$$\"3Qmm;a<.YNF1$\"3I()3qHUyu*)F17$$\"3
=LLe9tOcPF1$\"3!4+:t%=VY#*F17$$\"3u******\\Qk\\RF1$\"3#=36\\5\"\\g%*F1
7$$\"3CLL$3dg6<%F1$\"3U!HTok-Hm*F17$$\"3ImmmmxGpVF1$\"3F!*p'y,OV!)*F17
$$\"3A++D\"oK0e%F1$\"3vMecSjH8**F17$$\"3A++v=5s#y%F1$\"3eSwim<rw**F1Fc
[l-Fiz6&F[[lF_[lF_[lF\\[l-%*THICKNESSG6#\"\"$-F$6&7$Fc[lFez-%'SYMBOLG6
#%'CIRCLEGFgim-%&STYLEG6#%&POINTG-F$6&F_jm-Fajm6#%(DIAMONDGFgimFdjm-F$
6&F_jm-Fajm6#%&CROSSGFgimFdjm-%%TEXTG6&7$$\"#<F)$F)F)Q\"x6\"-%&COLORG6
&F[[l$Fg[l!\"#F^\\nF^\\n-%%FONTG6$%*HELVETICAG\"\"*-Fc[n6&7$$!\"&F_\\n
$\"#8F)Q\"yFj[nF[\\nF`\\n-%*AXESTICKSG6$7(/F)%#-pG/$Fi\\nF)%%-p/2G/F*%
\"0G/$\"\"&F)%$p/2G/Fg[l%\"pG/$\"#:F)%%3p/2GF\\jm-Fa\\n6$FajmFd\\n-%&T
ITLEG6$Q*y~=~sin~xFj[nF`\\n-%+AXESLABELSG6%%!GF[_nF`\\n-%%VIEWG6$;F(Ff
[n;F(Fj\\n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C
urve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "C
urve 8" }}{TEXT -1 7 "       " }}{PARA 257 "" 0 "" {TEXT -1 93 "The bl
ue curve is the graph of the sine function with its domain restricted \+
to the interval  " }{XPPEDIT 18 0 "-Pi/2 <= x;" "6#1,$*&%#PiG\"\"\"\"
\"#!\"\"F)%\"xG" }{XPPEDIT 18 0 "`` <= Pi/2;" "6#1%!G*&%#PiG\"\"\"\"\"
#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 108 "With this res
tricted domain the sine function is a one-to-one function, which then \+
has an inverse function. " }}{PARA 0 "" 0 "" {TEXT -1 29 "This inverse
 is denoted by   " }{XPPEDIT 18 0 "sin^(-1)" "6#)%$sinG,$\"\"\"!\"\"" 
}{TEXT -1 4 " or " }{TEXT 264 6 "arcsin" }{TEXT -1 17 ", and called th
e " }{TEXT 259 21 "inverse sine function" }{TEXT -1 8 " or the " }
{TEXT 259 16 "arcsine function" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "
y = arcsin*x;" "6#/%\"yG*&%'arcsinG\"\"\"%\"xGF'" }{TEXT -1 18 "   exa
ctly when   " }{XPPEDIT 18 0 "sin*y = x;" "6#/*&%$sinG\"\"\"%\"yGF&%\"
xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "-Pi/2 <= y;" "6#1,$*&%#PiG\"\"
\"\"\"#!\"\"F)%\"yG" }{XPPEDIT 18 0 "`` <= Pi/2;" "6#1%!G*&%#PiG\"\"\"
\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 
262 29 "_____________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "The graph of the arcs
ine function is obtained by reflecting the graph of the sine function \+
(restricted in the way just suggested) in the line " }{XPPEDIT 18 0 "y
=x" "6#/%\"yG%\"xG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{GLPLOT2D 476 476 476 {PLOTDATA 2 "6--%'CURVESG6$7S7$$!3++++++++]!#
=$!3#p!z$='))4$=$F*7$$!3MLLLe%G?y%F*$!3=N`9u'Qc<$F*7$$!3OmmT&esBf%F*$!
3'*\\#>D9Mq:$F*7$$!3KLL$3s%3zVF*$!3:Av`G3tAJF*7$$!33LL$e/$QkTF*$!3UG;_
ib/uIF*7$$!3!pm;/\"=q]RF*$!3USaI%[5<,$F*7$$!3SLL3_>f_PF*$!3DEpX53zTHF*
7$$!3))***\\(o1YZNF*$!33!oVN\\#RdGF*7$$!3]LL3-OJNLF*$!3O'*Q'z2Mwv#F*7$
$!3C++v$*o%Q7$F*$!3a(*pTr\"4gk#F*7$$!3ammm\"RFj!HF*$!3-5c&=l9!>DF*7$$!
3JLL$e4OZr#F*$!3'*HXyDsR(R#F*7$$!3=+++v'\\!*\\#F*$!3#GxR)3(=,D#F*7$$!3
3+++DwZ#G#F*$!3_@*41\\X=4#F*7$$!3-+++D.xt?F*$!3jc))pLA:I>F*7$$!3OLL3-T
C%)=F*$!3Yyr:)p8hx\"F*7$$!3!omm;4z)e;F*$!3SqUrO/![e\"F*7$$!3+nmmm`'zY
\"F*$!3*)[%Ge(3[;9F*7$$!3E++v=t)eC\"F*$!3w^GHg&=V@\"F*7$$!39nmm;1J\\5F
*$!39L<aD*3/.\"F*7$$!3&y***\\(=[jL)!#>$!35$)GUn\"y8C)Fiq7$$!3M****\\iX
g#G'Fiq$!31kTKJE*=C'Fiq7$$!3WlmmT&Q(RTFiq$!38f;E/&y!GTFiq7$$!3;nm;/'=>
<#Fiq$!35`c-*oL-<#Fiq7$$!3vDMLLe*e$\\!#@$!3A>=CbQ*e$\\F^s7$$\"3[em;zRQ
b@Fiq$\"3YP7LZst`@Fiq7$$\"3'[***\\(=>Y2%Fiq$\"3ebcF%\\+N1%Fiq7$$\"3Qhm
m\"zXu9'Fiq$\"3!>NZb5-$4hFiq7$$\"3]'******\\y))G)Fiq$\"3'R]1o)z^&>)Fiq
7$$\"3'*)***\\i_QQ5F*$\"3'oEgDzl+-\"F*7$$\"3@***\\7y%3T7F*$\"3wB\\*G%y
()47F*7$$\"35****\\P![hY\"F*$\"3uj(o<<`[T\"F*7$$\"3kKLL$Qx$o;F*$\"3OI)
)3D2.$f\"F*7$$\"3!)*****\\P+V)=F*$\"3?=&4hRghx\"F*7$$\"3?mm\"zpe*z?F*$
\"3Spz/X%p]$>F*7$$\"3%)*****\\#\\'QH#F*$\"3HN*36::/5#F*7$$\"3GKLe9S8&
\\#F*$\"3M>U]!RZtC#F*7$$\"3R***\\i?=bq#F*$\"3g&3EvNB8R#F*7$$\"3\"HLL$3
s?6HF*$\"35$)f6%*[*>_#F*7$$\"3a***\\7`Wl7$F*$\"3M_Hbgx]ZEF*7$$\"3#pmmm
'*RRL$F*$\"3g%Q7PuZpv#F*7$$\"3Qmm;a<.YNF*$\"3#)o,oYDwcGF*7$$\"3=LLe9tO
cPF*$\"3R:]#*o1BVHF*7$$\"3u******\\Qk\\RF*$\"3k5q%o%yO6IF*7$$\"3CLL$3d
g6<%F*$\"3#45@<W(zvIF*7$$\"3ImmmmxGpVF*$\"3R\")[&>3<37$F*7$$\"3A++D\"o
K0e%F*$\"3U*pr)H-]bJF*7$$\"3A++v=5s#y%F*$\"3sHP#p)fovJF*7$$\"3++++++++
]F*$\"3#p!z$='))4$=$F*-%'COLOURG6&%$RGBG$\"\"!F^[lF][l$\"*++++\"!\")-F
$6$7`q7$$!3%****za&))4$=$F*$!3iCPG9))z**\\F*7$$!3P2\"f)e)=e6$F*$!3\"*G
)[[z\"QWVF*7$$!3QcMa$*)ys0$F*$!3N+Pn/B/-TF*7$$!3sg@/!eW9*HF*$!3)=WiI#e
z*)QF*7$$!3_lj(R(Q<DHF*$!3fE3#)R'z(4PF*7$$!3W/%359=#fGF*$!3Sh)QV?4;b$F
*7$$!3!QgDOto!)z#F*$!3([WRH&*[\"=MF*7$$!3'4i#>k?vMFF*$!3G$*Hx;n0!H$F*7
$$!39)f*3u)p#pEF*$!3#fx@NC8h;$F*7$$!3[22?'o(*Rg#F*$!3!HY>kue&\\IF*7$$!
35^_)z@do`#F*$!3=xG,O$*oNHF*7$$!3Oed.d)>xZ#F*$!3F)HcdBx'RGF*7$$!3RvCD&
>X6T#F*$!3byG-&H9dt#F*7$$!3w[:^rrHWBF*$!3D+/y$=Z^j#F*7$$!3E\"e**Rrw)zA
F*$!3@ayi`TSTDF*7$$!3)3f))pww8A#F*$!33U&e(\\%p'eCF*7$$!3g2lvrY\"=:#F*$
!3#y\"zS`[$HO#F*7$$!3!)z)>EX')G4#F*$!3ea6l[*eQG#F*7$$!3-VM\"[#*QV-#F*$
!3aBj'RrwR>#F*7$$!3[ZJxZFmj>F*$!3p<=[4'Gh6#F*7$$!3c7K!\\O\"4(*=F*$!3l&
fpl`(RK?F*7$$!3#=M]cm*pL=F*$!3I\"Rsm(*eT&>F*7$$!3o<fqxqbn<F*$!3%p5%=xU
%R(=F*7$$!3+2;\\Mv\"oq\"F*$!3#HkF?Za9!=F*7$$!39/x]J<IT;F*$!3?x#z6M@Ws
\"F*7$$!3!GT)3D#\\Kd\"F*$!3kO(ylfvbk\"F*7$$!3@,y>=$4S^\"F*$!3gfGt=^$yd
\"F*7$$!3Sec#efG+X\"F*$!3X_gD<x`0:F*7$$!3AYq4\\-$RQ\"F*$!35?%z&\\#><V
\"F*7$$!3*RS'>7fE>8F*$!3!3G)=(*QHg8F*7$$!3[ynET)*pc7F*$!3_z@V$ou=H\"F*
7$$!3E/Ds')3B(=\"F*$!3W'**>t`Qm@\"F*7$$!3-,zIT*4[7\"F*$!3K_2`F9k\\6F*7
$$!3m'y$feA;e5F*$!3C)G0<;!py5F*7$$!3/_WhAbpx**Fiq$!3>\"fet#o([,\"F*7$$
!3_g2OCPW<$*Fiq$!3%RxM#G$4fX*Fiq7$$!3Ox)HoB)>'p)Fiq$!3+c$*Q_\"z\"3))Fi
q7$$!3e)\\4')p<o/)Fiq$!3i\"R0OY)3N\")Fiq7$$!3yUKaE&H>T(Fiq$!3?8O15\")f
![(Fiq7$$!3A4Lw[/EZnFiq$!3W*HM!R\"Q))z'Fiq7$$!3%\\Zn!H^52hFiq$!3s!owA@
2_9'Fiq7$$!3)p*p*\\&QX_aFiq$!3'RrmH9w%zaFiq7$$!3m<@+CMA.[Fiq$!3DJ>1-6k
@[Fiq7$$!36.?Q%3[m?%Fiq$!3\"Qd*[%G!**=UFiq7$$!3D8[o@a!H_$Fiq$!3Tc:O&QP
,`$Fiq7$$!3!3*H@LyN6HFiq$!3`:([<MKa\"HFiq7$$!3A\\k2_0KfAFiq$!3e5l8I>Ah
AFiq7$$!3q#QYfnP_j\"Fiq$!3G94=+y&fj\"Fiq7$$!3N:PTf%QFU*!#?$!3SE$er>:TU
*Fhjl7$$!3WvVRn3=AMFhjl$!3%po@pzYAU$Fhjl7$$\"3w'H<9n0&)Q$Fhjl$\"3A%*[n
t'p&)Q$Fhjl7$$\"3i]k?$32df*Fhjl$\"3u%zg(e5;(f*Fhjl7$$\"3m*p*p#*4aQ;Fiq
$\"3[kgP$\\l#R;Fiq7$$\"3;n9N@vPCAFiq$\"3I?4+4>>EAFiq7$$\"3cxc3Wpd\"*GF
iq$\"31KE]f(ob*GFiq7$$\"3<-::['eg`$Fiq$\"3CO\"QMAsLa$Fiq7$$\"3CI8\"zK>
,=%Fiq$\"3uY*e6GGA>%Fiq7$$\"3j15jC1\"=#[Fiq$\"3=h0\"=XW/%[Fiq7$$\"3\"4
cc>2v#QaFiq$\"3Yyi[=_3laFiq7$$\"3_&eY`Y*o/hFiq$\"3?^=,SeuUhFiq7$$\"3$=
t<MqI)RnFiq$\"3U=O8IeB\"z'Fiq7$$\"3Ct7i[pG3uFiq$\"3y\"HWxi_oZ(Fiq7$$\"
3Lp,5>3S8!)Fiq$\"3Bb$R,ia05)Fiq7$$\"3sfCiC\\;#o)Fiq$\"3gC249=f$z)Fiq7$
$\"3m*f:9SCFK*Fiq$\"3gdT)Q/K9Y*Fiq7$$\"3?;S^QZsh**Fiq$\"3/>O%R8&>85F*7
$$\"3[3'GnI;H1\"F*$\"3)H(4x)HKP3\"F*7$$\"3XM(yL\")*QC6F*$\"3%))y8sL#>
\\6F*7$$\"37Y`%H4Zt=\"F*$\"3h=z=#yjn@\"F*7$$\"3Od()H5%foD\"F*$\"3/CS8j
$[?H\"F*7$$\"30wj\"*>rz>8F*$\"3KjV)Gix3O\"F*7$$\"3-0.)oriTQ\"F*$\"3y?0
Z(Rx>V\"F*7$$\"37!=]5(3i\\9F*$\"3uHb@F(z]]\"F*7$$\"3y/p[_'*y4:F*$\"3'f
\\wF.SId\"F*7$$\"3!)))p3Dw#Rd\"F*$\"3upSjJgNY;F*7$$\"3Jv/FvtdP;F*$\"31
$*=jgj2?<F*7$$\"3qhhABGm0<F*$\"3u[\"**=l'3+=F*7$$\"3s#3%f\"GXdw\"F*$\"
3UV!y&Giwr=F*7$$\"3DYQ*f'z1N=F*$\"3=9\\QeK$e&>F*7$$\"3aUI6Wji(*=F*$\"3
[NN.?Q1L?F*7$$\"3O]vK*eC&f>F*$\"3*ysQi7u36#F*7$$\"39W&Gz3&3E?F*$\"3k%*
y,t+C'>#F*7$$\"3_!ymu$[\"H4#F*$\"37&3q<i'*QG#F*7$$\"3=t.D9+s`@F*$\"3gb
NAr9_lBF*7$$\"3io!Q5T#H=AF*$\"3K7G@'*eOaCF*7$$\"3e8PFf(38G#F*$\"3L\"Q)
G;pXVDF*7$$\"31ZAdX\">'\\BF*$\"3Rf'>oBDIk#F*7$$\"3\"*\\j/Ams3CF*$\"3j*
[Du<7?t#F*7$$\"3g:4k2PhwCF*$\"3uf+26e\"z$GF*7$$\"3*4pTq8J/a#F*$\"3oEsG
1OhTHF*7$$\"3O>.nf@j.EF*$\"3=#e#\\pK#*[IF*7$$\"3WKY=fM)om#F*$\"3[Tci*R
P<;$F*7$$\"3@:'H4y&RJFF*$\"37*=W')G4NG$F*7$$\"33B/Bhxx*z#F*$\"35C#yJVP
<U$F*7$$\"3Yo=(4MxC'GF*$\"3s\"Q]H!H/fNF*7$$\"3eTg=x)pV#HF*$\"3DQ-/ZAu2
PF*7$$\"3oNrxz>`!*HF*$\"39[tnS&Gr)QF*7$$\"3qCRD`+bcIF*$\"3'=XMB\"yU*4%
F*7$$\"33#e[Y>0e6$F*$\"3`:Pb&)\\JWVF*7$$\"33++me))4$=$F*$\"3Q8\"ROwd)*
*\\F*-Fjz6&F\\[lF_[lF][lF][l-F$6$7S7$F(F(7$F.F.7$F3F37$F8F87$F=F=7$FBF
B7$FGFG7$FLFL7$FQFQ7$FVFV7$FenFen7$FjnFjn7$F_oF_o7$FdoFdo7$FioFio7$F^p
F^p7$FcpFcp7$FhpFhp7$F]qF]q7$FbqFbq7$FgqFgq7$F]rF]r7$FbrFbr7$FgrFgr7$F
\\sF\\s7$FbsFbs7$FgsFgs7$F\\tF\\t7$FatFat7$FftFft7$F[uF[u7$F`uF`u7$Feu
Feu7$FjuFju7$F_vF_v7$FdvFdv7$FivFiv7$F^wF^w7$FcwFcw7$FhwFhw7$F]xF]x7$F
bxFbx7$FgxFgx7$F\\yF\\y7$FayFay7$FfyFfy7$F[zF[z7$F`zF`z7$FezFez-Fjz6&F
\\[lF][lF_[lF][l-%%TEXTG6&7$$\"\"&!\"\"$\"#F!\"#Q*y~=~sin~x6\"Fiz-%%FO
NTG6$%*HELVETICAG\"\"*-Fc^n6&7$$\"\"#Fh^n$\"#bF[_nQ-y~=~arcsin~xF]_nFj
jmF^_n-Fc^n6&7$$\"\"'Fh^n$!\"$F[_nQ\"xF]_n-Fjz6&F\\[lF^[lF^[lF^[lF^_n-
Fc^n6&7$F``nF^`nQ\"yF]_nFc`nF^_n-%*AXESTICKSG6$7'/$!\"&Fh^n%%-p/2G/$!&
J=$F_an%#-1G/F^[l%\"0G/$\"&J=$F_an%\"1G/Ff^n%$p/2GF\\an-F__n6$%'SYMBOL
GFb_n-%+AXESLABELSG6%%!GFcbn-F__n6#%(DEFAULTG-%%VIEWG6$;F^anF^`nFjbn" 
1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 "
 " }}{PARA 257 "" 0 "" {TEXT -1 17 "Since the graph  " }{XPPEDIT 18 0 
"y = sin*x;" "6#/%\"yG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 14 " has the li
ne " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 40 " as a tangent \+
at the origin, the graph  " }{XPPEDIT 18 0 "y = arcsin*x;" "6#/%\"yG*&
%'arcsinG\"\"\"%\"xGF'" }{TEXT -1 20 ", also has the line " }{XPPEDIT 
18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 30 "  as a tangent at the origin.
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The \+
graph of " }{XPPEDIT 18 0 "y = arcsin*x;" "6#/%\"yG*&%'arcsinG\"\"\"%
\"xGF'" }{TEXT -1 56 " on its own suggests the shape of \"half an hour
-glass\". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 298 369 369 
{PLOTDATA 2 "6*-%'CURVESG6$7`q7$$!3%****za&))4$=$!#=$!3iCPG9))z**\\F*7
$$!3P2\"f)e)=e6$F*$!3\"*G)[[z\"QWVF*7$$!3QcMa$*)ys0$F*$!3N+Pn/B/-TF*7$
$!3sg@/!eW9*HF*$!3)=WiI#ez*)QF*7$$!3_lj(R(Q<DHF*$!3fE3#)R'z(4PF*7$$!3W
/%359=#fGF*$!3Sh)QV?4;b$F*7$$!3!QgDOto!)z#F*$!3([WRH&*[\"=MF*7$$!3'4i#
>k?vMFF*$!3G$*Hx;n0!H$F*7$$!39)f*3u)p#pEF*$!3#fx@NC8h;$F*7$$!3[22?'o(*
Rg#F*$!3!HY>kue&\\IF*7$$!35^_)z@do`#F*$!3=xG,O$*oNHF*7$$!3Oed.d)>xZ#F*
$!3F)HcdBx'RGF*7$$!3RvCD&>X6T#F*$!3byG-&H9dt#F*7$$!3w[:^rrHWBF*$!3D+/y
$=Z^j#F*7$$!3E\"e**Rrw)zAF*$!3@ayi`TSTDF*7$$!3)3f))pww8A#F*$!33U&e(\\%
p'eCF*7$$!3g2lvrY\"=:#F*$!3#y\"zS`[$HO#F*7$$!3!)z)>EX')G4#F*$!3ea6l[*e
QG#F*7$$!3-VM\"[#*QV-#F*$!3aBj'RrwR>#F*7$$!3[ZJxZFmj>F*$!3p<=[4'Gh6#F*
7$$!3c7K!\\O\"4(*=F*$!3l&fpl`(RK?F*7$$!3#=M]cm*pL=F*$!3I\"Rsm(*eT&>F*7
$$!3o<fqxqbn<F*$!3%p5%=xU%R(=F*7$$!3+2;\\Mv\"oq\"F*$!3#HkF?Za9!=F*7$$!
39/x]J<IT;F*$!3?x#z6M@Ws\"F*7$$!3!GT)3D#\\Kd\"F*$!3kO(ylfvbk\"F*7$$!3@
,y>=$4S^\"F*$!3gfGt=^$yd\"F*7$$!3Sec#efG+X\"F*$!3X_gD<x`0:F*7$$!3AYq4
\\-$RQ\"F*$!35?%z&\\#><V\"F*7$$!3*RS'>7fE>8F*$!3!3G)=(*QHg8F*7$$!3[ynE
T)*pc7F*$!3_z@V$ou=H\"F*7$$!3E/Ds')3B(=\"F*$!3W'**>t`Qm@\"F*7$$!3-,zIT
*4[7\"F*$!3K_2`F9k\\6F*7$$!3m'y$feA;e5F*$!3C)G0<;!py5F*7$$!3/_WhAbpx**
!#>$!3>\"fet#o([,\"F*7$$!3_g2OCPW<$*F_v$!3%RxM#G$4fX*F_v7$$!3Ox)HoB)>'
p)F_v$!3+c$*Q_\"z\"3))F_v7$$!3e)\\4')p<o/)F_v$!3i\"R0OY)3N\")F_v7$$!3y
UKaE&H>T(F_v$!3?8O15\")f![(F_v7$$!3A4Lw[/EZnF_v$!3W*HM!R\"Q))z'F_v7$$!
3%\\Zn!H^52hF_v$!3s!owA@2_9'F_v7$$!3)p*p*\\&QX_aF_v$!3'RrmH9w%zaF_v7$$
!3m<@+CMA.[F_v$!3DJ>1-6k@[F_v7$$!36.?Q%3[m?%F_v$!3\"Qd*[%G!**=UF_v7$$!
3D8[o@a!H_$F_v$!3Tc:O&QP,`$F_v7$$!3!3*H@LyN6HF_v$!3`:([<MKa\"HF_v7$$!3
A\\k2_0KfAF_v$!3e5l8I>AhAF_v7$$!3q#QYfnP_j\"F_v$!3G94=+y&fj\"F_v7$$!3N
:PTf%QFU*!#?$!3SE$er>:TU*Ffz7$$!3WvVRn3=AMFfz$!3%po@pzYAU$Ffz7$$\"3w'H
<9n0&)Q$Ffz$\"3A%*[nt'p&)Q$Ffz7$$\"3i]k?$32df*Ffz$\"3u%zg(e5;(f*Ffz7$$
\"3m*p*p#*4aQ;F_v$\"3[kgP$\\l#R;F_v7$$\"3;n9N@vPCAF_v$\"3I?4+4>>EAF_v7
$$\"3cxc3Wpd\"*GF_v$\"31KE]f(ob*GF_v7$$\"3<-::['eg`$F_v$\"3CO\"QMAsLa$
F_v7$$\"3CI8\"zK>,=%F_v$\"3uY*e6GGA>%F_v7$$\"3j15jC1\"=#[F_v$\"3=h0\"=
XW/%[F_v7$$\"3\"4cc>2v#QaF_v$\"3Yyi[=_3laF_v7$$\"3_&eY`Y*o/hF_v$\"3?^=
,SeuUhF_v7$$\"3$=t<MqI)RnF_v$\"3U=O8IeB\"z'F_v7$$\"3Ct7i[pG3uF_v$\"3y
\"HWxi_oZ(F_v7$$\"3Lp,5>3S8!)F_v$\"3Bb$R,ia05)F_v7$$\"3sfCiC\\;#o)F_v$
\"3gC249=f$z)F_v7$$\"3m*f:9SCFK*F_v$\"3gdT)Q/K9Y*F_v7$$\"3?;S^QZsh**F_
v$\"3/>O%R8&>85F*7$$\"3[3'GnI;H1\"F*$\"3)H(4x)HKP3\"F*7$$\"3XM(yL\")*Q
C6F*$\"3%))y8sL#>\\6F*7$$\"37Y`%H4Zt=\"F*$\"3h=z=#yjn@\"F*7$$\"3Od()H5
%foD\"F*$\"3/CS8j$[?H\"F*7$$\"30wj\"*>rz>8F*$\"3KjV)Gix3O\"F*7$$\"3-0.
)oriTQ\"F*$\"3y?0Z(Rx>V\"F*7$$\"37!=]5(3i\\9F*$\"3uHb@F(z]]\"F*7$$\"3y
/p[_'*y4:F*$\"3'f\\wF.SId\"F*7$$\"3!)))p3Dw#Rd\"F*$\"3upSjJgNY;F*7$$\"
3Jv/FvtdP;F*$\"31$*=jgj2?<F*7$$\"3qhhABGm0<F*$\"3u[\"**=l'3+=F*7$$\"3s
#3%f\"GXdw\"F*$\"3UV!y&Giwr=F*7$$\"3DYQ*f'z1N=F*$\"3=9\\QeK$e&>F*7$$\"
3aUI6Wji(*=F*$\"3[NN.?Q1L?F*7$$\"3O]vK*eC&f>F*$\"3*ysQi7u36#F*7$$\"39W
&Gz3&3E?F*$\"3k%*y,t+C'>#F*7$$\"3_!ymu$[\"H4#F*$\"37&3q<i'*QG#F*7$$\"3
=t.D9+s`@F*$\"3gbNAr9_lBF*7$$\"3io!Q5T#H=AF*$\"3K7G@'*eOaCF*7$$\"3e8PF
f(38G#F*$\"3L\"Q)G;pXVDF*7$$\"31ZAdX\">'\\BF*$\"3Rf'>oBDIk#F*7$$\"3\"*
\\j/Ams3CF*$\"3j*[Du<7?t#F*7$$\"3g:4k2PhwCF*$\"3uf+26e\"z$GF*7$$\"3*4p
Tq8J/a#F*$\"3oEsG1OhTHF*7$$\"3O>.nf@j.EF*$\"3=#e#\\pK#*[IF*7$$\"3WKY=f
M)om#F*$\"3[Tci*RP<;$F*7$$\"3@:'H4y&RJFF*$\"37*=W')G4NG$F*7$$\"33B/Bhx
x*z#F*$\"35C#yJVP<U$F*7$$\"3Yo=(4MxC'GF*$\"3s\"Q]H!H/fNF*7$$\"3eTg=x)p
V#HF*$\"3DQ-/ZAu2PF*7$$\"3oNrxz>`!*HF*$\"39[tnS&Gr)QF*7$$\"3qCRD`+bcIF
*$\"3'=XMB\"yU*4%F*7$$\"33#e[Y>0e6$F*$\"3`:Pb&)\\JWVF*7$$\"33++me))4$=
$F*$\"3Q8\"ROwd)**\\F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F`[mF_[m-%
%TEXTG6&7$$\"\"#!\"\"$\"#[!\"#Q-y~=~arcsin~x6\"Fhjl-%%FONTG6$%*HELVETI
CAG\"\"*-Fb[m6&7$$\"#PFj[m$!\"$Fj[mQ\"xF\\\\m-Fijl6&F[[mF`[mF`[mF`[mF]
\\m-Fb[m6&7$$Fj[mFj[m$\"\"'Fg[mQ\"yF\\\\mFj\\mF]\\m-%*AXESTICKSG6$7%/$
!&J=$!\"&%#-1G/F`[m%\"0G/$\"&J=$Fj]m%\"1G7%/$Fj]mFg[m%%-p/2GF\\^m/$\"
\"&Fg[m%$p/2G-F^\\m6$%'SYMBOLGFa\\m-%+AXESLABELSG6%%!GF`_m-F^\\m6#%(DE
FAULTG-%%VIEWG6$;$!#LFj[mFe\\m;Fd^mF`]m" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" }}{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 35 "The domain of the
 arcsine function " }{XPPEDIT 18 0 "arcsin*x;" "6#*&%'arcsinG\"\"\"%\"
xGF%" }{TEXT -1 17 " is the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$
\"\"\"!\"\"F%" }{TEXT -1 32 ", and its range is the interval " }
{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F
(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 55 "The arcsine funct
ion has the following special values. " }}{PARA 256 "" 0 "" {XPPEDIT 
18 0 "arcsin*0 = 0;" "6#/*&%'arcsinG\"\"\"\"\"!F&F'" }{TEXT -1 2 ", " 
}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(1/2) = Pi/6;
" "6#/-%'arcsinG6#*&\"\"\"F(\"\"#!\"\"*&%#PiGF(\"\"'F*" }{TEXT -1 10 "
,         " }{XPPEDIT 18 0 "arcsin(-1/2) = -Pi/6;" "6#/-%'arcsinG6#,$*
&\"\"\"F)\"\"#!\"\"F+,$*&%#PiGF)\"\"'F+F+" }{TEXT -1 2 ", " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(1/sqrt(2)) = Pi/4;" 
"6#/-%'arcsinG6#*&\"\"\"F(-%%sqrtG6#\"\"#!\"\"*&%#PiGF(\"\"%F-" }
{TEXT -1 10 ",         " }{XPPEDIT 18 0 "arcsin(-1/sqrt(2)) = -Pi/4;" 
"6#/-%'arcsinG6#,$*&\"\"\"F)-%%sqrtG6#\"\"#!\"\"F.,$*&%#PiGF)\"\"%F.F.
" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
arcsin(sqrt(3)/2)=Pi/3" "6#/-%'arcsinG6#*&-%%sqrtG6#\"\"$\"\"\"\"\"#!
\"\"*&%#PiGF,F+F." }{TEXT -1 11 ",         a" }{XPPEDIT 18 0 "rcsin(-s
qrt(3)/2)=-Pi/3" "6#/-%&rcsinG6#,$*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F/,
$*&%#PiGF-F,F/F/" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "arcsin*1 = Pi/2;" "6#/*&%'arcsinG\"\"\"F&F&*&%#PiGF&\"
\"#!\"\"" }{TEXT -1 11 ",         a" }{XPPEDIT 18 0 "rcsin(-1) = -Pi/2
;" "6#/-%&rcsinG6#,$\"\"\"!\"\",$*&%#PiGF(\"\"#F)F)" }{TEXT -1 2 ". " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The arc
sine function is an " }{TEXT 259 12 "odd function" }{TEXT -1 38 ", tha
t is, it satisfies the relation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "arcsin(-x) = -arcsin*x;" "6#/-%'arcsinG6#,$%\"xG!\"\",
$*&F%\"\"\"F(F,F)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 269 12 "____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 28 "The inverse cosine function " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{GLPLOT2D 737 220 220 {PLOTDATA 2 "6/-%'CURVESG6$7S7$$!3++++++++]!
#=$\"3Mint&*RBBh!#M7$$!3%pmm\"HU,\"*[F*$\"3wJVVP+ABM!#>7$$!3=L$3FH'='z
%F*$\"3EKszX@i)R'F37$$!3SmmTgBa*o%F*$\"3O>HfN$eyt*F37$$!3_mm\"H_\">#e%
F*$\"3N3(*>`#=)38F*7$$!3YL$3_!4NvWF*$\"349I*[,\"yS;F*7$$!3Um;/wfHwVF*$
\"35-o7h)4p%>F*7$$!3%***\\PM.ttUF*$\"3!Rg$oW#)*=E#F*7$$!3-n;/,olnTF*$
\"3SMs1=d=&e#F*7$$!3%)**\\(oWB>1%F*$\"3\"fk@i-!e/HF*7$$!3ELL$epjJ&RF*$
\"3V6y++%p(HKF*7$$!3mmm\"z/ot&QF*$\"3^d2Z&)e38NF*7$$!35++]P[_\\PF*$\"3
^%)R!zT8#GQF*7$$!3/++]7)Q7k$F*$\"3&\\7=b]8-9%F*7$$!3u****\\i^)o`$F*$\"
3vb2P:VNOWF*7$$!3om;/^?7UMF*$\"31.;zvj:,ZF*7$$!3oLL$eaR%HLF*$\"3%Q9E.3
!f5]F*7$$!3]LLL$o#)RB$F*$\"35!)\\-qS#yE&F*7$$!3S+]PfO%H7$F*$\"3sj)*p<E
2hbF*7$$!3&QLL$3`lCIF*$\"3k@C3]G,:eF*7$$!3*)**\\P4u\"o\"HF*$\"3'*yg=t&
Qs3'F*7$$!3C+]7G-89GF*$\"32I6&p-t*RjF*7$$!3bLL3Fp)pq#F*$\"3GC.=Qxj'f'F
*7$$!3OL$3-$ff3EF*$\"3HPEO*Qgd#oF*7$$!3rmm\"z%zY-DF*$\"3#*)[P_G$elqF*7
$$!33n;/,3B#R#F*$\"3*oMuftsjI(F*7$$!3E+]iS!piH#F*$\"3Rn/S!)o&)3vF*7$$!
3%pm;/rFE>#F*$\"3E[F3a\\()>xF*7$$!3;+++vgb&3#F*$\"3$z\\a!>0GHzF*7$$!3_
++vot!3)>F*$\"3ydN'HVja7)F*7$$!3Q+]P4wXz=F*$\"3VD(R4x2pI)F*7$$!3W++D\"
)f#pw\"F*$\"3eo>W6w\\)\\)F*7$$!3oLLL38\"em\"F*$\"3wf)y)=tfh')F*7$$!35+
+]7)\\yb\"F*$\"3&ex^C')*3E))F*7$$!3!pmT5l?+Y\"F*$\"378nwU;Qm*)F*7$$!33
++]Pv1`8F*$\"3V=A=!zi+6*F*7$$!3'QL3F*HV_7F*$\"3)pH:lfneB*F*7$$!3I+](o*
3CZ6F*$\"3/4v%fq+vN*F*7$$!3aLL$eR'RW5F*$\"3QQc\"eCTlY*F*7$$!3K-+vVtFn$
*F3$\"3_)Q6\"[t5q&*F*7$$!3Slmmm,II$)F3$\"3]0[.Z[]f'*F*7$$!35om;H7%)psF
3$\"3fA7())[C.u*F*7$$!39ML3FM;=iF3$\"35A)QN9*z4)*F*7$$!3M,++]2y^_F3$\"
3eZ^y.2?k)*F*7$$!3xLL$e9(>WTF3$\"3OVD&*GvO:**F*7$$!3aommm6c`JF3$\"3s\"
*z!pxj4&**F*7$$!3#))**\\PfOt4#F3$\"3u(=<=a+$y**F*7$$!3#*)**\\i!\\R'3\"
F3$\"3s\"H^eCwT***F*7$$\"\"!Ffz$\"\"\"Ffz-%'COLOURG6&%$RGBG$\"*++++\"!
\")FezFez-F$6$7S7$Fgz$!\"\"Ffz7$$\"3hmm;arz@5!#<$!3G;1:HJcw**F*7$$\"3O
L$e9ui2/\"Fi[l$!3fF1Fp_6=**F*7$$\"3smm\"z_\"4i5Fi[l$!34[=2I#[.\")*F*7$
$\"3qmmT&phN3\"Fi[l$!3s_#=g&*)Rd'*F*7$$\"3UL$e*=)H\\5\"Fi[l$!3EHLBKvch
%*F*7$$\"3sm;z/3uC6Fi[l$!3oV%H&)R3>C*F*7$$\"3-+]7LRDX6Fi[l$!3'pWRhEjn(
*)F*7$$\"3em;zR'ok;\"Fi[l$!3/S-igHOj')F*7$$\"3-+]i5`h(=\"Fi[l$!3f`[DuH
o7$)F*7$$\"3YLL$3En$47Fi[l$!3>NYbCzr8zF*7$$\"3cmmT!RE&G7Fi[l$!3eL7E7bk
JvF*7$$\"3)*****\\K]4]7Fi[l$!3!3)fR0k&*oqF*7$$\"3))****\\PAvr7Fi[l$!3;
p]uDksrlF*7$$\"3/++]nHi#H\"Fi[l$!3uc#)px?vjgF*7$$\"3bm;z*ev:J\"Fi[l$!3
%yzqguD)zbF*7$$\"3ELL$347TL\"Fi[l$!3?[uQ4uzy\\F*7$$\"3=LLLjM?`8Fi[l$!3
#y4`M\"f+]WF*7$$\"3#***\\7o7Tv8Fi[l$!3wdP8oU*[\"QF*7$$\"3ALLLQ*o]R\"Fi
[l$!3Vi_n5^7PKF*7$$\"3-+]7=lj;9Fi[l$!3m2Ji5`5*e#F*7$$\"3&***\\PaR<P9Fi
[l$!3uEk].%[4'>F*7$$\"3GLLe9Ege9Fi[l$!3?WIC5T(oH\"F*7$$\"3WL$eR\"3Gy9F
i[l$!3'*=g,9-*z\"oF37$$\"3mmmT5k]*\\\"Fi[l$!3/MDX(yc1b\"!#?7$$\"3em;zR
Qb@:Fi[l$\"3mv%yR6lhw'F37$$\"3%)**\\(=>Y2a\"Fi[l$\"3M#[n+L'ew7F*7$$\"3
imm\"zXu9c\"Fi[l$\"30<[LhQH>>F*7$$\"3'******\\y))Ge\"Fi[l$\"3cH9(*4zpu
DF*7$$\"3!****\\i_QQg\"Fi[l$\"3L&e&3+7j/KF*7$$\"3#***\\7y%3Ti\"Fi[l$\"
3nb(zHMV4!QF*7$$\"3#****\\P![hY;Fi[l$\"32jpLIB*[W%F*7$$\"3ELLLQx$om\"F
i[l$\"34')=)Gi`Y+&F*7$$\"3')****\\P+V)o\"Fi[l$\"3I&z9?Xs*zbF*7$$\"3im;
zpe*zq\"Fi[l$\"3Ax([F&**>zgF*7$$\"3)*****\\#\\'QH<Fi[l$\"3wAD@3)[')f'F
*7$$\"37L$e9S8&\\<Fi[l$\"3AUA2_+DgqF*7$$\"3%***\\i?=bq<Fi[l$\"3.I;O#*Q
c7vF*7$$\"3GLL$3s?6z\"Fi[l$\"3=RduJ13BzF*7$$\"3&***\\7`Wl7=Fi[l$\"30<^
(3$4R<$)F*7$$\"3emmm'*RRL=Fi[l$\"31zx(zx17m)F*7$$\"3_mmTvJga=Fi[l$\"3=
')3qHUyu*)F*7$$\"3KL$e9tOc(=Fi[l$\"3y**\\JZ=VY#*F*7$$\"3'******\\Qk\\*
=Fi[l$\"3#H36\\5\"\\g%*F*7$$\"3@LL3dg6<>Fi[l$\"3I*GTok-Hm*F*7$$\"3_mmm
w(Gp$>Fi[l$\"3F!*p'y,OV!)*F*7$$\"3-+]7oK0e>Fi[l$\"3kLecSjH8**F*7$$\"3-
+](=5s#y>Fi[l$\"3eSwim<rw**F*7$$\"\"#FfzFgzFiz-F$6%7SFdz7$$\"3emmm;arz
@F3$\"3G;1:HJcw**F*7$$\"3[LL$e9ui2%F3$\"3fF1Fp_6=**F*7$$\"3nmmm\"z_\"4
iF3$\"34[=2I#[.\")*F*7$$\"3[mmmT&phN)F3$\"3s_#=g&*)Rd'*F*7$$\"3CLLe*=)
H\\5F*$\"3OILBKvch%*F*7$$\"3gmm\"z/3uC\"F*$\"3zW%H&)R3>C*F*7$$\"3%)***
\\7LRDX\"F*$\"3'eWRhEjn(*)F*7$$\"3]mm\"zR'ok;F*$\"3/S-igHOj')F*7$$\"3w
***\\i5`h(=F*$\"3qa[DuHo7$)F*7$$\"3WLLL3En$4#F*$\"3IOYbCzr8zF*7$$\"3qm
m;/RE&G#F*$\"3DI7E7bkJvF*7$$\"3\")*****\\K]4]#F*$\"3!3)fR0k&*oqF*7$$\"
3$******\\PAvr#F*$\"3#e1XdUE<d'F*7$$\"3)******\\nHi#HF*$\"3'yD)px?vjgF
*7$$\"3jmm\"z*ev:JF*$\"3I#zqguD)zbF*7$$\"3?LLL347TLF*$\"3kZuQ4uzy\\F*7
$$\"3,LLLLY.KNF*$\"3g!4`M\"f+]WF*7$$\"3w***\\7o7Tv$F*$\"3lcP8oU*[\"QF*
7$$\"3'GLLLQ*o]RF*$\"3Kh_n5^7PKF*7$$\"3A++D\"=lj;%F*$\"3!\\5B1J0\"*e#F
*7$$\"31++vV&R<P%F*$\"3ODk].%[4'>F*7$$\"3WLL$e9Ege%F*$\"3QQIC5T(oH\"F*
7$$\"3GLLeR\"3Gy%F*$\"33tg,9-*z\"oF37$$\"3cmm;/T1&*\\F*$\"3eLEX(yc1b\"
F_cl7$$\"3&em;zRQb@&F*$!3M5&yR6lhw'F37$$\"3\\***\\(=>Y2aF*$!3=)[n+L'ew
7F*7$$\"39mm;zXu9cF*$!3m?[LhQH>>F*7$$\"3l******\\y))GeF*$!3KK9(*4zpuDF
*7$$\"3'*)***\\i_QQgF*$!3A%e&3+7j/KF*7$$\"3@***\\7y%3TiF*$!3+f(zHMV4!Q
F*7$$\"35****\\P![hY'F*$!3=kpLIB*[W%F*7$$\"3kKLL$Qx$omF*$!3k\"*=)Gi`Y+
&F*7$$\"3!)*****\\P+V)oF*$!3w*z9?Xs*zbF*7$$\"3?mm\"zpe*zqF*$!3My([F&**
>zgF*7$$\"3%)*****\\#\\'QH(F*$!3'Q_7#3)[')f'F*7$$\"3GKLe9S8&\\(F*$!3mY
A2_+DgqF*7$$\"3R***\\i?=bq(F*$!3!yihB*Qc7vF*7$$\"3\"HLL$3s?6zF*$!3^Udu
J13BzF*7$$\"3a***\\7`Wl7)F*$!3Q?^(3$4R<$)F*7$$\"3#pmmm'*RRL)F*$!3G\"yx
zx17m)F*7$$\"3Qmm;a<.Y&)F*$!3=')3qHUyu*)F*7$$\"3=LLe9tOc()F*$!3+-]JZ=V
Y#*F*7$$\"3u******\\Qk\\*)F*$!3#=36\\5\"\\g%*F*7$$\"3CLL$3dg6<*F*$!3I*
GTok-Hm*F*7$$\"3ImmmmxGp$*F*$!3F!*p'y,OV!)*F*7$$\"3A++D\"oK0e*F*$!3kLe
cSjH8**F*7$$\"3A++v=5s#y*F*$!3eSwim<rw**F*Fc[l-Fjz6&F\\[lFezFezF][l-%*
THICKNESSG6#\"\"$-F$6&7$FdzFc[l-%'SYMBOLG6#%'CIRCLEGFdim-%&STYLEG6#%&P
OINTG-F$6&F\\jm-F^jm6#%(DIAMONDGFdimFajm-F$6&F\\jm-F^jm6#%&CROSSGFdimF
ajm-%%TEXTG6&7$$\"#AFe[l$Fe[lFe[lQ\"x6\"-%&COLORG6&F\\[l$Fhz!\"#F[\\nF
[\\n-%%FONTG6$%*HELVETICAG\"\"*-F`[n6&7$$!\"&F\\\\n$\"#8Fe[lQ\"yFg[nFh
[nF]\\n-%*AXESTICKSG6$7(/$Ff\\nFe[l%%-p/2G/Ffz%\"0G/$\"\"&Fe[l%$p/2G/F
hz%\"pG/$\"#:Fe[l%%3p/2G/Fejl%#2pGFiim-F^\\n6$F^jmFa\\n-%&TITLEG6$Q*y~
=~cos~xFg[nF]\\n-%+AXESLABELSG6%%!GFh^nF]\\n-%%VIEWG6$;F_]nFc[n;Fd[lFg
\\n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1
" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8
" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 95 "The blue curve is the graph of the cosine function with i
ts domain restricted to the interval  " }{XPPEDIT 18 0 "0 <= x;" "6#1
\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= Pi;" "6#1%!G%#PiG" }{TEXT -1 2 ". " 
}}{PARA 0 "" 0 "" {TEXT -1 110 "With this restricted domain the cosine
 function is a one-to-one function, which then has an inverse function
. " }}{PARA 0 "" 0 "" {TEXT -1 29 "This inverse is denoted by   " }
{XPPEDIT 18 0 "cos^(-1);" "6#)%$cosG,$\"\"\"!\"\"" }{TEXT -1 4 " or " 
}{TEXT 264 6 "arccos" }{TEXT -1 17 ", and called the " }{TEXT 259 23 "
inverse cosine function" }{TEXT -1 8 " or the " }{TEXT 259 18 "arccosi
ne function" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus  " }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "y = arccos*x;" "6#/%
\"yG*&%'arccosG\"\"\"%\"xGF'" }{TEXT -1 17 "   exactly when  " }
{XPPEDIT 18 0 "cos*y = x;" "6#/*&%$cosG\"\"\"%\"yGF&%\"xG" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "0 <= y;" "6#1\"\"!%\"yG" }{XPPEDIT 18 0 "`` \+
<= Pi;" "6#1%!G%#PiG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{TEXT 263 27 "___________________________" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 152 "The grap
h of the arccosine function is obtained by reflecting the graph of the
 cosine function (restricted in the way we've just suggested) in the l
ine " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 2 ". " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 405 382 382 {PLOTDATA 2 "6--%'CU
RVESG6$7S7$$\"\"!F)$\"3#p!z$='))4$=$!#=7$$\"3emmm;arz@!#>$\"3=N`9u'Qc<
$F,7$$\"3[LL$e9ui2%F0$\"3'*\\#>D9Mq:$F,7$$\"3nmmm\"z_\"4iF0$\"3:Av`G3t
AJF,7$$\"3[mmmT&phN)F0$\"3'*G;_ib/uIF,7$$\"3CLLe*=)H\\5F,$\"3USaI%[5<,
$F,7$$\"3gmm\"z/3uC\"F,$\"3\"o#pX53zTHF,7$$\"3%)***\\7LRDX\"F,$\"33!oV
N\\#RdGF,7$$\"3]mm\"zR'ok;F,$\"3O'*Q'z2Mwv#F,7$$\"3w***\\i5`h(=F,$\"3a
(*pTr\"4gk#F,7$$\"3WLLL3En$4#F,$\"3e5c&=l9!>DF,7$$\"3qmm;/RE&G#F,$\"3'
*HXyDsR(R#F,7$$\"3\")*****\\K]4]#F,$\"3#GxR)3(=,D#F,7$$\"3$******\\PAv
r#F,$\"3_@*41\\X=4#F,7$$\"3)******\\nHi#HF,$\"3jc))pLA:I>F,7$$\"3jmm\"
z*ev:JF,$\"3uyr:)p8hx\"F,7$$\"3?LLL347TLF,$\"3SqUrO/![e\"F,7$$\"3,LLLL
Y.KNF,$\"3*)[%Ge(3[;9F,7$$\"3w***\\7o7Tv$F,$\"3\\^GHg&=V@\"F,7$$\"3'GL
LLQ*o]RF,$\"3FL<aD*3/.\"F,7$$\"3A++D\"=lj;%F,$\"3(e)GUn\"y8C)F07$$\"31
++vV&R<P%F,$\"3OjTKJE*=C'F07$$\"3WLL$e9Ege%F,$\"3Hj;E/&y!GTF07$$\"3GLL
eR\"3Gy%F,$\"3#[lD!*oL-<#F07$$\"3cmm;/T1&*\\F,$\"3@.<CbQ*e$\\!#@7$$\"3
&em;zRQb@&F,$!3oM7LZst`@F07$$\"3\\***\\(=>Y2aF,$!37_cF%\\+N1%F07$$\"39
mm;zXu9cF,$!37\\ta0@I4hF07$$\"3l******\\y))GeF,$!3'R]1o)z^&>)F07$$\"3'
*)***\\i_QQgF,$!3'oEgDzl+-\"F,7$$\"3@***\\7y%3TiF,$!3.C\\*G%y()47F,7$$
\"35****\\P![hY'F,$!3=j(o<<`[T\"F,7$$\"3kKLL$Qx$omF,$!3kI))3D2.$f\"F,7
$$\"3!)*****\\P+V)oF,$!3?=&4hRghx\"F,7$$\"3?mm\"zpe*zqF,$!3Spz/X%p]$>F
,7$$\"3%)*****\\#\\'QH(F,$!3XM*36::/5#F,7$$\"3GKLe9S8&\\(F,$!31>U]!RZt
C#F,7$$\"3R***\\i?=bq(F,$!3K&3EvNB8R#F,7$$\"3\"HLL$3s?6zF,$!3l$)f6%*[*
>_#F,7$$\"3a***\\7`Wl7)F,$!3M_Hbgx]ZEF,7$$\"3#pmmm'*RRL)F,$!3/%Q7PuZpv
#F,7$$\"3Qmm;a<.Y&)F,$!3#)o,oYDwcGF,7$$\"3=LLe9tOc()F,$!3&f,D*o1BVHF,7
$$\"3u******\\Qk\\*)F,$!3k5q%o%yO6IF,7$$\"3CLL$3dg6<*F,$!3O+6sTuzvIF,7
$$\"3ImmmmxGp$*F,$!3R\")[&>3<37$F,7$$\"3A++D\"oK0e*F,$!3U*pr)H-]bJF,7$
$\"3A++v=5s#y*F,$!3sHP#p)fovJF,7$$\"\"\"F)$!3#p!z$='))4$=$F,-%'COLOURG
6&%$RGBGF(F($\"*++++\"!\")-F$6$7`q7$$!3%****za&))4$=$F,$\"3uDPG9))z***
*F,7$$!3P2\"f)e)=e6$F,$\"3ZH)[[z\"QW$*F,7$$!3QcMa$*)ys0$F,$\"3!4qtYIU?
5*F,7$$!3sg@/!eW9*HF,$\"3)HWiI#ez*)))F,7$$!3_lj(R(Q<DHF,$\"3/E3#)R'z(4
()F,7$$!3W/%359=#fGF,$\"3_i)QV?4;b)F,7$$!3!QgDOto!)z#F,$\"3)fWRH&*[\"=
%)F,7$$!3'4i#>k?vMFF,$\"3G$*Hx;n0!H)F,7$$!39)f*3u)p#pEF,$\"3Ov<_VK6m\"
)F,7$$!3[22?'o(*Rg#F,$\"3!HY>kue&\\!)F,7$$!35^_)z@do`#F,$\"3uxG,O$*oNz
F,7$$!3Oed.d)>xZ#F,$\"3F)HcdBx'RyF,7$$!3RvCD&>X6T#F,$\"3byG-&H9dt(F,7$
$!3w[:^rrHWBF,$\"3!3S!y$=Z^j(F,7$$!3E\"e**Rrw)zAF,$\"3@ayi`TSTvF,7$$!3
)3f))pww8A#F,$\"3_T&e(\\%p'euF,7$$!3g2lvrY\"=:#F,$\"3a<zS`[$HO(F,7$$!3
!)z)>EX')G4#F,$\"3-a6l[*eQG(F,7$$!3-VM\"[#*QV-#F,$\"3aBj'RrwR>(F,7$$!3
[ZJxZFmj>F,$\"3p<=[4'Gh6(F,7$$!3c7K!\\O\"4(*=F,$\"3x'fpl`(RKqF,7$$!3#=
M]cm*pL=F,$\"39#Rsm(*eT&pF,7$$!3o<fqxqbn<F,$\"3]2T=xU%R(oF,7$$!3+2;\\M
v\"oq\"F,$\"3kUw-sWX,oF,7$$!39/x]J<IT;F,$\"3[x#z6M@Ws'F,7$$!3!GT)3D#\\
Kd\"F,$\"3kO(ylfvbk'F,7$$!3@,y>=$4S^\"F,$\"3')fGt=^$yd'F,7$$!3Sec#efG+
X\"F,$\"3c`gD<x`0lF,7$$!3AYq4\\-$RQ\"F,$\"3a>%z&\\#><V'F,7$$!3*RS'>7fE
>8F,$\"3a!G)=(*QHgjF,7$$!3[ynET)*pc7F,$\"3Cz@V$ou=H'F,7$$!3E/Ds')3B(=
\"F,$\"3;'**>t`Qm@'F,7$$!3-,zIT*4[7\"F,$\"3;`2`F9k\\hF,7$$!3m'y$feA;e5
F,$\"33*G0<;!pygF,7$$!3/_WhAbpx**F0$\"3k!fet#o([,'F,7$$!3_g2OCPW<$*F0$
\"3%zZBG$4fXfF,7$$!3Ox)HoB)>'p)F0$\"3;O*Q_\"z\"3)eF,7$$!3e)\\4')p<o/)F
0$\"3*)Q0OY)3N\"eF,7$$!3yUKaE&H>T(F0$\"3wgj+6)f![dF,7$$!3A4Lw[/EZnF0$
\"3AIM!R\"Q))zcF,7$$!3%\\Zn!H^52hF0$\"3@owA@2_9cF,7$$!3)p*p*\\&QX_aF0$
\"3;smH9w%za&F,7$$!3m<@+CMA.[F0$\"3a$>1-6k@[&F,7$$!36.?Q%3[m?%F0$\"3md
*[%G!**=U&F,7$$!3D8[o@a!H_$F0$\"3Obh`QP,``F,7$$!3!3*H@LyN6HF0$\"3Or[<M
Ka\"H&F,7$$!3A\\k2_0KfAF0$\"3W^O,$>AhA&F,7$$!3q#QYfnP_j\"F0$\"31#4=+y&
fj^F,7$$!3N:PTf%QFU*!#?$\"3c$er>:TU4&F,7$$!3WvVRn3=AMFfjl$\"38<#pzYAU.
&F,7$$\"3w'H<9n0&)Q$Ffjl$\"3)4DjKI9h'\\F,7$$\"3i]k?$32df*Ffjl$\"3h#R7%
*QGS!\\F,7$$\"3m*p*p#*4aQ;F0$\"3D%Ri1Xtg$[F,7$$\"3;n9N@vPCAF0$\"393**4
43QxZF,7$$\"3cxc3Wpd\"*GF0$\"3UP(\\S7V/r%F,7$$\"3<-::['eg`$F0$\"3_'=cw
xick%F,7$$\"3CI8\"zK>,=%F0$\"3g0T)=<x2e%F,7$$\"3j15jC1\"=#[F0$\"3^W*=[
bbf^%F,7$$\"3\"4cc>2v#QaF0$\"3Os8:y9\\`WF,7$$\"3_&eY`Y*o/hF0$\"3q:))*f
TDdQ%F,7$$\"3$=t<MqI)RnF0$\"3;Qm)pTw3K%F,7$$\"3Ct7i[pG3uF0$\"3'4dDst9B
D%F,7$$\"3Lp,5>3S8!)F0$\"3/lg)z`W**=%F,7$$\"3sfCiC\\;#o)F0$\"3oF4f=3k?
TF,7$$\"3m*f:9SCFK*F0$\"3_%e6czcQ0%F,7$$\"3?;S^QZsh**F0$\"3'4Qcg'[!o)R
F,7$$\"3[3'GnI;H1\"F,$\"3,F!H7qni\"RF,7$$\"3XM(yL\")*QC6F,$\"3W6iyiw!3
&QF,7$$\"37Y`%H4Zt=\"F,$\"33#37y@OKy$F,7$$\"3Od()H5%foD\"F,$\"3&f(f'oj
^zq$F,7$$\"30wj\"*>rz>8F,$\"3pOc6xB7ROF,7$$\"3-0.)oriTQ\"F,$\"3yz%HDgA
!oNF,7$$\"37!=]5(3i\\9F,$\"3EqWys-#\\\\$F,7$$\"3y/p[_'*y4:F,$\"3g/NAn*
fpU$F,7$$\"3!)))p3Dw#Rd\"F,$\"3EIfOoRk`LF,7$$\"3Jv/FvtdP;F,$\"3]2\"o$R
O#*zKF,7$$\"3qhhABGm0<F,$\"3E^35[L\"**>$F,7$$\"3s#3%f\"GXdw\"F,$\"3ec>
UrPBGJF,7$$\"3DYQ*f'z1N=F,$\"3#e3:;umT/$F,7$$\"3aUI6Wji(*=F,$\"3_kk'*z
h$p'HF,7$$\"3O]vK*eC&f>F,$\"3Qs7wte7*)GF,7$$\"39W&Gz3&3E?F,$\"3j0@)p#*
fP!GF,7$$\"3_!ymu$[\"H4#F,$\"3V:*H#yL5;FF,7$$\"3=t.D9+s`@F,$\"3RWkxG&y
Wj#F,7$$\"3io!Q5T#H=AF,$\"3o(=(y.TjXDF,7$$\"3e8PFf(38G#F,$\"3A>;r$3VlX
#F,7$$\"31ZAdX\">'\\BF,$\"3)3M!=jZ(pN#F,7$$\"3\"*\\j/Ams3CF,$\"3#4^uD#
y)zE#F,7$$\"3g:4k2PhwCF,$\"3!3%*H*)=%3i@F,7$$\"3*4pTq8J/a#F,$\"3)Qx7PR
'Qe?F,7$$\"3O>.nf@j.EF,$\"3a<u]In2^>F,7$$\"3WKY=fM)om#F,$\"3zeVP+EEQ=F
,7$$\"3@:'H4y&RJFF,$\"3W6eN62\\;<F,7$$\"33B/Bhxx*z#F,$\"3=w<#oci#y:F,7
$$\"3Yo=(4MxC'GF,$\"3b='\\q4d4W\"F,7$$\"3eTg=x)pV#HF,$\"3ei(fHvdAH\"F,
7$$\"3oNrxz>`!*HF,$\"3s^EKf9(G6\"F,7$$\"3qCRD`+bcIF,$\"3jyalw=s0!*F07$
$\"33#e[Y>0e6$F,$\"3/YGYW,&ob'F07$$\"33++me))4$=$F,$\"3d.$p)3OOA9!#A-F
jz6&F\\[lF][lF(F(-F$6$7S7$$!35+++++++?F,F_[n7$$!3=++]7c#Q!=F,Fb[n7$$!3
8+](oKNJj\"F,Fe[n7$$!3;++v[i<T9F,Fh[n7$$!39++DTZ%zC\"F,F[\\n7$$!3C+]PH
;jb5F,F^\\n7$$!3[,+vovKt()F0Fa\\n7$$!3C-+v=g9FpF0Fd\\n7$$!3!4+](=C#y,&
F0Fg\\n7$$!32,+vV?i9JF0Fj\\n7$$!3S*****\\_Yp:\"F0F]]n7$$\"3=!)***\\P^P
n&FfjlF`]n7$$\"3#))****\\#Hb3DF0Fc]n7$$\"3u(****\\P,xX%F0Ff]n7$$\"3W,+
+vq1OjF0Fi]n7$$\"3b)**\\73.=/)F0F\\^n7$$\"3r****\\<)3q+\"F,F_^n7$$\"3#
)******p6$)y6F,Fb^n7$$\"3s**\\789qy8F,Fe^n7$$\"3U*****\\W?cb\"F,Fh^n7$
$\"3w**\\7j'G(\\<F,F[_n7$$\"3B+]P*elX$>F,F^_n7$$\"31++DJNUF@F,Fa_n7$$
\"3%)**\\iDt_/BF,Fd_n7$$\"3e***\\Ppdb\\#F,Fg_n7$$\"3O**\\7eX)Rp#F,Fj_n
7$$\"3K**\\(os:n'GF,F]`n7$$\"3k***\\77qK0$F,F``n7$$\"3!)*****\\1**fC$F
,Fc`n7$$\"3%)***\\itYXV$F,Ff`n7$$\"3j**\\7.j(ph$F,Fi`n7$$\"3J***\\PBL&
>QF,F\\an7$$\"3.*****\\kR:+%F,F_an7$$\"3/++]P.(e>%F,Fban7$$\"3[**\\7GG
'>P%F,Fean7$$\"3S++]K%yWc%F,Fhan7$$\"3]**\\781iXZF,F[bn7$$\"3z**\\i&Qm
\\$\\F,F^bn7$$\"3h****\\(['3?^F,Fabn7$$\"3q)*\\7y+*QJ&F,Fdbn7$$\"3B***
***pfa+bF,Fgbn7$$\"3k***\\(y&G9p&F,Fjbn7$$\"33+]7$eI2)eF,F]cn7$$\"35**
***\\YzY0'F,F`cn7$$\"3#*)**\\P^WSD'F,Fccn7$$\"3++++!**eBV'F,Ffcn7$$\"3
x**\\78%zCi'F,Ficn7$$\"3'))*\\(o\"*[W!oF,F\\dn7$$\"3a**************pF,
F_dn-Fjz6&F\\[lF(F][lF(-%%TEXTG6&7$$\"\"&!\"\"$\"#F!\"#Q*y~=~cos~x6\"F
iz-%%FONTG6$%*HELVETICAG\"\"*-Fddn6&7$$\"\"#Fidn$\"#bF\\enQ-y~=~arccos
~xF^enFijmF_en-Fddn6&7$$\"$8\"F\\en$!\"$F\\enQ\"xF^en-Fjz6&F\\[lF)F)F)
F_en-Fddn6&7$FafnF_fnQ\"yF^enFdfnF_en-%*AXESTICKSG6$7(/$!&J=$!\"&%#-1G
/F)%\"0G/$\"&J=$Fagn%\"1G/Fgdn%$p/2G/$\"&iO'Fagn%\"2G/Ffz%\"pGF]gn-F`e
n6$%'SYMBOLGFcen-%+AXESLABELSG6%%!GFghn-F`en6#%(DEFAULTG-%%VIEWG6$;$!#
LF\\enF_fnF^in" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 
0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7
" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }}{PARA 0 "" 0 
"" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "y = arccos*x;" "6#/%\"y
G*&%'arccosG\"\"\"%\"xGF'" }{TEXT -1 62 " on its own again suggests th
e shape of \"half an hour-glass\". " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{GLPLOT2D 310 401 401 {PLOTDATA 2 "6*-%'CURVESG6$7`q7$$!3%****za&))
4$=$!#=$\"3uDPG9))z****F*7$$!3P2\"f)e)=e6$F*$\"3ZH)[[z\"QW$*F*7$$!3QcM
a$*)ys0$F*$\"3!4qtYIU?5*F*7$$!3sg@/!eW9*HF*$\"3)HWiI#ez*)))F*7$$!3_lj(
R(Q<DHF*$\"3/E3#)R'z(4()F*7$$!3W/%359=#fGF*$\"3_i)QV?4;b)F*7$$!3!QgDOt
o!)z#F*$\"3)fWRH&*[\"=%)F*7$$!3'4i#>k?vMFF*$\"3G$*Hx;n0!H)F*7$$!39)f*3
u)p#pEF*$\"3Ov<_VK6m\")F*7$$!3[22?'o(*Rg#F*$\"3!HY>kue&\\!)F*7$$!35^_)
z@do`#F*$\"3uxG,O$*oNzF*7$$!3Oed.d)>xZ#F*$\"3F)HcdBx'RyF*7$$!3RvCD&>X6
T#F*$\"3byG-&H9dt(F*7$$!3w[:^rrHWBF*$\"3!3S!y$=Z^j(F*7$$!3E\"e**Rrw)zA
F*$\"3@ayi`TSTvF*7$$!3)3f))pww8A#F*$\"3_T&e(\\%p'euF*7$$!3g2lvrY\"=:#F
*$\"3a<zS`[$HO(F*7$$!3!)z)>EX')G4#F*$\"3-a6l[*eQG(F*7$$!3-VM\"[#*QV-#F
*$\"3aBj'RrwR>(F*7$$!3[ZJxZFmj>F*$\"3p<=[4'Gh6(F*7$$!3c7K!\\O\"4(*=F*$
\"3x'fpl`(RKqF*7$$!3#=M]cm*pL=F*$\"39#Rsm(*eT&pF*7$$!3o<fqxqbn<F*$\"3]
2T=xU%R(oF*7$$!3+2;\\Mv\"oq\"F*$\"3kUw-sWX,oF*7$$!39/x]J<IT;F*$\"3[x#z
6M@Ws'F*7$$!3!GT)3D#\\Kd\"F*$\"3kO(ylfvbk'F*7$$!3@,y>=$4S^\"F*$\"3')fG
t=^$yd'F*7$$!3Sec#efG+X\"F*$\"3c`gD<x`0lF*7$$!3AYq4\\-$RQ\"F*$\"3a>%z&
\\#><V'F*7$$!3*RS'>7fE>8F*$\"3a!G)=(*QHgjF*7$$!3[ynET)*pc7F*$\"3Cz@V$o
u=H'F*7$$!3E/Ds')3B(=\"F*$\"3;'**>t`Qm@'F*7$$!3-,zIT*4[7\"F*$\"3;`2`F9
k\\hF*7$$!3m'y$feA;e5F*$\"33*G0<;!pygF*7$$!3/_WhAbpx**!#>$\"3k!fet#o([
,'F*7$$!3_g2OCPW<$*F_v$\"3%zZBG$4fXfF*7$$!3Ox)HoB)>'p)F_v$\"3;O*Q_\"z
\"3)eF*7$$!3e)\\4')p<o/)F_v$\"3*)Q0OY)3N\"eF*7$$!3yUKaE&H>T(F_v$\"3wgj
+6)f![dF*7$$!3A4Lw[/EZnF_v$\"3AIM!R\"Q))zcF*7$$!3%\\Zn!H^52hF_v$\"3@ow
A@2_9cF*7$$!3)p*p*\\&QX_aF_v$\"3;smH9w%za&F*7$$!3m<@+CMA.[F_v$\"3a$>1-
6k@[&F*7$$!36.?Q%3[m?%F_v$\"3md*[%G!**=U&F*7$$!3D8[o@a!H_$F_v$\"3Obh`Q
P,``F*7$$!3!3*H@LyN6HF_v$\"3Or[<MKa\"H&F*7$$!3A\\k2_0KfAF_v$\"3W^O,$>A
hA&F*7$$!3q#QYfnP_j\"F_v$\"31#4=+y&fj^F*7$$!3N:PTf%QFU*!#?$\"3c$er>:TU
4&F*7$$!3WvVRn3=AMFfz$\"38<#pzYAU.&F*7$$\"3w'H<9n0&)Q$Ffz$\"3)4DjKI9h'
\\F*7$$\"3i]k?$32df*Ffz$\"3h#R7%*QGS!\\F*7$$\"3m*p*p#*4aQ;F_v$\"3D%Ri1
Xtg$[F*7$$\"3;n9N@vPCAF_v$\"393**443QxZF*7$$\"3cxc3Wpd\"*GF_v$\"3UP(\\
S7V/r%F*7$$\"3<-::['eg`$F_v$\"3_'=cwxick%F*7$$\"3CI8\"zK>,=%F_v$\"3g0T
)=<x2e%F*7$$\"3j15jC1\"=#[F_v$\"3^W*=[bbf^%F*7$$\"3\"4cc>2v#QaF_v$\"3O
s8:y9\\`WF*7$$\"3_&eY`Y*o/hF_v$\"3q:))*fTDdQ%F*7$$\"3$=t<MqI)RnF_v$\"3
;Qm)pTw3K%F*7$$\"3Ct7i[pG3uF_v$\"3'4dDst9BD%F*7$$\"3Lp,5>3S8!)F_v$\"3/
lg)z`W**=%F*7$$\"3sfCiC\\;#o)F_v$\"3oF4f=3k?TF*7$$\"3m*f:9SCFK*F_v$\"3
_%e6czcQ0%F*7$$\"3?;S^QZsh**F_v$\"3'4Qcg'[!o)RF*7$$\"3[3'GnI;H1\"F*$\"
3,F!H7qni\"RF*7$$\"3XM(yL\")*QC6F*$\"3W6iyiw!3&QF*7$$\"37Y`%H4Zt=\"F*$
\"33#37y@OKy$F*7$$\"3Od()H5%foD\"F*$\"3&f(f'oj^zq$F*7$$\"30wj\"*>rz>8F
*$\"3pOc6xB7ROF*7$$\"3-0.)oriTQ\"F*$\"3yz%HDgA!oNF*7$$\"37!=]5(3i\\9F*
$\"3EqWys-#\\\\$F*7$$\"3y/p[_'*y4:F*$\"3g/NAn*fpU$F*7$$\"3!)))p3Dw#Rd
\"F*$\"3EIfOoRk`LF*7$$\"3Jv/FvtdP;F*$\"3]2\"o$RO#*zKF*7$$\"3qhhABGm0<F
*$\"3E^35[L\"**>$F*7$$\"3s#3%f\"GXdw\"F*$\"3ec>UrPBGJF*7$$\"3DYQ*f'z1N
=F*$\"3#e3:;umT/$F*7$$\"3aUI6Wji(*=F*$\"3_kk'*zh$p'HF*7$$\"3O]vK*eC&f>
F*$\"3Qs7wte7*)GF*7$$\"39W&Gz3&3E?F*$\"3j0@)p#*fP!GF*7$$\"3_!ymu$[\"H4
#F*$\"3V:*H#yL5;FF*7$$\"3=t.D9+s`@F*$\"3RWkxG&yWj#F*7$$\"3io!Q5T#H=AF*
$\"3o(=(y.TjXDF*7$$\"3e8PFf(38G#F*$\"3A>;r$3VlX#F*7$$\"31ZAdX\">'\\BF*
$\"3)3M!=jZ(pN#F*7$$\"3\"*\\j/Ams3CF*$\"3#4^uD#y)zE#F*7$$\"3g:4k2PhwCF
*$\"3!3%*H*)=%3i@F*7$$\"3*4pTq8J/a#F*$\"3)Qx7PR'Qe?F*7$$\"3O>.nf@j.EF*
$\"3a<u]In2^>F*7$$\"3WKY=fM)om#F*$\"3zeVP+EEQ=F*7$$\"3@:'H4y&RJFF*$\"3
W6eN62\\;<F*7$$\"33B/Bhxx*z#F*$\"3=w<#oci#y:F*7$$\"3Yo=(4MxC'GF*$\"3b=
'\\q4d4W\"F*7$$\"3eTg=x)pV#HF*$\"3ei(fHvdAH\"F*7$$\"3oNrxz>`!*HF*$\"3s
^EKf9(G6\"F*7$$\"3qCRD`+bcIF*$\"3jyalw=s0!*F_v7$$\"33#e[Y>0e6$F*$\"3/Y
GYW,&ob'F_v7$$\"33++me))4$=$F*$\"3d.$p)3OOA9!#A-%'COLOURG6&%$RGBG$\"*+
+++\"!\")$\"\"!Fa[mF`[m-%%TEXTG6&7$$!\"#!\"\"$\"#'*Fg[mQ-y~=~arccos~x6
\"Fijl-%%FONTG6$%*HELVETICAG\"\"*-Fc[m6&7$$\"#PFg[m$!\"$Fg[mQ\"xF\\\\m
-Fjjl6&F\\[mFa[mFa[mFa[mF]\\m-Fc[m6&7$$Fg[mFg[m$\"$8\"Fg[mQ\"yF\\\\mFj
\\mF]\\m-%*AXESTICKSG6$7%/$!&J=$!\"&%#-1G/Fa[m%\"0G/$\"&J=$Fj]m%\"1G7%
F\\^m/$\"\"&Fh[m%$p/2G/\"\"\"%\"pG-F^\\m6$%'SYMBOLGFa\\m-%+AXESLABELSG
6%%!GF`_m-F^\\m6#%(DEFAULTG-%%VIEWG6$;$!#LFg[mFe\\m;Fg\\mF`]m" 1 2 0 
1 10 0 2 9 1 4 2 1.000000 49.000000 38.000000 0 0 "Curve 1" "Curve 2" 
"Curve 3" "Curve 4" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+
 " }}{PARA 0 "" 0 "" {TEXT -1 37 "The domain of the arccosine function
 " }{XPPEDIT 18 0 "arccos*x;" "6#*&%'arccosG\"\"\"%\"xGF%" }{TEXT -1 
17 " is the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }
{TEXT -1 32 ", and its range is the interval " }{XPPEDIT 18 0 "[0, Pi]
;" "6#7$\"\"!%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "Th
e arccosine function has the following special values. " }}{PARA 256 "
" 0 "" {XPPEDIT 18 0 "arccos*0 = Pi/2;" "6#/*&%'arccosG\"\"\"\"\"!F&*&
%#PiGF&\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "arccos(1/2) = Pi/3;" "6#/-%'arccosG6#*&\"\"\"F(\"\"#
!\"\"*&%#PiGF(\"\"$F*" }{TEXT -1 10 ",         " }{XPPEDIT 18 0 "arcco
s(-1/2) = 2*Pi/3;" "6#/-%'arccosG6#,$*&\"\"\"F)\"\"#!\"\"F+*(F*F)%#PiG
F)\"\"$F+" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "arccos(1/sqrt(2)) = Pi/4;" "6#/-%'arccosG6#*&\"\"\"F(-%
%sqrtG6#\"\"#!\"\"*&%#PiGF(\"\"%F-" }{TEXT -1 10 ",         " }
{XPPEDIT 18 0 "arccos(-1/sqrt(2)) = 3*Pi/4;" "6#/-%'arccosG6#,$*&\"\"
\"F)-%%sqrtG6#\"\"#!\"\"F.*(\"\"$F)%#PiGF)\"\"%F." }{TEXT -1 2 ", " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arccos(sqrt(3)/2) = P
i/6;" "6#/-%'arccosG6#*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"*&%#PiGF,\"\"'F
." }{TEXT -1 11 ",          " }{XPPEDIT 18 0 "arccos(-sqrt(3)/2) = 5*P
i/6;" "6#/-%'arccosG6#,$*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F/*(\"\"&F-%#
PiGF-\"\"'F/" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "arccos*1 = 0;" "6#/*&%'arccosG\"\"\"F&F&\"\"!" }{TEXT 
-1 16 ",               " }{XPPEDIT 18 0 "arccos(-1)=Pi" "6#/-%'arccosG
6#,$\"\"\"!\"\"%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 40 "The following symmetry formula applie
s: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arccos(-x) = P
i-arccos*x;" "6#/-%'arccosG6#,$%\"xG!\"\",&%#PiG\"\"\"*&F%F,F(F,F)" }
{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 268 15 "____
___________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 67 "The following formula relates the arccosine and
 arcsine functions: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 
18 0 "arcsin*x+arccos*x = Pi/2;" "6#/,&*&%'arcsinG\"\"\"%\"xGF'F'*&%'a
rccosGF'F(F'F'*&%#PiGF'\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{TEXT 265 13 "_____________" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 39 "This formula holds for any real number " 
}{TEXT 266 1 "x" }{TEXT -1 9 " between " }{XPPEDIT 18 0 "-1" "6#,$\"\"
\"!\"\"" }{TEXT -1 8 " and 1. " }}{PARA 0 "" 0 "" {TEXT -1 24 "Here ar
e some examples: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
arcsin*0+arccos*0 = 0+Pi/2;" "6#/,&*&%'arcsinG\"\"\"\"\"!F'F'*&%'arcco
sGF'F(F'F',&F(F'*&%#PiGF'\"\"#!\"\"F'" }{XPPEDIT 18 0 "`` = Pi/2" "6#/
%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(1/2)+arccos(1/2) = Pi/6+Pi/3" "6
#/,&-%'arcsinG6#*&\"\"\"F)\"\"#!\"\"F)-%'arccosG6#*&F)F)F*F+F),&*&%#Pi
GF)\"\"'F+F)*&F2F)\"\"$F+F)" }{XPPEDIT 18 0 "`` = Pi/2" "6#/%!G*&%#PiG
\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "arcsin(-1/2)+arccos(-1/2) = -Pi/6+2*Pi/3;" "6#/,&-%'ar
csinG6#,$*&\"\"\"F*\"\"#!\"\"F,F*-%'arccosG6#,$*&F*F*F+F,F,F*,&*&%#PiG
F*\"\"'F,F,*(F+F*F4F*\"\"$F,F*" }{XPPEDIT 18 0 "`` = Pi/2" "6#/%!G*&%#
PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "arcsin(1/sqrt(2))+arccos(1/sqrt(2)) = Pi/4+Pi/4" "6
#/,&-%'arcsinG6#*&\"\"\"F)-%%sqrtG6#\"\"#!\"\"F)-%'arccosG6#*&F)F)-F+6
#F-F.F),&*&%#PiGF)\"\"%F.F)*&F7F)F8F.F)" }{XPPEDIT 18 0 "`` = Pi/2" "6
#/%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(-1/sqrt(2))+arccos(-1/sqrt(2)) =
 -Pi/4+3*Pi/4;" "6#/,&-%'arcsinG6#,$*&\"\"\"F*-%%sqrtG6#\"\"#!\"\"F/F*
-%'arccosG6#,$*&F*F*-F,6#F.F/F/F*,&*&%#PiGF*\"\"%F/F/*(\"\"$F*F9F*F:F/
F*" }{XPPEDIT 18 0 "`` = Pi/2" "6#/%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 
-1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(sq
rt(3)/2)+arccos(sqrt(3)/2) = Pi/3+Pi/6;" "6#/,&-%'arcsinG6#*&-%%sqrtG6
#\"\"$\"\"\"\"\"#!\"\"F--%'arccosG6#*&-F*6#F,F-F.F/F-,&*&%#PiGF-F,F/F-
*&F8F-\"\"'F/F-" }{XPPEDIT 18 0 "`` = Pi/2" "6#/%!G*&%#PiG\"\"\"\"\"#!
\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "arcsin(-sqrt(3)/2)+arccos(-sqrt(3)/2) = -Pi/3+5*Pi/6;" "6#/,&-%'
arcsinG6#,$*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F0F.-%'arccosG6#,$*&-F+6#F
-F.F/F0F0F.,&*&%#PiGF.F-F0F0*(\"\"&F.F:F.\"\"'F0F." }{XPPEDIT 18 0 "``
 = Pi/2" "6#/%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "'arc
sin(0)+arccos(0)'=arcsin(0)+arccos(0);\n'arcsin(1/2)+arccos(1/2)'=arcs
in(1/2)+arccos(1/2);\n'arcsin(sqrt(2)/2)+arccos(sqrt(2)/2)'=arcsin(sqr
t(2)/2)+arccos(sqrt(2)/2);\n'arcsin(sqrt(3)/2)+arccos(sqrt(3)/2)'=arcs
in(sqrt(3)/2)+arccos(sqrt(3)/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,
&-%'arcsinG6#\"\"!\"\"\"-%'arccosGF'F),$*&\"\"#!\"\"%#PiGF)F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%'arcsinG6##\"\"\"\"\"#F)-%'arccos
GF'F),$*&F*!\"\"%#PiGF)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%'arc
sinG6#,$*&#\"\"\"\"\"#F+-%%sqrtG6#F,F+F+F+-%'arccosGF'F+,$*&F,!\"\"%#P
iGF+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%'arcsinG6#,$*&#\"\"\"\"
\"#F+-%%sqrtG6#\"\"$F+F+F+-%'arccosGF'F+,$*&F,!\"\"%#PiGF+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The formula " }{XPPEDIT 18 0 "a
rcsin*x+arccos*x = Pi/2;" "6#/,&*&%'arcsinG\"\"\"%\"xGF'F'*&%'arccosGF
'F(F'F'*&%#PiGF'\"\"#!\"\"" }{TEXT -1 34 " is a consequence of the for
mula  " }{XPPEDIT 18 0 "sin(Pi/2-theta) = cos*theta;" "6#/-%$sinG6#,&*
&%#PiG\"\"\"\"\"#!\"\"F*%&thetaGF,*&%$cosGF*F-F*" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "alpha = arcsin*x;" "
6#/%&alphaG*&%'arcsinG\"\"\"%\"xGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "beta = arccos*x;" "6#/%%betaG*&%'arccosG\"\"\"%\"xGF'" }{TEXT -1 2 
". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "sin*alpha = \+
x;" "6#/*&%$sinG\"\"\"%&alphaGF&%\"xG" }{TEXT -1 7 ", with " }
{XPPEDIT 18 0 "-Pi/2<=alpha" "6#1,$*&%#PiG\"\"\"\"\"#!\"\"F)%&alphaG" 
}{XPPEDIT 18 0 "``<=Pi/2" "6#1%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 6 
", and " }{XPPEDIT 18 0 "cos*beta = x;" "6#/*&%$cosG\"\"\"%%betaGF&%\"
xG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "0<=beta" "6#1\"\"!%%betaG" }
{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "sin*alpha = cos*beta;" "6#/*&%$si
nG\"\"\"%&alphaGF&*&%$cosGF&%%betaGF&" }{TEXT -1 10 ", so that " }
{XPPEDIT 18 0 "sin*alpha = sin(Pi/2-beta);" "6#/*&%$sinG\"\"\"%&alphaG
F&-F%6#,&*&%#PiGF&\"\"#!\"\"F&%%betaGF." }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 15 "The inequality " }{XPPEDIT 18 0 "0<=beta" "6#1\"\"
!%%betaG" }{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 14 " implies
 that " }{XPPEDIT 18 0 "-Pi/2<=Pi/2-beta" "6#1,$*&%#PiG\"\"\"\"\"#!\"
\"F),&*&F&F'F(F)F'%%betaGF)" }{XPPEDIT 18 0 "``<=Pi/2" "6#1%!G*&%#PiG
\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 54 "Sinc
e the sine function is one-to-one on the interval " }{XPPEDIT 18 0 "[-
Pi/2,Pi/2]" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 15 "
, the equation " }{XPPEDIT 18 0 "sin*alpha = sin(Pi/2-beta);" "6#/*&%$
sinG\"\"\"%&alphaGF&-F%6#,&*&%#PiGF&\"\"#!\"\"F&%%betaGF." }{TEXT -1 
14 " implies that " }{XPPEDIT 18 0 "alpha=Pi/2-beta" "6#/%&alphaG,&*&%
#PiG\"\"\"\"\"#!\"\"F(%%betaGF*" }{TEXT -1 11 ", that is  " }{XPPEDIT 
18 0 "alpha+beta=Pi/2" "6#/,&%&alphaG\"\"\"%%betaGF&*&%#PiGF&\"\"#!\"
\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "This shows that " }
{XPPEDIT 18 0 "arcsin*x+arccos*x = Pi/2;" "6#/,&*&%'arcsinG\"\"\"%\"xG
F'F'*&%'arccosGF'F(F'F'*&%#PiGF'\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "The inverse tangent function
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{GLPLOT2D 476 476 476 {PLOTDATA 2 "60-%'CURVESG6$7ao7$
$!3)******p******\\\"!#<$!3U%Rtg*G.h5!\"*7$$!3[yj$f$)=$*\\\"F*$!3E=l%H
/@In%!#:7$$!3AdF([nP')\\\"F*$!3k*3nq'\\]OBF37$$!3'f84Q^cz\\\"F*$!3IHC)
GBfwb\"F37$$!3q9bu_`F(\\\"F*$!3MA`SqBBo6F37$$!3'>F=1.8f\\\"F*$!3ua%es)
3)zy(!#;7$$!3AH5\\32b%\\\"F*$!3r]C6rqtSeFH7$$!3uVlBkg#=\\\"F*$!3c%ol#e
)\\L*QFH7$$!3Ee?)*>95*[\"F*$!3hB\\[$G8&>HFH7$$!3G(3t98_O[\"F*$!37TejX5
RX>FH7$$!3I;T'H%G?y9F*$!35JN\")GU/e9FH7$$!3wz^P\\+so9F*$!3#ouQCfQV,\"F
H7$$!3?ViybsBf9F*$!3)[CrP,5hw(F*7$$!3d#)eXp%3zV\"F*$!3#Gbv))eu71&F*7$$
!3c.Z3-$QkT\"F*$!3A\\%3k[p8s$F*7$$!3gX7ny,2&R\"F*$!37X$R`jjG#HF*7$$!37
yn&H>f_P\"F*$!3_,aQeZy>CF*7$$!3qBlukgua8F*$!3.'4V3')Qr.#F*7$$!33Xr?e8`
L8F*$!3Mst7Y>aM<F*7$$!3$>p+vo%Q78F*$!3Y'4PBsKa\\\"F*7$$!3KqGUPFj!H\"F*
$!3U)[d$*R(\\%H\"F*7$$!3s\"\\azgt9F\"F*$!3!4DGk**=\\9\"F*7$$!35q0+m\\!
*\\7F*$!3'faS(>1.%***!#=7$$!3P808hxCG7F*$!3RliicH#)=()Fhr7$$!3qPdDJqP2
7F*$!3;_<9(HZci(Fhr7$$!3!pyx!4WU)=\"F*$!3x2d,N^(Qs'Fhr7$$!39R8<3z)e;\"
F*$!3g\\ENz!G4u&Fhr7$$!3O()eyNlzY6F*$!3#H<u6ZH\"p\\Fhr7$$!3gnu7J()eC6F
*$!3**GZ\\Il+FTFhr7$$!3R!3P51J\\5\"F*$!3!zl;(ydM@MFhr7$$!3'*=[P\"[jL3
\"F*$!3Au;^zq]!o#Fhr7$$!3!Q/[_/EG1\"F*$!3_EzZPTx**>Fhr7$$!3H#Go^Q(RT5F
*$!3:yUDx&>zI\"Fhr7$$!3Z^8\"f=><-\"F*$!3l=](eM#*Q$o!#>7$$!3rr.e*e$\\+5
F*$!3UboXU'e1b\"!#?7$$!37klP.;Y%y*Fhr$\"3eR.NVlq\"y'F`v7$$!3crZp$3QDf*
Fhr$\"3COv2.t6(G\"Fhr7$$!3;,=_CaD&Q*Fhr$\"35??T')=lb>Fhr7$$!3mFL(\\:76
<*Fhr$\"3%*Qo5$*)GXm#Fhr7$$!3u6.tVZhh*)Fhr$\"3uHB[q)\\IQ$Fhr7$$!3k3l>E
_\"*e()Fhr$\"3njT1Q,O4TFhr7$$!3b))oHr>&Q`)Fhr$\"3kdx9DC,i\\Fhr7$$!3@Kp
nEEiJ$)Fhr$\"3i&R?!32n!y&Fhr7$$!30-eIO'*p:\")Fhr$\"3/?#R2&=8CnFhr7$$!3
e'3jXJT+#zFhr$\"3#oZbGdRkl(Fhr7$$!3k*=j()3Nhq(Fhr$\"3/Dr#fL*)>y)Fhr7$$
!3?ruQ+g'[](Fhr$\"3%eG'>hEZp**Fhr7$$!3!)3J)*4=[%H(Fhr$\"3]G*pb;\\#Q6F*
7$$!3I5R84Gz)3(Fhr$\"3=t:P\\vg)H\"F*7$$!3%zE4v[bM(oFhr$\"3%=OuV%*y\")
\\\"F*7$$!3%H(pL`+1mmFhr$\"3CIF2_L\"Gt\"F*7$$!3UC&4rEoRX'Fhr$\"3[S*=C8
H[.#F*7$$!3Nq[&zqKOC'Fhr$\"3+K6Zm!RzU#F*7$$!3#G'ypthN]gFhr$\"3j4yH\")[
p>HF*7$$!3[HO>a%R)GeFhr$\"3+#[Z))*zB`PF*7$$!3'*f!\\&fArIcFhr$\"31=%z&=
0h!)\\F*7$$!3f'>LiMn%>aFhr$\"3*4H48=`Wa(F*7$$!3Sh(*3yJP=`Fhr$\"3exH[.)
\\Y'**F*7$$!3@Ej%*4!zs@&Fhr$\"3Yarlrhqi9FH7$$!3cX(f\\EfH;&Fhr$\"3C!*31
KFg^>FH7$$!3zjJ(*>&R'3^Fhr$\"3a:%p^FE)GHFH7$$!3!H()zukz93&Fhr$\"3:7A:i
]w0RFH7$$!3-#e')\\x>V0&Fhr$\"3;`)oKgd$feFH7$$!3.O*R(Q)R2/&Fhr$\"3=Y;f'
>2G\"yFH7$$!39\"H$\\-*fr-&Fhr$\"3)yAU2?c><\"F37$$!3:o*pV$*p.-&Fhr$\"3'
*fVY$GCEc\"F37$$!3EYmCm*zN,&Fhr$\"3Xa%G`Z_RM#F37$$!3EBL7)***y1]Fhr$\"3
].CxDf\"zo%F37$$!3E+++I+++]Fhr$\"3g'*[k[H.h5F--%'COLOURG6&%$RGBG$\"*++
++\"!\")$\"\"!FialFhal-F$6$7ao7$$\"3E+++I+++]Fhr$!3g'*[k[H.h5F-7$$\"3)
R@O1k6o+&Fhr$!3gQv%H/@In%F37$$\"3pFCF^Ki8]Fhr$!3'[Mnq'\\]OBF37$$\"3TT'
3>'[V?]Fhr$!3+*[#)GBfwb\"F37$$\"38b[askCF]Fhr$!3_M_SqBBo6F37$$\"3c#G<Q
pp3/&Fhr$!3OIzD()3)zy(FH7$$\"3+5(*3:H\\a]Fhr$!3CfC6rqtSeFH7$$\"3'[cMwN
R<3&Fhr$!3K@cEe)\\L*QFH7$$\"3s>%z,!e)*3^Fhr$!3*)o[[$G8&>HFH7$$\"3YH\"p
_oyM;&Fhr$!3mLejX5RX>FH7$$\"32Q)e.drz@&Fhr$!3/AN\")GU/e9FH7$$\"3j.#[i]
*z7`Fhr$!3J\\(QCfQV,\"FH7$$\"3>pv8Uui2aFhr$!3)3ErP,5hw(F*7$$\"3Nv6W0`
\"4i&Fhr$!3Q[b())eu71&F*7$$\"3^lH:zphNeFhr$!36Y%3k[p8s$F*7$$\"3#Ra(G8#
)H\\gFhr$!3,]$R`jjG#HF*7$$\"3!)=AVq!3uC'Fhr$!3u.aQeZy>CF*7$$\"33kZ`_$R
DX'Fhr$!3[#4V3')Qr.#F*7$$\"35[&GzT'okmFhr$!3Bvt7Y>aM<F*7$$\"3%=3$*\\7`
h(oFhr$!3+'4PBsKa\\\"F*7$$\"3'yHrdisO4(Fhr$!3?)[d$*R(\\%H\"F*7$$\"3'G3
b/#RE&G(Fhr$!3z^#Gk**=\\9\"F*7$$\"3%*)H%**R.&4](Fhr$!3S]0u>1.%***Fhr7$
$\"3Hm[p)QAvr(Fhr$!3soiicH#)=()Fhr7$$\"3-BEW(oHi#zFhr$!3Qa<9(HZci(Fhr7
$$\"3(48A#4fv:\")Fhr$!3@7d,N^(Qs'Fhr7$$\"3i3mG=47T$)Fhr$!3$Hl_$z!G4u&F
hr7$$\"3RE69UY.K&)Fhr$!3EwT<r%H\"p\\Fhr7$$\"3+C`s)o7Tv)Fhr$!3@JZ\\Il+F
TFhr7$$\"39'>H'*Q*o]*)Fhr$!37gmrydM@MFhr7$$\"3=4=D'=lj;*Fhr$!3W\"o6&zq
]!o#Fhr7$$\"31j&>vaR<P*Fhr$!3-HzZPTx**>Fhr7$$\"39xrJ[h-'e*Fhr$!3l!Gasd
>zI\"Fhr7$$\"3K&[')393Gy*Fhr$!3jV](eM#*Q$oF`v7$$\"3w\"G'>/T1&***Fhr$!3
aWvXU'e1b\"Ffv7$$\"3fVBmRQb@5F*$\"3g9.NVlq\"y'F`v7$$\"3&H_I;>Y2/\"F*$
\"3uQv2.t6(G\"Fhr7$$\"3))>yadWZh5F*$\"3;=?T')=lb>Fhr7$$\"3MnE]%y))G3\"
F*$\"3;To5$*)GXm#Fhr7$$\"3ropiD&QQ5\"F*$\"3)pK#[q)\\IQ$Fhr7$$\"39\\.Qx
%3T7\"F*$\"3+iT1Q,O4TFhr7$$\"396.(G![hY6F*$\"3Kax9DC,i\\Fhr7$$\"3)oIKt
tPo;\"F*$\"3'*)R?!32n!y&Fhr7$$\"3y>%pj.I%)=\"F*$\"3r;#R2&=8CnFhr7$$\"3
M\"pV&oe*z?\"F*$\"3]ta&GdRkl(Fhr7$$\"3:\"oB6\\'QH7F*$\"3QGr#fL*)>y)Fhr
7$$\"3)GDh**R8&\\7F*$\"3]#G'>hEZp**Fhr7$$\"37*o,!>=bq7F*$\"3%y#*pb;\\#
Q6F*7$$\"3'*3m3>27\"H\"F*$\"3Is:P\\vg)H\"F*7$$\"3@t!\\7XaEJ\"F*$\"3GiV
PW*y\")\\\"F*7$$\"3q-jm%*RRL8F*$\"38FF2_L\"Gt\"F*7$$\"3cZ!*GtJga8F*$\"
3PT*=C8H[.#F*7$$\"3'G^/#Hnjv8F*$\"3yD6Zm!RzU#F*7$$\"3g8-j#Qk\\R\"F*$\"
3I+yH\")[p>HF*7$$\"3%pj!eag6<9F*$\"3))yu%))*zB`PF*7$$\"3+%4XSxGpV\"F*$
\"3%=Tz&=0h!)\\F*7$$\"3A!ow`E`!e9F*$\"3ZR#48=`Wa(F*7$$\"3'Q-\">#oi\"o9
F*$\"3!e(H[.)\\Y'**F*7$$\"3]n`+*4s#y9F*$\"37arlrhqi9FH7$$\"3oDS]tSq$[
\"F*$\"3e141KFg^>FH7$$\"3%Qo-![g8*[\"F*$\"3z*[p^FE)GHFH7$$\"3#G,__._=
\\\"F*$\"3g3A:i]w0RFH7$$\"3!=M,D-oX\\\"F*$\"3m?*oKgd$feFH7$$\"3S1g7;g#
f\\\"F*$\"3(>j\"f'>2G\"yFH7$$\"3+r1v4SG(\\\"F*$\"3;bAu+i&><\"F37$$\"3=
.Ic1I'z\\\"F*$\"3q\">kMGCEc\"F37$$\"3QN`P.?k)\\\"F*$\"3_(>G`Z_RM#F37$$
\"3ynw=+5K*\\\"F*$\"3#*GLxDf\"zo%F37$$\"3)******p******\\\"F*$\"3U%Rtg
*G.h5F-Faal-F$6%7ao7$$!3++++++++]Fhr$!3+Q&>`$R7L;!\"\"7$$!3u;HK*Q)=$*
\\Fhr$!3a..5'fTIn%F37$$!3]LekynP')\\Fhr$!3C+2Q)45lL#F37$$!3E](oz;l&z\\
Fhr$!3)4&*p.^hwb\"F37$$!3-n;HdNvs\\Fhr$!3-POv\\OBo6F37$$!3_+v$fLI\"f\\
Fhr$!3!f#*[yb')zy(FH7$$!3[LLe9r]X\\Fhr$!3:qR>_-uSeFH7$$!3$***\\(=ng#=
\\Fhr$!3#zViWE^L*QFH7$$!3%pmm\"HU,\"*[Fhr$!3G6KAqS^>HFH7$$!3()***\\PM@
l$[Fhr$!3[v0d\"R\"RX>FH7$$!3MLLLe%G?y%Fhr$!3-X5M@W/e9FH7$$!39+](=_+so%
Fhr$!3XkdA%oQV,\"FH7$$!3OmmT&esBf%Fhr$!3%>k<Xa5hw(F*7$$!3KLL$3s%3zVFhr
$!31r#y&3[Fh]F*7$$!33LL$e/$QkTFhr$!3M\"4lHgp8s$F*7$$!3!pm;/\"=q]RFhr$!
3SvgS1P'G#HF*7$$!3SLL3_>f_PFhr$!3#3dwo![y>CF*7$$!3))***\\(o1YZNFhr$!3$
pqz_*)Qr.#F*7$$!3]LL3-OJNLFhr$!3IR$H8(>aM<F*7$$!3C++v$*o%Q7$Fhr$!3UaNR
TFV&\\\"F*7$$!3ammm\"RFj!HFhr$!3!*)*e,9u\\%H\"F*7$$!3JLL$e4OZr#Fhr$!3;
!=`#3!>\\9\"F*7$$!3=+++v'\\!*\\#Fhr$!3s%G'*QrIS***Fhr7$$!33+++DwZ#G#Fh
r$!3#flbB.B)=()Fhr7$$!3-+++D.xt?Fhr$!3bPA'*etkDwFhr7$$!3OLL3-TC%)=Fhr$
!3Vj.f'=vQs'Fhr7$$!3!omm;4z)e;Fhr$!3Bav#47G4u&Fhr7$$!3+nmmm`'zY\"Fhr$!
3%G9xc]H\"p\\Fhr7$$!3E++v=t)eC\"Fhr$!3Yl!zzb1q7%Fhr7$$!39nmm;1J\\5Fhr$
!3_^4\"3!eM@MFhr7$$!3&y***\\(=[jL)F`v$!3MqVN'420o#Fhr7$$!3M****\\iXg#G
'F`v$!3i[Rz\\Tx**>Fhr7$$!3WlmmT&Q(RTF`v$!3W*)4>&e>zI\"Fhr7$$!3;nm;/'=>
<#F`v$!3(y%f+(Q#*Q$oF`v7$$!3vDMLLe*e$\\!#@$!3#HVh<le1b\"Ffv7$$\"3[em;z
RQb@F`v$\"3i'>lTe1<y'F`v7$$\"3'[***\\(=>Y2%F`v$\"3))\\_)3J<rG\"Fhr7$$
\"3Qhmm\"zXu9'F`v$\"3/gGW)*=lb>Fhr7$$\"3]'******\\y))G)F`v$\"3cy-%)4*G
Xm#Fhr7$$\"3'*)***\\i_QQ5Fhr$\"3SvbH#*)\\IQ$Fhr7$$\"3@***\\7y%3T7Fhr$
\"3([a3a;g$4TFhr7$$\"35****\\P![hY\"Fhr$\"3Qn%)efC,i\\Fhr7$$\"3kKLL$Qx
$o;Fhr$\"3?St(*\\2n!y&Fhr7$$\"3!)*****\\P+V)=Fhr$\"3$zj;B!>8CnFhr7$$\"
3?mm\"zpe*z?Fhr$\"3)f#\\/N'Rkl(Fhr7$$\"3%)*****\\#\\'QH#Fhr$\"3O/B^7%*
)>y)Fhr7$$\"3GKLe9S8&\\#Fhr$\"3G**Q(\\vs%p**Fhr7$$\"3R***\\i?=bq#Fhr$
\"3nUqFx\"\\#Q6F*7$$\"3\"HLL$3s?6HFhr$\"3)438Tc2')H\"F*7$$\"3a***\\7`W
l7$Fhr$\"3)ps&\\j*y\")\\\"F*7$$\"3#pmmm'*RRL$Fhr$\"3v[mAxL\"Gt\"F*7$$
\"3Qmm;a<.YNFhr$\"3)*=)yn;H[.#F*7$$\"3=LLe9tOcPFhr$\"3hw4H:\"RzU#F*7$$
\"3u******\\Qk\\RFhr$\"3$)pw?_\\p>HF*7$$\"3CLL$3dg6<%Fhr$\"3!Hpmu6QKv$
F*7$$\"3ImmmmxGpVFhr$\"3gk$>6t51)\\F*7$$\"3A++D\"oK0e%Fhr$\"3[geQ\"o`W
a(F*7$$\"3C+++]oi\"o%Fhr$\"3eiba)o]Y'**F*7$$\"3A++v=5s#y%Fhr$\"3A0,Wlj
qi9FH7$$\"3;+D1k2/P[Fhr$\"3Or)R-3.;&>FH7$$\"35+]P40O\"*[Fhr$\"3bq^)p1F
)GHFH7$$\"33]7.#Q?&=\\Fhr$\"3yP.Sxkw0RFH7$$\"31+voa-oX\\Fhr$\"33/<x/3O
feFH7$$\"3KDc,\">g#f\\Fhr$\"32km`.H\"G\"yFH7$$\"3-]PMF,%G(\\Fhr$\"3Mk9
G)[d><\"F37$$\"357y]&4I'z\\Fhr$\"3WdJ_vlii:F37$$\"3uu=nj+U')\\Fhr$\"3&
>&*\\%Rw&RM#F37$$\"3OPf$=.5K*\\Fhr$\"3+]gC5m$zo%F37$$\"3++++++++]Fhr$
\"3+Q&>`$R7L;F^hm-Fbal6&FdalFhalFhalFeal-%*THICKNESSG6#\"\"#-F$6%7$7$$
!3++++++++:F*$!\"'Fial7$F^^o$\"\"'Fial-Fbal6&FdalFialFialFial-%*LINEST
YLEG6#\"\"$-F$6%7$7$FjgmF`^o7$FjgmFc^oFe^oFg^o-F$6%7$7$F`]oF`^o7$F`]oF
c^oFe^oFg^o-F$6%7$7$$\"3++++++++:F*F`^o7$Fi_oFc^oFe^oFg^o-%%TEXTG6&7$$
\"#<F^hm$!\"#F^hmQ\"x6\"-%&COLORG6&Fdal$\"\"\"Fc`oFi`oFi`o-%%FONTG6$%*
HELVETICAG\"\"*-F]`o6&7$$Fa^oFc`oFc^oQ\"yFe`oFf`oF[ao-%*AXESTICKSG6$7)
/$F3F^hm%&-3p/2G/F^hm%#-pG/$!\"&F^hm%%-p/2G/Fial%\"0G/$\"\"&F^hm%$p/2G
/Fj`o%\"pG/$\"#:F^hm%%3p/2GFj^o-F\\ao6$%'SYMBOLGF_ao-%&TITLEG6$Q*y~=~t
an~xFe`oF[ao-%+AXESLABELSG6%%!GFhcoF[ao-%%VIEWG6$;$FHF^hmF``o;F`^oFc^o
" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "C
urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "C
urve 9" }}{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 96 "The blue cur
ve is the graph of the tangent function with its domain restricted to \+
the interval  " }{XPPEDIT 18 0 "-Pi/2 < x;" "6#2,$*&%#PiG\"\"\"\"\"#!
\"\"F)%\"xG" }{XPPEDIT 18 0 "`` < Pi/2;" "6#2%!G*&%#PiG\"\"\"\"\"#!\"
\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 111 "With this restric
ted domain the tangent function is a one-to-one function, which then h
as an inverse function. " }}{PARA 0 "" 0 "" {TEXT -1 29 "This inverse \+
is denoted by   " }{XPPEDIT 18 0 "tan^(-1);" "6#)%$tanG,$\"\"\"!\"\"" 
}{TEXT -1 4 " or " }{TEXT 264 6 "arctan" }{TEXT -1 16 " and called the
 " }{TEXT 259 24 "inverse tangent function" }{TEXT -1 8 " or the " }
{TEXT 259 19 "arctangent function" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 
"y = arctan*x;" "6#/%\"yG*&%'arctanG\"\"\"%\"xGF'" }{TEXT -1 16 "  exa
ctly when  " }{XPPEDIT 18 0 "tan*y = x;" "6#/*&%$tanG\"\"\"%\"yGF&%\"x
G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "-Pi/2 < y;" "6#2,$*&%#PiG\"\"\"
\"\"#!\"\"F)%\"yG" }{XPPEDIT 18 0 "`` < Pi/2;" "6#2%!G*&%#PiG\"\"\"\"
\"#!\"\"" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 
267 28 "____________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "The graph of the arcta
ngent function is obtained by reflecting the graph of the tangent func
tion (restricted in the way we've just suggested) in the line " }
{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 
"" {TEXT -1 1 " " }{GLPLOT2D 476 476 476 {PLOTDATA 2 "65-%'CURVESG6$7a
o7$$!3G+++!*******\\!#=$!3]H>x_=@85!\"*7$$!3-SlLz$)=$*\\F*$!3eXJ#y:tu[
\"!#:7$$!3yzInonP')\\F*$!3YrxK!>PtV(!#;7$$!33?'4!e^cz\\F*$!3?$y&Hk\")=
e\\F97$$!3#)fhMZNvs\\F*$!3&pyfM%=g=PF97$$!3')R#>gKI\"f\\F*$!3O16;8D**y
CF97$$!3O>Bp/r]X\\F*$!3C\"e\"QF];f=F97$$!3*)y%Q?mg#=\\F*$!3B4Cm[?HR7F9
7$$!3UQYQ>U,\"*[F*$!3EG=Hz55$H*!#<7$$!3]dp2M8_O[F*$!3')f$*GdFP#>'FX7$$
!3ew#p([%G?y%F*$!3XL<!QL*4TYFX7$$!3)))f+D^+so%F*$!39T\"))Q@S(GKFX7$$!3
u@>BwDP#f%F*$!3G0A?2(H?Z#FX7$$!39k^27Z3zVF*$!3$RD_\\y`5h\"FX7$$!3inX]P
IQkTF*$!3]$f+Cs[X=\"FX7$$!3#HE:D!=q]RF*$!3WiVwhSw.$*F*7$$!3q\\\"yX%>f_
PF*$!3PX2@8UT-xF*7$$!3cy]lh1YZNF*$!3A%=KCY9W['F*7$$!361FT&f8`L$F*$!32$
Q3![&=7_&F*7$$!3@1B]()o%Q7$F*$!3WKa'oB5,w%F*7$$!3u6S&eQFj!HF*$!3*\\1`'
HK^?TF*7$$!3)3'QS!4OZr#F*$!3#Gk?&z.RWOF*7$$!3[+>+q'\\!*\\#F*$!3S+u1p()
>\"=$F*7$$!3+X]V?wZ#G#F*$!3I/10fvGvFF*7$$!3QfC&3KqP2#F*$!3!yj\\j$*=tU#
F*7$$!31X[J)4WU)=F*$!39_`))RfFS@F*7$$!3_3*[$)3z)e;F*$!33R0NsTRF=F*7$$!
3/O2tj`'zY\"F*$!3'*GPsVIs\"e\"F*7$$!39D#eiJ()eC\"F*$!3*\\*Q$[)pm88F*7$
$!3IX!oXh5$\\5F*$!3uXs5!>[!*3\"F*7$$!3H,t#3<[jL)!#>$!3%yP$op!>B`)F_u7$
$!3!QyM*\\Xg#G'F_u$!3]tU22)yaO'F_u7$$!3z#=(QL&Q(RTF_u$!3uHedMtBjTF_u7$
$!3?DG#)*f=><#F_u$!3u3g`.bHv@F_u7$$!3hA8YBe*e$\\!#@$!3U%z['z(**e$\\Fdv
7$$\"3i&*e&[(RQb@F_u$\"3u)*ogYUoe@F_u7$$\"3-e2Nz\">Y2%F_u$\"31lk/R;-(4
%F_u7$$\"3;&yr$zdWZhF_u$\"3@1&>$>L.DiF_u7$$\"3AYAU$[y))G)F_u$\"3!*o?\\
A*e9[)F_u7$$\"35HKUg_QQ5F*$\"37Li&QAeo2\"F*7$$\"3IIywyZ3T7F*$\"3Do!QO'
*\\!38F*7$$\"3q/xcM![hY\"F*$\"31&))*f<wXz:F*7$$\"3Uyl**ztPo;F*$\"3.'[R
SkW+%=F*7$$\"3#**RJ7P+V)=F*$\"3(=nQ]ld.9#F*7$$\"3M\\nv$pe*z?F*$\"3o4d?
A/7PCF*7$$\"3uqAT?\\'QH#F*$\"3%z5QA/%R&z#F*7$$\"3=lIf4S8&\\#F*$\"3'3.)
*zs\"QtJF*7$$\"3Wk*Q3?=bq#F*$\"3@coD['fJi$F*7$$\"3D>4^-s?6HF*$\"39@7mV
ifLTF*7$$\"3\\6p*\\_Wl7$F*$\"3UYVCL<&)oZF*7$$\"3X)y)**f*RRL$F*$\"3_7&Q
@E;d^&F*7$$\"3)QguquJga$F*$\"3Yy7NoB1xkF*7$$\"3_+122tOcPF*$\"3K\"*\\p#
=q$GxF*7$$\"3'Hr+@%Qk\\RF*$\"3![U[SgxOH*F*7$$\"3385\\i0;rTF*$\"3_*f0ay
#p%>\"FX7$$\"3)>4Gzv(GpVF*$\"3W&=(R7wP&e\"FX7$$\"35O*)3sE`!e%F*$\"3%z/
\"eRTZ,CFX7$$\"3GZnjSoi\"o%F*$\"3'G6CEtY=<$FX7$$\"3YeX=45s#y%F*$\"3cj`
Uh(Qfl%FX7$$\"3j=%)Qa2/P[F*$\"3;m9/u^97iFX7$$\"3OzAf*\\g8*[F*$\"3NfyHj
_uA$*FX7$$\"3Y4U>s._=\\F*$\"3IU#>F$RCV7F97$$\"35ShzW-oX\\F*$\"3sIl=J@4
l=F97$$\"3)[5(4\"=g#f\\F*$\"3I-Sd!)\\*o[#F97$$\"3?q!)R<,%G(\\F*$\"34$p
[4=b/t$F97$$\"39`&[b3I'z\\F*$\"3C*Qt?W#*R(\\F97$$\"3_N!*p`+U')\\F*$\"3
g\\)=]TV5Y(F97$$\"3!z^\\=-5K*\\F*$\"3#)*4ZTP9A\\\"F37$$\"3G+++!*******
\\F*$\"3]H>x_=@85F--%'COLOURG6&%$RGBG$\"\"!FfalFeal$\"*++++\"!\")-F$6$
7V7$$!33+++++++@FX$!33ezA;J;@XF*7$$!3>++]_>X3?FX$!3ev?]hro*\\%F*7$$!3(
***\\(e['zG>FX$!3-].zwYQzWF*7$$!32++v#e:#R=FX$!3#*e\\#[**4XX%F*7$$!33+
+Dz3/\\<FX$!3l$)z7@d(pU%F*7$$!37+]PgZHf;FX$!3QG$p.I,nR%F*7$$!3;+]()>')
3w:FX$!3syRl\"*ymlVF*7$$!3++](3[L**[\"FX$!3;H@YJM.IVF*7$$!33+](GrJ3S\"
FX$!3e>)pof$y)G%F*7$$!3'***\\P&p:?J\"FX$!3H.^#4E#QUUF*7$$!35++]/vl?7FX
$!37(Q%)RLG!)=%F*7$$!3$)***\\-;*=S6FX$!3zRk(\\CGM8%F*7$$!3#)****\\j3g
\\5FX$!3y1[&*e1tiSF*7$$!3[+++DgS'e*F*$!3YES1&=M&zRF*7$$!3A)****\\ON)4(
)F*$!3A?CUdFq%)QF*7$$!3%****\\(G_#Q\"zF*$!3y())[:VBFy$F*7$$!3E+++&=#Hn
pF*$!3k\"4*4NP*ej$F*7$$!3Q******RXXlhF*$!3!zc5.[yH[$F*7$$!3$)***\\(Qns
K_F*$!3-(\\b\\IA/E$F*7$$!3V*******e/rS%F*$!36lBO)4H*3IF*7$$!3@)**\\(Qi
E,NF*$!3[^LhOsR^EF*7$$!3%*)**\\i\"RpQEF*$!3<vU6Nl?.AF*7$$!3E)***\\(=!p
Q<F*$!393(=.,h8f\"F*7$$!3Ub**\\P\"e?7*F_u$!3c.[6BF'R)))F_u7$$!3yq*\\7)
GokYF_u$!3#eiO9Z9<j%F_u7$$!3=h)***\\i2t?!#?$!3Ij?<Sp/t?F]jl7$$\"3G0+vV
DlAWF_u$\"3#\\YS'G#=XR%F_u7$$\"3m'***\\7Fh_!*F_u$\"3\"4=2Ox](>))F_u7$$
\"3`*****\\m+$38F*$\"39U\"Q\\D*HT7F*7$$\"3R***\\(e+M6<F*$\"3%)\\==.)>-
d\"F*7$$\"3)3++DBF>e#F*$\"35H?%\\tk#p@F*7$$\"3k,++q*G8[$F*$\"3WHLBcsMU
EF*7$$\"3k****\\-\"=7O%F*$\"3wTl]j(*3$*HF*7$$\"3e,+D\"3cD@&F*$\"3%[771
+f\\D$F*7$$\"3g,+]d<#y:'F*$\"3y!R)\\(eq8[$F*7$$\"3$))*****4]=2qF*$\"34
1_!y)*\\Fk$F*7$$\"3H/++v:19zF*$\"3'*yk^Ujv#y$F*7$$\"3k,+DJl#et)F*$\"35
X?Y5rv()QF*7$$\"3'f++]oKUj*F*$\"3aGx/:BE%)RF*7$$\"3!)**\\7'Gcz/\"FX$\"
3#\\7bV_V81%F*7$$\"3B+]iYwJO6FX$\"3X_z:--iITF*7$$\"3F++]FqqA7FX$\"3s)G
2dBJ$*=%F*7$$\"3)***\\7.([JJ\"FX$\"3M]jkn:,VUF*7$$\"3`+++'ya-S\"FX$\"3
=JB;E-])G%F*7$$\"37++vOLL*[\"FX$\"3plV6o8xHVF*7$$\"3M+]7sUnx:FX$\"3#)p
#e.r)GmVF*7$$\"3Y+++</&)e;FX$\"3+V)yKaVlR%F*7$$\"3Q++vRu)=v\"FX$\"3HHU
1jo)yU%F*7$$\"3W+++i35N=FX$\"3UD*3\\w5LX%F*7$$\"3>+]7EP#Q#>FX$\"3CoM^N
H1yWF*7$$\"3!3+vy#Gu3?FX$\"3!Ges?Ue(*\\%F*7$$\"33+++++++@FX$\"33ezA;J;
@XF*-Fbal6&FdalFgalFealFeal-F$6$7S7$$!\"#FfalFhbm7$$!3MLLL$Q6G\">FXF[c
m7$$!3bmm;M!\\p$=FXF^cm7$$!3MLLL))Qj^<FXFacm7$$!3ALLL=Kvl;FXFdcm7$$!3w
mm;C2G!e\"FXFgcm7$$!3OLL$3yO5]\"FXFjcm7$$!3&*****\\nU)*=9FXF]dm7$$!3SL
L$3WDTL\"FXF`dm7$$!35++]d(Q&\\7FXFcdm7$$!3gmmmc4`i6FXFfdm7$$!3KLLLQW*e
3\"FXFidm7$$!3w++++()>'***F*F\\em7$$!3E++++0\"*H\"*F*F_em7$$!35++++83&
H)F*Fbem7$$!3\\LLL3k(p`(F*Feem7$$!3Anmmmj^NmF*Fhem7$$!3)zmmmYh=(eF*F[f
m7$$!3+,++v#\\N)\\F*F^fm7$$!3commmCC(>%F*Fafm7$$!39*****\\FRXL$F*Fdfm7
$$!3t*****\\#=/8DF*Fgfm7$$!3=mmm;a*el\"F*Fjfm7$$!3komm;Wn(o)F_uF]gm7$$
!3IqLLL$eV(>F]jlF`gm7$$\"3)Qjmm\"f`@')F_uFcgm7$$\"3%z****\\nZ)H;F*Ffgm
7$$\"3ckmm;$y*eCF*Figm7$$\"3f)******R^bJ$F*F\\hm7$$\"3'e*****\\5a`TF*F
_hm7$$\"3'o****\\7RV'\\F*Fbhm7$$\"3Y'*****\\@fkeF*Fehm7$$\"3_ILLL&4Nn'
F*Fhhm7$$\"3A*******\\,s`(F*F[im7$$\"3%[mm;zM)>$)F*F^im7$$\"3M*******p
fa<*F*Faim7$$\"39HLLeg`!)**F*Fdim7$$\"3w****\\#G2A3\"FXFgim7$$\"3;LLL$
)G[k6FXFjim7$$\"3#)****\\7yh]7FXF]jm7$$\"3xmmm')fdL8FXF`jm7$$\"3bmmm,F
T=9FXFcjm7$$\"3FLL$e#pa-:FXFfjm7$$\"3!*******Rv&)z:FXFijm7$$\"3ILLLGUY
o;FXF\\[n7$$\"3_mmm1^rZ<FXF_[n7$$\"34++]sI@K=FXFb[n7$$\"34++]2%)38>FXF
e[n7$$\"\"#FfalFh[n-Fbal6&FdalFealFgalFeal-F$6%7$7$$!3++++++++]F*F^bl7
$F`\\nF^bm-%&COLORG6&Fdal$\"\"$!\"\"Feal$\"\"(Fh\\n-%*LINESTYLEG6#Fg\\
n-F$6%7$7$$\"3++++++++]F*F^bl7$Fb]nF^bmFc\\nF[]n-F$6%7$7$F^blF`\\n7$F^
bmF`\\n-Fbal6&Fdal$\")#)eqkFial$\"))eqk\"FialF^^nF[]n-F$6%7$7$F^blFb]n
7$F^bmFb]nFj]nF[]n-%%TEXTG6&7$$\"\")Fh\\n$\"#=Fh\\nQ*y~=~tan~x6\"Faal-
%%FONTG6$%*HELVETICAG\"\"*-Ff^n6&7$F[_n$\"#NFibmQ-y~=~arctan~xF^_nFbbm
F__n-Ff^n6&7$$\"#@Fh\\n$!\"(FibmQ\"xF^_n-Fbal6&FdalFfalFfalFfalF__n-Ff
^n6&7$F_`nF]`nQ\"yF^_nFb`nF__n-Ff^n6&7$$!\"%Fh\\nF_`nQ%-p/2F^_nFb`n-F`
_n6$%'SYMBOLGFc_n-Ff^n6&7$$\"\"%Fh\\nF_`nQ$p/2F^_nFb`nF^an-Ff^n6&7$$Fh
\\nFh\\nF[anF]anFb`nF^an-Ff^n6&7$FjanFdanFfanFb`nF^an-%*AXESTICKSG6$7)
/$!&*4>F\\an%#-6G/$!&KF\"F\\an%#-4G/$!&iO'!\"&%#-2G/Ffal%\"0G/$\"&iO'F
]cn%\"2G/$\"&KF\"F\\an%\"4G/$\"&*4>F\\an%\"6GFabnF^an-%+AXESLABELSG6%%
!GF`dn-F`_n6#%(DEFAULTG-%%VIEWG6$;$FdvFh\\nF]`nFgdn" 1 2 0 1 10 0 2 9 
1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "
Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" 
"Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" }}{TEXT -1 1 " \+
" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }}{PARA 257 "" 0 "" {TEXT -1 16 
"Since the graph " }{XPPEDIT 18 0 "y = tan*x;" "6#/%\"yG*&%$tanG\"\"\"
%\"xGF'" }{TEXT -1 14 " has the line " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%
\"xG" }{TEXT -1 39 " as a tangent at the origin, the graph " }
{XPPEDIT 18 0 "y = arctan*x;" "6#/%\"yG*&%'arctanG\"\"\"%\"xGF'" }
{TEXT -1 19 " also has the line " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG
" }{TEXT -1 30 "  as a tangent at the origin. " }}{PARA 0 "" 0 "" 
{TEXT -1 14 "The graph of  " }{XPPEDIT 18 0 "y = arctan*x;" "6#/%\"yG*
&%'arctanG\"\"\"%\"xGF'" }{TEXT -1 15 " has the lines " }{XPPEDIT 18 
0 "y = Pi/2;" "6#/%\"yG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y = -Pi/2;" "6#/%\"yG,$*&%#PiG\"\"\"\"\"#!\"\"F*" }
{TEXT -1 27 " as horizontal asymptotes. " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{GLPLOT2D 554 174 174 {PLOTDATA 2 "6.-%'CURVESG6$7V7$$!33+++
++++@!#<$!33ezA;J;@X!#=7$$!3>++]_>X3?F*$!3ev?]hro*\\%F-7$$!3(***\\(e['
zG>F*$!3-].zwYQzWF-7$$!32++v#e:#R=F*$!3#*e\\#[**4XX%F-7$$!33++Dz3/\\<F
*$!3l$)z7@d(pU%F-7$$!37+]PgZHf;F*$!3QG$p.I,nR%F-7$$!3;+]()>')3w:F*$!3s
yRl\"*ymlVF-7$$!3++](3[L**[\"F*$!3;H@YJM.IVF-7$$!33+](GrJ3S\"F*$!3e>)p
of$y)G%F-7$$!3'***\\P&p:?J\"F*$!3H.^#4E#QUUF-7$$!35++]/vl?7F*$!37(Q%)R
LG!)=%F-7$$!3$)***\\-;*=S6F*$!3zRk(\\CGM8%F-7$$!3#)****\\j3g\\5F*$!3y1
[&*e1tiSF-7$$!3[+++DgS'e*F-$!3YES1&=M&zRF-7$$!3A)****\\ON)4()F-$!3A?CU
dFq%)QF-7$$!3%****\\(G_#Q\"zF-$!3y())[:VBFy$F-7$$!3E+++&=#HnpF-$!3k\"4
*4NP*ej$F-7$$!3Q******RXXlhF-$!3!zc5.[yH[$F-7$$!3$)***\\(QnsK_F-$!3-(
\\b\\IA/E$F-7$$!3V*******e/rS%F-$!36lBO)4H*3IF-7$$!3@)**\\(QiE,NF-$!3[
^LhOsR^EF-7$$!3%*)**\\i\"RpQEF-$!3<vU6Nl?.AF-7$$!3E)***\\(=!pQ<F-$!393
(=.,h8f\"F-7$$!3Ub**\\P\"e?7*!#>$!3c.[6BF'R)))Fir7$$!3yq*\\7)GokYFir$!
3#eiO9Z9<j%Fir7$$!3=h)***\\i2t?!#?$!3Ij?<Sp/t?Fds7$$\"3G0+vVDlAWFir$\"
3#\\YS'G#=XR%Fir7$$\"3m'***\\7Fh_!*Fir$\"3\"4=2Ox](>))Fir7$$\"3`*****
\\m+$38F-$\"39U\"Q\\D*HT7F-7$$\"3R***\\(e+M6<F-$\"3%)\\==.)>-d\"F-7$$
\"3)3++DBF>e#F-$\"35H?%\\tk#p@F-7$$\"3k,++q*G8[$F-$\"3WHLBcsMUEF-7$$\"
3k****\\-\"=7O%F-$\"3wTl]j(*3$*HF-7$$\"3e,+D\"3cD@&F-$\"3%[771+f\\D$F-
7$$\"3g,+]d<#y:'F-$\"3y!R)\\(eq8[$F-7$$\"3$))*****4]=2qF-$\"341_!y)*\\
Fk$F-7$$\"3H/++v:19zF-$\"3'*yk^Ujv#y$F-7$$\"3k,+DJl#et)F-$\"35X?Y5rv()
QF-7$$\"3'f++]oKUj*F-$\"3aGx/:BE%)RF-7$$\"3!)**\\7'Gcz/\"F*$\"3#\\7bV_
V81%F-7$$\"3B+]iYwJO6F*$\"3X_z:--iITF-7$$\"3F++]FqqA7F*$\"3s)G2dBJ$*=%
F-7$$\"3)***\\7.([JJ\"F*$\"3M]jkn:,VUF-7$$\"3`+++'ya-S\"F*$\"3=JB;E-])
G%F-7$$\"37++vOLL*[\"F*$\"3plV6o8xHVF-7$$\"3M+]7sUnx:F*$\"3#)p#e.r)GmV
F-7$$\"3Y+++</&)e;F*$\"3+V)yKaVlR%F-7$$\"3Q++vRu)=v\"F*$\"3HHU1jo)yU%F
-7$$\"3W+++i35N=F*$\"3UD*3\\w5LX%F-7$$\"3>+]7EP#Q#>F*$\"3CoM^NH1yWF-7$
$\"3!3+vy#Gu3?F*$\"3!Ges?Ue(*\\%F-7$$\"33+++++++@F*$\"33ezA;J;@XF--%'C
OLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fa\\lF`\\l-F$6%7$7$F($!3++++++++]F-7
$Fe[lFf\\l-Fj[l6&F\\\\lFa\\lFa\\lFa\\l-%*LINESTYLEG6#\"\"$-F$6%7$7$F($
\"3++++++++]F-7$Fe[lFc]lFi\\lF[]l-%%TEXTG6&7$$\"#=!\"\"$\"\"%F\\^lQ-y~
=~arctan~x6\"Fi[l-%%FONTG6$%*HELVETICAG\"\"*-Fg]l6&7$$\"#@F\\^l$!\"(!
\"#Q\"xF`^lFi\\lFa^l-Fg]l6&7$F[_l$\"\"'F\\^lQ\"yF`^lFi\\lFa^l-Fg]l6&7$
$!#8F]_l$!#WF]_lQ%-p/2F`^lFi\\l-Fb^l6$%'SYMBOLGFe^l-Fg]l6&7$$F\\^lF\\^
l$\"#WF]_lQ$p/2F`^lFi\\lF]`l-%*AXESTICKSG6$7)/$!&*4>!\"%%#-6G/$!&KF\"F
^al%#-4G/$!&iO'!\"&%#-2G/Fa\\l%\"0G/$\"&iO'Fgal%\"2G/$\"&KF\"F^al%\"4G
/$\"&*4>F^al%\"6GFa\\lF]`l-%+AXESLABELSG6%%!GFjbl-Fb^l6#%(DEFAULTG-%%V
IEWG6$;$!#@F\\^lFi^l;$!\"'F\\^lFb_l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
47.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" "Curve 8" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 63 "A \"nice\" feature of \+
the inverse tangent function is that it is " }{TEXT 259 28 "defined fo
r all real numbers" }{TEXT -1 86 ", unlike the inverse sine and cosine
 functions which are only defined on the interval " }{XPPEDIT 18 0 "[-
1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 107 "The domain of the arctangent function is the set of all real n
umbers |R, and its range is the open interval" }{XPPEDIT 18 0 "``(-Pi/
2,Pi/2) =``" "6#/-%!G6$,$*&%#PiG\"\"\"\"\"#!\"\"F,*&F)F*F+F,F%" }
{TEXT -1 1 "\{" }{XPPEDIT 18 0 "y*`|`-Pi/2 < y" "6#2,&*&%\"yG\"\"\"%\"
|grGF'F'*&%#PiGF'\"\"#!\"\"F,F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``< P
i/2" "6#2%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 "\}." }}{PARA 0 "" 0 
"" {TEXT -1 58 "The arctangent function has the following special valu
es. " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "arctan*0 = 0;" "6#/*&%'arctan
G\"\"\"\"\"!F&F'" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "arctan(1/sqrt(3)) = Pi/6;" "6#/-%'arctanG6#*&\"\"\"F(-
%%sqrtG6#\"\"$!\"\"*&%#PiGF(\"\"'F-" }{TEXT -1 10 ",         " }
{XPPEDIT 18 0 "arctan(-1/sqrt(3)) = -Pi/6;" "6#/-%'arctanG6#,$*&\"\"\"
F)-%%sqrtG6#\"\"$!\"\"F.,$*&%#PiGF)\"\"'F.F." }{TEXT -1 2 ", " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*1 = Pi/4;" "6#
/*&%'arctanG\"\"\"F&F&*&%#PiGF&\"\"%!\"\"" }{TEXT -1 10 ",         " }
{XPPEDIT 18 0 "arctan(-1) = -Pi/4;" "6#/-%'arctanG6#,$\"\"\"!\"\",$*&%
#PiGF(\"\"%F)F)" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "arctan(sqrt(3)) = Pi/3;" "6#/-%'arctanG6#-%%sqrtG6#\"\"
$*&%#PiG\"\"\"F*!\"\"" }{TEXT -1 10 ",         " }{XPPEDIT 18 0 "arcta
n(-sqrt(3)) = -Pi/3;" "6#/-%'arctanG6#,$-%%sqrtG6#\"\"$!\"\",$*&%#PiG
\"\"\"F+F,F," }{TEXT -1 3 ".  " }}{PARA 256 "" 0 "" {TEXT -1 4 " As " 
}{XPPEDIT 18 0 "x->infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"%)
infinityGF*F*F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "arctan(x)->Pi/2" "6#
f*6#-%'arctanG6#%\"xG7\"6$%)operatorG%&arrowG6\"*&%#PiG\"\"\"\"\"#!\"
\"F-F-F-" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 4 " As " }
{XPPEDIT 18 0 "proc (x) options operator, arrow; -infinity end proc;" 
"6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\",$%)infinityG!\"\"F*F*F*" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "arctan(x)->-Pi/2" "6#f*6#-%'arctanG6#%
\"xG7\"6$%)operatorG%&arrowG6\",$*&%#PiG\"\"\"\"\"#!\"\"F3F-F-F-" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 30 "The arctangent function is an " }{TEXT 259 12 "odd functi
on" }{TEXT -1 38 ", that is, it satisfies the relation: " }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(-x) = -arctan*x;" "6#/-%
'arctanG6#,$%\"xG!\"\",$*&F%\"\"\"F(F,F)" }{TEXT -1 2 ". " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{TEXT 270 13 "_____________" }{TEXT -1 1 "
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "The genera
l solution of the equation " }{XPPEDIT 18 0 "tan*theta=a" "6#/*&%$tanG
\"\"\"%&thetaGF&%\"aG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 \+
" }}{PARA 0 "" 0 "" {TEXT 283 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 
"" 0 "" {TEXT -1 42 "Find the general solution of the equation " }
{XPPEDIT 18 0 "tan*theta = sqrt(3);" "6#/*&%$tanG\"\"\"%&thetaGF&-%%sq
rtG6#\"\"$" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 284 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" 
{TEXT -1 29 "One solution of the equation " }{XPPEDIT 18 0 "tan*theta \+
= sqrt(3);" "6#/*&%$tanG\"\"\"%&thetaGF&-%%sqrtG6#\"\"$" }{TEXT -1 4 "
 is " }{XPPEDIT 18 0 "theta = arctan(sqrt(3));" "6#/%&thetaG-%'arctanG
6#-%%sqrtG6#\"\"$" }{XPPEDIT 18 0 "``=Pi/3" "6#/%!G*&%#PiG\"\"\"\"\"$!
\"\"" }{TEXT -1 47 ", and this is the only solution in the interval" }
{XPPEDIT 18 0 "``(-Pi/2,Pi/2)" "6#-%!G6$,$*&%#PiG\"\"\"\"\"#!\"\"F+*&F
(F)F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 51 "Since the tangent function is periodic with period
 " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 15 ", the numbers: " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(Pi/3+Pi) = 4*Pi/3;
" "6#/-%!G6#,&*&%#PiG\"\"\"\"\"$!\"\"F*F)F**(\"\"%F*F)F*F+F," }{TEXT 
-1 5 ",    " }{XPPEDIT 18 0 "``(Pi/3+2*Pi) = 7*Pi/3;" "6#/-%!G6#,&*&%#
PiG\"\"\"\"\"$!\"\"F**&\"\"#F*F)F*F**(\"\"(F*F)F*F+F," }{TEXT -1 5 ", \+
   " }{XPPEDIT 18 0 "``(Pi/3+3*Pi) = 10*Pi/3;" "6#/-%!G6#,&*&%#PiG\"\"
\"\"\"$!\"\"F**&F+F*F)F*F**(\"#5F*F)F*F+F," }{TEXT -1 5 ",    " }
{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT -1 8 ", etc., " }}
{PARA 0 "" 0 "" {TEXT -1 40 "are also solutions, as are the numbers: \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``(Pi/3-Pi) = -2
*Pi/3" "6#/-%!G6#,&*&%#PiG\"\"\"\"\"$!\"\"F*F)F,,$*(\"\"#F*F)F*F+F,F,
" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "``(Pi/3-2*Pi) = -5*Pi/3" "6#/-%!
G6#,&*&%#PiG\"\"\"\"\"$!\"\"F**&\"\"#F*F)F*F,,$*(\"\"&F*F)F*F+F,F," }
{TEXT -1 5 ",    " }{XPPEDIT 18 0 "``(Pi/3-3*Pi) = -8*Pi/3" "6#/-%!G6#
,&*&%#PiG\"\"\"\"\"$!\"\"F**&F+F*F)F*F,,$*(\"\")F*F)F*F+F,F," }{TEXT 
-1 5 ",    " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT -1 6 ", \+
etc." }}{PARA 0 "" 0 "" {TEXT -1 38 "In general, any number of the for
m:   " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta = Pi/3
+k*Pi;" "6#/%&thetaG,&*&%#PiG\"\"\"\"\"$!\"\"F(*&%\"kGF(F'F(F(" }
{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 286 7 "______
_" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 6 "where " }{TEXT 285 1 "k" }{TEXT -1 25 " is an integer, tha
t is, " }{XPPEDIT 18 0 "k=` . . . `,-4,-3,-2,-1,0,1,2,3,4,` . . . `" "
6-/%\"kG%(~.~.~.~G,$\"\"%!\"\",$\"\"$F(,$\"\"#F(,$\"\"\"F(\"\"!F.F,F*F
'F%" }{TEXT -1 32 ", is a solution of the equation " }{XPPEDIT 18 0 "t
an*theta = sqrt(3)" "6#/*&%$tanG\"\"\"%&thetaGF&-%%sqrtG6#\"\"$" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 75 "The following picture s
hows the points of intersection P and Q of the line " }{XPPEDIT 18 0 "
y=sqrt(3)*x" "6#/%\"yG*&-%%sqrtG6#\"\"$\"\"\"%\"xGF*" }{TEXT -1 22 " w
ith the unit circle " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"\"#\"
\"\"*$%\"yGF'F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 127 "All
 the solutions are associated with either P or Q, more precisely, the \+
\"wrapping function\" W, which associates a real number " }{XPPEDIT 
18 0 "theta" "6#%&thetaG" }{TEXT -1 16 " with the point " }{XPPEDIT 
18 0 "W(theta)=``(cos*theta,sin*theta)" "6#/-%\"WG6#%&thetaG-%!G6$*&%$
cosG\"\"\"F'F-*&%$sinGF-F'F-" }{TEXT -1 40 ", maps each solution to ei
ther P or Q.  " }}{PARA 256 "" 0 "" {GLPLOT2D 222 220 220 {PLOTDATA 2 
"60-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4hRPij!**!#=$\"3Ikwb#=y_O
\"F/7$$\"3E8J#))4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\"3-N5')yke[#*F/$\"3gLj&[K
5J!QF/7$$\"3?goz=42`')F/$\"3j9NXR4U7]F/7$$\"3Sb](G._U!zF/$\"3t_H-qceDh
F/7$$\"3_\\$R'oTd#3(F/$\"3!GO#*3qU&fqF/7$$\"3?H$>jubk6'F/$\"3!\\@@&\\!
>8\"zF/7$$\"3VGuIc:x5]F/$\"3-$>p(Qh-a')F/7$$\"37C3nW'R,#QF/$\"3aK+16bc
T#*F/7$$\"3$oY@iF'QDDF/$\"3s&Q\"[M\"oen*F/7$$\"3OB^hAo8X8F/$\"33)fVPO<
\"4**F/7$$!3+qB/u(p5(f!#@$\"2%HhJ<#)******!#<7$$!33)\\T#fB[i8F/$\"3ap%
>wGZn!**F/7$$!3)e:d.:#=YEF/$\"3PA>=\\D`V'*F/7$$!3%*yV1J*3Jx$F/$\"3IzF#
e`m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'yI2&oN')F/7$$!3[igNw%*[RgF/$\"3!G;)RQ
+BqzF/7$$!3/XUwYjJ*3(F/$\"3AfvOg?x_qF/7$$!3&=e_b?/U!zF/$\"3u3<J%QZc7'F
/7$$!3w!G5)RnIf')F/$\"3D%H$\\4/k,]F/7$$!3/`k8ti$4B*F/$\"3In'[*>HvXQF/7
$$!3!p!RS4Nij'*F/$\"3o$p9m#Q%=d#F/7$$!3#p(pT>+.2**F/$\"31V?00]Ug8F/7$$
!3/gKG4>&*****F/$\"3vCA\\O%485$!#?7$$!3__Is=!Q%3**F/$!3q#4*>c=8]8F/7$$
!3)>b`sYlSn*F/$!3UNbFGIGKDF/7$$!37_J.8AEj#*F/$!3q4&=j`Bsw$F/7$$!3%\\F)
4Kh=u')F/$!3kENX')3zv\\F/7$$!3EB*z7xng%zF/$!37W(H!pUCrgF/7$$!3XD84Pfc5
rF/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[gF/$!3)4j2yeGL'zF/7$$!3g@UD=%)o!*\\F/
$!3I_Mh6Mil')F/7$$!3o+1s\"[\"ysPF/$!3#\\IN,%***4E*F/7$$!30yCH\"el'3EF/
$!3=V>(fp[Pl*F/7$$!3g67Cvnc\"H\"F/$!3;$RoI*>C;**F/7$$!3EzyOHMQdIFas$!3
W1O#>E`*****F/7$$\"3GIAj`Ks(G\"F/$!3lYZ3X=u;**F/7$$\"3EKRqX8/bDF/$!3U$
[o%H'z!o'*F/7$$\"3@%)yb&Q)zNQF/$!36.2<#>x]B*F/7$$\"3O7w4w0I.]F/$!3wHgb
5wMe')F/7$$\"3@\\#)[*R]$4hF/$!3/a*[=H2o\"zF/7$$\"33/wY'Q+$*4(F/$!3)4d)
eg?sUqF/7$$\"3[f=s$Ry,!zF/$!3D1$H<bQ38'F/7$$\"3%*R,@;vLu')F/$!3e>)zD(p
_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2ndo*fQF/7$$\"31mGAp))oa'*F/$!3%)f]nRQ=
0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4O\"F/7$F($\"36YKhSr8/#)!#F-%'COLOURG
6&%$RGBG$\"*++++\"!\")F+F+-%*THICKNESSG6#\"\"#-F$6%7$7$$!3++++++++]F/$
!3a+++SSDg')F/7$$\"3++++++++]F/$\"3a+++SSDg')F/-Fjz6&F\\[l$\")#)eqkF_[
l$\"))eqk\"F_[lFe\\lF`[l-F$6&7$F\\\\lFg[l-Fjz6&F\\[lF+F][lF][l-%'SYMBO
LG6$%&CROSSG\"#5-%&STYLEG6#%&POINTG-F$6&Fi\\lFj\\l-F]]l6$%(DIAMONDGF`]
lFa]l-F$6&Fi\\lFj\\l-F]]l6$%'CIRCLEGF`]lFa]l-F$6&Fi\\l-Fjz6&F\\[lF*F*F
*-F]]l6$F^^l\"#7Fa]l-%%TEXTG6%7$$\"#e!\"#F(Q\"P6\"-%&COLORG6&F\\[l$F)F
\\_lFb_lFb_l-Fg^l6%7$$!#eF\\_l$!\"\"F*Q\"QF^_lF__l-Fg^l6%7$$\"$D\"F\\_
l$!\"'F\\_lQ\"xF^_lF__l-Fg^l6%7$F``lF^`lQ\"yF^_lF__l-%(SCALINGG6#%,CON
STRAINEDG-%*AXESTICKSG6$F*F*-%+AXESLABELSG6%Q!F^_lFaal-%%FONTG6#%(DEFA
ULTG-%%VIEWG6$;$!#6Fi_l$\"#8Fi_l;FjalF^`l" 1 2 0 1 10 0 2 9 1 4 1 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }}
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The following picture sho
ws the graphs of " }{XPPEDIT 18 0 "y=tan*theta" "6#/%\"yG*&%$tanG\"\"
\"%&thetaGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=sqrt(3)" "6#/%\"yG
-%%sqrtG6#\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "The so
lutions of the equation " }{XPPEDIT 18 0 "tan*theta=sqrt(3)" "6#/*&%$t
anG\"\"\"%&thetaGF&-%%sqrtG6#\"\"$" }{TEXT -1 18 " are given by the " 
}{TEXT 391 1 "x" }{TEXT -1 62 " coordinates of the points of intersect
ion of the two graphs. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{GLPLOT2D 826 534 534 {PLOTDATA 2 "6[p-%'CURVESG6-7gn7$$!3#)**********
**\\N!#<$\"35(4vY^^PJ'F*7$$!3SYUWU,\"*[NF*$\"3%o4pcRbnX'F*7$$!3OAC@j='
za$F*$\"3)[b:AAFke'F*7$$!3K2)QSU&*oa$F*$\"3@C>hcMZQnF*7$$!3W)49e\">#ea
$F*$\"3s/iALCi)*oF*7$$!34#f'y4NvWNF*$\"3+H[.(f%elqF*7$$!33-OjgHwVNF*$
\"3&yl&[(\\8wA(F*7$$!3DF0O/ttUNF*$\"35)R0F*)[KS(F*7$$!31(pv\"plnTNF*$
\"3IZPF[&GRf(F*7$$!3&p0#yN#>1a$F*$\"3y\"yi2c\\Qz(F*7$$!3</RUQ;`RNF*$\"
3)\\4(>6')e5!)F*7$$!3:^)y?ot&QNF*$\"3Aj]LYDc6#)F*7$$!3Nlc7]_\\PNF*$\"3
k4*GbhP+X)F*7$$!3[ls-!R7k`$F*$\"3q+oqRsj.()F*7$$!3^gLn`)o``$F*$\"3cv*
\\&4Vli*)F*7$$!3wX9pA7UMNF*$\"3*\\a4(z_T6#*F*7$$!32=rz(R%HLNF*$\"3Ud)y
/dsb_*F*7$$!3IddIH)RB`$F*$\"3MW*[Woo(3)*F*7$$!3-H;AR%H7`$F*$\"3q0YW9)*
)f,\"!#;7$$!3'e\")[ebY-`$F*$\"3%*GS%oe9#\\5Faq7$$!3Z0-,x\"o\"HNF*$\"3i
L$>]ZN#)3\"Faq7$$!3Un9M089GNF*$\"3k?fSRU;G6Faq7$$!3U^5[s)pq_$F*$\"3'QK
mcNYI<\"Faq7$$!39\\+lif3ENF*$\"3RF?aY9]<7Faq7$$!3q6d(HoC]_$F*$\"3;(4D>
.i$p7Faq7$$!3W$Hh;JAR_$F*$\"3=bknvA4G8Faq7$$!3Mt9>%piH_$F*$\"3gt:bc*)z
$Q\"Faq7$$!3z([M5GE>_$F*$\"3%yNDn\"3V\\9Faq7$$!3L@-$[cb3_$F*$\"3j6l$*
\\T2C:Faq7$$!3*)pV\"z23)>NF*$\"3C?[k]_*[g\"Faq7$$!3P4DY!e%z=NF*$\"3]>Q
4)Hd;p\"Faq7$$!3X.)QVEpw^$F*$\"3;D\"[S[Q'*z\"Faq7$$!3u(>^x6em^$F*$\"3F
#o,3q%44>Faq7$$!345S%H]yb^$F*$\"3F\"y/;+K;/#Faq7$$!3o(Qm9@+Y^$F*$\"3WZ
l?xRky@Faq7$$!3Q02[!oIN^$F*$\"3gr.7Sv3^BFaq7$$!3CxO<NV_7NF*$\"3L>'>0_>
-a#Faq7$$!3f7EO9CZ6NF*$\"3?&4(>egOtFFaq7$$!3Vyh\\pRW5NF*$\"3)y\"4NsEpY
IFaq7$$!3.JB.$Gn$4NF*$\"3))pS\\\"e@rR$Faq7$$!3]C/+1.L3NF*$\"3[?)Q#yVB?
QFaq7$$!3,*Q6s%)ps]$F*$\"3CvO#G`LxP%Faq7$$!3USlbp\"=i]$F*$\"32B&=*\\vP
=^Faq7$$!3g]Z,(y^_]$F*$\"3Um8I:CVggFaq7$$!3#eklN?WT]$F*$\"3CwCCJ=T!o(F
aq7$$!3Ro^snN:.NF*$\"3*y!eaL8L45!#:7$$!3,B,IVt4-NF*$\"3Y@\"z2Tfw^\"Fiy
7$$!3))Qr_s=f,NF*$\"3G,\\k(Rx&**>Fiy7$$!3vaTv,k3,NF*$\"3B\\9B!)\\$*HHF
iy7$$!3)ehlI![\"3]$F*$\"3xo>m>#yl!RFiy7$$!3WxqP/Ka+NF*$\"3odh')o#\\)fe
Fiy7$$!3B3G.0uS+NF*$\"3;2s_:@58yFiy7$$!3,R&)o0;F+NF*$\"3O>(HYgb><\"!#9
7$$!3S/k,1P?+NF*$\"31\\3o9Ufi:F]\\l7$$!3!)pUM1e8+NF*$\"3/w%4'G7&QM#F]
\\l7$$!3\\-#3l&=5+NF*$\"3CE()Qh83DJF]\\l7$$!3>N@n1z1+NF*$\"3E%4D,1huo%
F]\\l7$$!3Wng$o&R.+NF*$\"3q7lnMf&RP*F]\\l7$$!39+++2+++NF*$\"3#\\<J7?%G
ZX!#57ao7$$!3')*****H******\\$F*$!3?E-(R-%GZXF\\^l7$$!3-G=%>$)=$*\\$F*
$!3s:^-RO*Hn%Fiy7$$!3ubO)3nP')\\$F*$!3m)H]`7)\\OBFiy7$$!3W$[D)4l&z\\$F
*$!3KoEC'>cwb\"Fiy7$$!3;6tw[`F(\\$F*$!3U^8hk1Bo6Fiy7$$!3/n4lEI\"f\\$F*
$!3)4i\"[EL(zy(Faq7$$!3!HiMXq]X\\$F*$!3uy_nHGtSeFaq7$$!3mM>Igg#=\\$F*$
!3Kn!QL)zM$*QFaq7$$!3SY#pgT,\"*[$F*$!3%**)4]MA^>HFaq7$$!3!*pQgF@l$[$F*
$!3Kr'*Q%e!RX>Faq7$$!3Q$\\Q\"RG?yMF*$!3/7-6sR/e9Faq7$$!3E>aiX+soMF*$!3
-yg0q%QV,\"Faq7$$!37XB6_sBfMF*$!3_k&4hI4hw(F*7$$!3X9E&fY3zV$F*$!3#GbQf
Hu71&F*7$$!3:(>`()HQkT$F*$!3X$e**4Lp8s$F*7$$!3y$o5b<q]R$F*$!3!4q$eSN'G
#HF*7$$!3k/Z&**=f_P$F*$!32?0t$p%y>CF*7$$!3=b&3>1YZN$F*$!3%3hF\\\")Qr.#
F*7$$!37%*)QbNJNL$F*$!3*4vCD\">aM<F*7$$!3K9;+&o%Q7LF*$!3!G[GppKa\\\"F*
7$$!3S3y4NFj!H$F*$!3mQH\")zt\\%H\"F*7$$!3_-Fy0OZrKF*$!3Ix;m!)*=\\9\"F*
7$$!3AI8+k\\!*\\KF*$!3Cn&*>%\\IS***!#=7$$!3MJXIfxCGKF*$!3#Rv`c&G#)=()F
ael7$$!3m@nfHqP2KF*$!3%\\Y9Z@Zci(Fael7$$!3e\"RqvSC%)=$F*$!30&\\\\i1vQs
'Fael7$$!3#eBWo!z)e;$F*$!3*[Y>R-G4u&Fael7$$!3?::hMlzYJF*$!3/N-<D%H\"p
\\Fael7$$!3yd28I()eCJF*$!3ZT*[Q\\1q7%Fael7$$!3)=j(>g5$\\5$F*$!3\\!fd#
\\dM@MFael7$$!356zq![jL3$F*$!3+aZ0dq]!o#Fael7$$!3=NauWg#G1$F*$!3w\\l0@
Tx**>Fael7$$!3U.r$[Q(RTIF*$!343?nm&>zI\"Fael7$$!3z(fPd=><-$F*$!3?IP.\"
H#*Q$o!#>7$$!3(GUw&*e$\\+IF*$!3yO=0I'e1b\"!#?7$$!3%p350;Y%yHF*$\"3ib0$
*)[1<y'Fihl7$$!3_Yap3QDfHF*$\"3G]sm#H<rG\"Fael7$$!3$e(R%HaD&QHF*$\"3;%
*3Pq=lb>Fael7$$!36V/;;76<HF*$\"3u(f&zq)GXm#Fael7$$!3ARP?v9;'*GF*$\"3C/
!)RT)\\IQ$Fael7$$!3))=DhB:*e(GF*$\"3np\\g,,O4TFael7$$!3]2EI)>&Q`GF*$\"
31!zE#zB,i\\Fael7$$!3B&R-SEiJ$GF*$\"33&zx?lq1y&Fael7$$!3g?!Q^'*p:\"GF*
$\"3F?$p>yJTs'Fael7$$!3iv-7LT+#z#F*$\"3[;i$**[Rkl(Fael7$$!3%4T62^81x#F
*$\"3y]N\"QB*)>y)Fael7$$!3Sa[.-m[]FF*$\"31Y%fh`s%p**Fael7$$!3`DF;$=[%H
FF*$\"3\">Wg*\\\"\\#Q6F*7$$!3scBC$Gz)3FF*$\"3sIirHvg)H\"F*7$$!3oi@D^bM
(o#F*$\"3u1#z)=*y\")\\\"F*7$$!37\\3+3ggmEF*$\"3?tT`=L\"Gt\"F*7$$!3)yxZ
&HoRXEF*$\"3+pdg'3H[.#F*7$$!35\"e+QFjVi#F*$\"360!y8+RzU#F*7$$!3b,&H0iN
]g#F*$\"34M8v'y%p>HF*7$$!3c\"Hc([R)Ge#F*$\"3%y`)oSyB`PF*7$$!3QO.XH72jD
F*$\"3I_G>N-h!)\\F*7$$!3(eu(GQn%>a#F*$\"37r1a9DXWvF*7$$!3wxUb@t$=`#F*$
\"310)*RB'[Y'**F*7$$!3m43#[!zs@DF*$\"3)3'*zK\"fqi9Faq7$$!3<2cOIfH;DF*$
\"3upC#yE-;&>Faq7$$!3o//\"f&R'3^#F*$\"3r#z:%>_#)GHFaq7$$!3V.Gooz93DF*$
\"3J&Rc^<jd!RFaq7$$!3=-_X\")>V0DF*$\"3A2UgMLNfeFaq7$$!3c,9%y)R2/DF*$\"
3Pc\\M(e*z7yFaq7$$!3%4gFU*fr-DF*$\"3Cfx-%[a><\"Fiy7$$!3j+2U(*p.-DF*$\"
33&p'RF7ii:Fiy7$$!3J+Qh+!e8]#F*$\"3*[Dp)*eXRM#Fiy7$$!3X+p!Q+z1]#F*$\"3
iNyvY$))yo%Fiy7$$!39+++2+++DF*$\"3o!\\_A*RGZXF\\^l7ao7$$!3-+++&******
\\#F*$!3E-bSOx>mjF\\^l7$$!3E.\"RR$)=$*\\#F*$!3?Tc%4M2In%Fiy7$$!3#p?yGn
P')\\#F*$!3'y*[?Y:]OBFiy7$$!3;5t\"=^cz\\#F*$!3#**\\hXrdwb\"Fiy7$$!3Q8k
v]`F(\\#F*$!3)o&z]<:Bo6Fiy7$$!3%)>YjGI\"f\\#F*$!3EV(oo5xzy(Faq7$$!3GEG
^12b%\\#F*$!3&*\\NR]\\tSeFaq7$$!3?R#pA1E=\\#F*$!3a$e,3#*[L*QFaq7$$!3a_
c-=95*[#F*$!3Y+H**eF^>HFaq7$$!3!)y%Q&H@l$[#F*$!3!Gw7]\"3RX>Faq7$$!3108
0TG?yCF*$!37z=Y+T/e9Faq7$$!3]*H+v/?(oCF*$!3S8uCJ&QV,\"Faq7$$!3%RH\\RDP
#fCF*$!3+2.%*f'4hw(F*7$$!3&)[Uqn%3zV#F*$!3Y$32CWu71&F*7$$!3Z]*=/IQkT#F
*$!3+@Sq3%p8s$F*7$$!3-l44x,2&R#F*$!3cM:'zejG#HF*7$$!3ETdX\">f_P#F*$!3
\"*fz0EZy>CF*7$$!3ARvKjguaBF*$!3L[`)y$)Qr.#F*7$$!3#)>I(oNJNL#F*$!3mmgK
H>aM<F*7$$!3,`6D'o%Q7BF*$!3=)yK'4FV&\\\"F*7$$!3OR.EOFj!H#F*$!3w9_e*Q(
\\%H\"F*7$$!3Y(foogt9F#F*$!3wn\\a))*=\\9\"F*7$$!3;]4+l\\!*\\AF*$!3q_+(
pbIS***Fael7$$!3OAv@gxCGAF*$!3a8+91H#)=()Fael7$$!3oHiUIqP2AF*$!3A0\"Gf
DZci(Fael7$$!37*3C$3WU)=#F*$!323Ej+^(Qs'Fael7$$!3q(y2v!z)e;#F*$!3!R1O;
0G4u&Fael7$$!3],()>NlzY@F*$!3!)4A<[%H\"p\\Fael7$$!3\"H6H1t)eC@F*$!3LY=
<7l+FTFael7$$!38cthg5$\\5#F*$!3(p7()RwX8U$Fael7$$!3El8/\"[jL3#F*$!3QCK
Goq]!o#Fael7$$!3]Rn*\\/EG1#F*$!3sUsEHTx**>Fael7$$!3u#p-]Q(RT?F*$!3uJJ'
>d>zI\"Fael7$$!3'\\ZCe=><-#F*$!39'[a%=B*Q$oFihl7$$!3H(Ry&*e$\\+?F*$!3%
R7aije1b\"F_il7$$!3orQUghWy>F*$\"35p/9;lq\"y'Fihl7$$!3yhC`3QDf>F*$\"3C
)RsyH<rG\"Fael7$$!3'y2)pUb_Q>F*$\"3?k9Ry=lb>Fael7$$!3)y))Ge@6r\">F*$\"
3187&>))GXm#Fael7$$!3C&Q)yu9;'*=F*$\"3]p,%f&)\\IQ$Fael7$$!3'[3;J_\"*e(
=F*$\"3RrX$)>,O4TFael7$$!3'z9;x>&Q`=F*$\"3Y(G(=-C,i\\Fael7$$!3<W]LjA;L
=F*$\"3y/\"\\+oq1y&Fael7$$!3S+VQk*p:\"=F*$\"3fzUN;=8CnFael7$$!3-#H)GKT
+#z\"F*$\"3?_eRJ&Rkl(Fael7$$!3!\\'Qz4Nhq<F*$\"3-M.([G*)>y)Fael7$$!3w+o
.,m[]<F*$\"3%4(yn)fs%p**Fael7$$!34=03#=[%H<F*$\"3l&=lx:\\#Q6F*7$$!3)R(
y2#Gz)3<F*$\"3S**QaRvg)H\"F*7$$!3SW:+]bM(o\"F*$\"3S'yE;$*y\")\\\"F*7$$
!3?tsm1ggm;F*$\"3s_MINL\"Gt\"F*7$$!3%\\OH\"GoRX;F*$\"3&)fB^4\"H[.#F*7$
$!3!R.)HsKOC;F*$\"3*=dCR.RzU#F*7$$!3vV'\\*=c.0;F*$\"3'=dCS$[p>HF*7$$!3
5Fy3ZR)Ge\"F*$\"3e9!o(>zB`PF*7$$!33@EqF72j:F*$\"3iJh)oP51)\\F*7$$!3gKb
XOn%>a\"F*$\"3E**\\#z%GXWvF*7$$!3%oi\"o>t$=`\"F*$\"3_89W8#\\Y'**F*7$$!
3'3s2H!zs@:F*$\"37e&oC/1FY\"Faq7$$!3k!zI%GfH;:F*$\"37#oT**\\-;&>Faq7$$
!3UgQ&R&R'3^\"F*$\"3_cDHZd#)GHFaq7$$!3U&R:n'z93:F*$\"3?DSl=Tw0RFaq7$$!
3>IpZz>V0:F*$\"3[wc$*oaNfeFaq7$$!3p(pde)R2/:F*$\"3o:l'>R.G\"yFaq7$$!3(
\\YQA*fr-:F*$\"3_MVQU`&><\"Fiy7$$!3h[)Ga*p.-:F*$\"3#y3HavAEc\"Fiy7$$!3
ZK#>')*zN,:F*$\"3[\")RfK!\\RM#Fiy7$$!3L;'4=+z1]\"F*$\"3%RJCi8-zo%Fiy7$
$!3(******\\+++]\"F*$\"3mqx$\\$z>mjF\\^l7ao7$$!3)******p******\\\"F*$!
3U%Rtg*G.h5!\"*7$$!3[yj$f$)=$*\\\"F*$!3E=l%H/@In%Fiy7$$!3AdF([nP')\\\"
F*$!3k*3nq'\\]OBFiy7$$!3'f84Q^cz\\\"F*$!3IHC)GBfwb\"Fiy7$$!3q9bu_`F(\\
\"F*$!3MA`SqBBo6Fiy7$$!3'>F=1.8f\\\"F*$!3ua%es)3)zy(Faq7$$!3AH5\\32b%
\\\"F*$!3r]C6rqtSeFaq7$$!3uVlBkg#=\\\"F*$!3c%ol#e)\\L*QFaq7$$!3Ee?)*>9
5*[\"F*$!3hB\\[$G8&>HFaq7$$!3G(3t98_O[\"F*$!37TejX5RX>Faq7$$!3I;T'H%G?
y9F*$!35JN\")GU/e9Faq7$$!3wz^P\\+so9F*$!3#ouQCfQV,\"Faq7$$!3?ViybsBf9F
*$!3)[CrP,5hw(F*7$$!3d#)eXp%3zV\"F*$!3#Gbv))eu71&F*7$$!3c.Z3-$QkT\"F*$
!3A\\%3k[p8s$F*7$$!3gX7ny,2&R\"F*$!37X$R`jjG#HF*7$$!37yn&H>f_P\"F*$!3_
,aQeZy>CF*7$$!3qBlukgua8F*$!3.'4V3')Qr.#F*7$$!33Xr?e8`L8F*$!3Mst7Y>aM<
F*7$$!3$>p+vo%Q78F*$!3Y'4PBsKa\\\"F*7$$!3KqGUPFj!H\"F*$!3U)[d$*R(\\%H
\"F*7$$!3s\"\\azgt9F\"F*$!3!4DGk**=\\9\"F*7$$!35q0+m\\!*\\7F*$!3'faS(>
1.%***Fael7$$!3P808hxCG7F*$!3RliicH#)=()Fael7$$!3qPdDJqP27F*$!3;_<9(HZ
ci(Fael7$$!3!pyx!4WU)=\"F*$!3x2d,N^(Qs'Fael7$$!39R8<3z)e;\"F*$!3g\\ENz
!G4u&Fael7$$!3O()eyNlzY6F*$!3#H<u6ZH\"p\\Fael7$$!3gnu7J()eC6F*$!3**GZ
\\Il+FTFael7$$!3R!3P51J\\5\"F*$!3!zl;(ydM@MFael7$$!3'*=[P\"[jL3\"F*$!3
Au;^zq]!o#Fael7$$!3!Q/[_/EG1\"F*$!3_EzZPTx**>Fael7$$!3H#Go^Q(RT5F*$!3:
yUDx&>zI\"Fael7$$!3Z^8\"f=><-\"F*$!3l=](eM#*Q$oFihl7$$!3rr.e*e$\\+5F*$
!3UboXU'e1b\"F_il7$$!37klP.;Y%y*Fael$\"3eR.NVlq\"y'Fihl7$$!3crZp$3QDf*
Fael$\"3COv2.t6(G\"Fael7$$!3;,=_CaD&Q*Fael$\"35??T')=lb>Fael7$$!3mFL(
\\:76<*Fael$\"3%*Qo5$*)GXm#Fael7$$!3u6.tVZhh*)Fael$\"3uHB[q)\\IQ$Fael7
$$!3k3l>E_\"*e()Fael$\"3njT1Q,O4TFael7$$!3b))oHr>&Q`)Fael$\"3kdx9DC,i
\\Fael7$$!3@KpnEEiJ$)Fael$\"3i&R?!32n!y&Fael7$$!30-eIO'*p:\")Fael$\"3/
?#R2&=8CnFael7$$!3e'3jXJT+#zFael$\"3#oZbGdRkl(Fael7$$!3k*=j()3Nhq(Fael
$\"3/Dr#fL*)>y)Fael7$$!3?ruQ+g'[](Fael$\"3%eG'>hEZp**Fael7$$!3!)3J)*4=
[%H(Fael$\"3]G*pb;\\#Q6F*7$$!3I5R84Gz)3(Fael$\"3=t:P\\vg)H\"F*7$$!3%zE
4v[bM(oFael$\"3%=OuV%*y\")\\\"F*7$$!3%H(pL`+1mmFael$\"3CIF2_L\"Gt\"F*7
$$!3UC&4rEoRX'Fael$\"3[S*=C8H[.#F*7$$!3Nq[&zqKOC'Fael$\"3+K6Zm!RzU#F*7
$$!3#G'ypthN]gFael$\"3j4yH\")[p>HF*7$$!3[HO>a%R)GeFael$\"3+#[Z))*zB`PF
*7$$!3'*f!\\&fArIcFael$\"31=%z&=0h!)\\F*7$$!3f'>LiMn%>aFael$\"3*4H48=`
Wa(F*7$$!3Sh(*3yJP=`Fael$\"3exH[.)\\Y'**F*7$$!3@Ej%*4!zs@&Fael$\"3Yarl
rhqi9Faq7$$!3cX(f\\EfH;&Fael$\"3C!*31KFg^>Faq7$$!3zjJ(*>&R'3^Fael$\"3a
:%p^FE)GHFaq7$$!3!H()zukz93&Fael$\"3:7A:i]w0RFaq7$$!3-#e')\\x>V0&Fael$
\"3;`)oKgd$feFaq7$$!3.O*R(Q)R2/&Fael$\"3=Y;f'>2G\"yFaq7$$!39\"H$\\-*fr
-&Fael$\"3)yAU2?c><\"Fiy7$$!3:o*pV$*p.-&Fael$\"3'*fVY$GCEc\"Fiy7$$!3EY
mCm*zN,&Fael$\"3Xa%G`Z_RM#Fiy7$$!3EBL7)***y1]Fael$\"3].CxDf\"zo%Fiy7$$
!3E+++I+++]Fael$\"3g'*[k[H.h5Fjin7ao7$$!3G+++!*******\\Fael$!3c'R)>8*)
4$=$Fjin7$$!3-SlLz$)=$*\\Fael$!3%\\?H]uMIn%Fiy7$$!3yzInonP')\\Fael$!30
)QSzQ3lL#Fiy7$$!33?'4!e^cz\\Fael$!3U\\n?^2md:Fiy7$$!3#)fhMZNvs\\Fael$!
3%G(QIBKBo6Fiy7$$!3')R#>gKI\"f\\Fael$!3uH5lnY)zy(Faq7$$!3O>Bp/r]X\\Fae
l$!3&=/L=>R2%eFaq7$$!3*)y%Q?mg#=\\Fael$!33p+t&z]L*QFaq7$$!3UQYQ>U,\"*[
Fael$!3meq(z!Q^>HFaq7$$!3]dp2M8_O[Fael$!3K#)*eiF\"RX>Faq7$$!3ew#p([%G?
y%Fael$!3#3?lrNW!e9Faq7$$!3)))f+D^+so%Fael$!3:#4IOlQV,\"Faq7$$!3u@>BwD
P#f%Fael$!3mv@gn.6mxF*7$$!39k^27Z3zVFael$!31JSMNZFh]F*7$$!3inX]PIQkTFa
el$!3nzG6k&p8s$F*7$$!3#HE:D!=q]RFael$!3Ylrr#ojG#HF*7$$!3q\\\"yX%>f_PFa
el$!3Y[Gr!z%y>CF*7$$!3cy]lh1YZNFael$!3uN3!Q))Qr.#F*7$$!361FT&f8`L$Fael
$!3,%oGH'>aM<F*7$$!3@1B]()o%Q7$Fael$!3=,9/NFV&\\\"F*7$$!3u6S&eQFj!HFae
l$!33i(H\"4u\\%H\"F*7$$!3)3'QS!4OZr#Fael$!3$p`6V+>\\9\"F*7$$!3[+>+q'\\
!*\\#Fael$!35Q5^#oIS***Fael7$$!3+X]V?wZ#G#Fael$!3LGD62I#)=()Fael7$$!3Q
fC&3KqP2#Fael$!3U7aNQtkDwFael7$$!31X[J)4WU)=Fael$!3\">\"))Rp^(Qs'Fael7
$$!3_3*[$)3z)e;Fael$!3;a#pq5G4u&Fael7$$!3/O2tj`'zY\"Fael$!3Oah<%\\H\"p
\\Fael7$$!39D#eiJ()eC\"Fael$!3')=w\")[l+FTFael7$$!3IX!oXh5$\\5Fael$!3$
Q=YMzX8U$Fael7$$!3H,t#3<[jL)Fihl$!3gR,u!420o#Fael7$$!3!QyM*\\Xg#G'Fihl
$!3g0')oXTx**>Fael7$$!3z#=(QL&Q(RTFihl$!3!zTXDe>zI\"Fael7$$!3?DG#)*f=>
<#Fihl$!3VMcHtB*Q$oFihl7$$!3hA8YBe*e$\\!#@$!3)H,g'['e1b\"F_il7$$\"3i&*
e&[(RQb@Fihl$\"3G]-cqlq\"y'Fihl7$$\"3-e2Nz\">Y2%Fihl$\"3myEG3t6(G\"Fae
l7$$\"3;&yr$zdWZhFihl$\"3e%eKW*=lb>Fael7$$\"3AYAU$[y))G)Fihl$\"3ooCE/*
GXm#Fael7$$\"35HKUg_QQ5Fael$\"3+&\\C]))\\IQ$Fael7$$\"3IIywyZ3T7Fael$\"
3^^PHc,O4TFael7$$\"3q/xcM![hY\"Fael$\"3gN#3\"[C,i\\Fael7$$\"3Uyl**ztPo
;Fael$\"3M&p\"*ftq1y&Fael7$$\"3#**RJ7P+V)=Fael$\"3EoT7&)=8CnFael7$$\"3
M\\nv$pe*z?Fael$\"3c7^J9'Rkl(Fael7$$\"3uqAT?\\'QH#Fael$\"3%\\\"R)pQ*)>
y)Fael7$$\"3=lIf4S8&\\#Fael$\"3G'p9Pss%p**Fael7$$\"3Wk*Q3?=bq#Fael$\"3
!=nuL<\\#Q6F*7$$\"3D>4^-s?6HFael$\"3kX#*>fvg)H\"F*7$$\"3\\6p*\\_Wl7$Fa
el$\"3/R>7d*y\")\\\"F*7$$\"3X)y)**f*RRL$Fael$\"3V5?%)oL\"Gt\"F*7$$\"3)
QguquJga$Fael$\"3AGbKb\"H[.#F*7$$\"3_+122tOcPFael$\"3A*p<!*4RzU#F*7$$
\"3'Hr+@%Qk\\RFael$\"3G_5dG\\p>HF*7$$\"3385\\i0;rTFael$\"3uep#z2QKv$F*
7$$\"3)>4Gzv(GpVFael$\"3/CFFg1h!)\\F*7$$\"35O*)3sE`!e%Fael$\"3>!p$p9NX
WvF*7$$\"3GZnjSoi\"o%Fael$\"3],Z_$R]Y'**F*7$$\"3YeX=45s#y%Fael$\"37)yX
3I1FY\"Faq7$$\"3j=%)Qa2/P[Fael$\"3q&>!=kHg^>Faq7$$\"3OzAf*\\g8*[Fael$
\"3\\fl/.o#)GHFaq7$$\"3Y4U>s._=\\Fael$\"3da3l0gw0RFaq7$$\"35ShzW-oX\\F
ael$\"39NPgP(f$feFaq7$$\"3)[5(4\"=g#f\\Fael$\"3X;1A,5\"G\"yFaq7$$\"3?q
!)R<,%G(\\Fael$\"3cR95fq&><\"Fiy7$$\"39`&[b3I'z\\Fael$\"3]\"*G]6eii:Fi
y7$$\"3_N!*p`+U')\\Fael$\"3,sP2=f&RM#Fiy7$$\"3!z^\\=-5K*\\Fael$\"3W>;S
:(Hzo%Fiy7$$\"3G+++!*******\\Fael$\"3c'R)>8*)4$=$Fjin7ao7$$\"3E+++I+++
]Fael$!3g'*[k[H.h5Fjin7$$\"3)R@O1k6o+&Fael$!3gQv%H/@In%Fiy7$$\"3pFCF^K
i8]Fael$!3'[Mnq'\\]OBFiy7$$\"3TT'3>'[V?]Fael$!3+*[#)GBfwb\"Fiy7$$\"38b
[askCF]Fael$!3_M_SqBBo6Fiy7$$\"3c#G<Qpp3/&Fael$!3OIzD()3)zy(Faq7$$\"3+
5(*3:H\\a]Fael$!3CfC6rqtSeFaq7$$\"3'[cMwNR<3&Fael$!3K@cEe)\\L*QFaq7$$
\"3s>%z,!e)*3^Fael$!3*)o[[$G8&>HFaq7$$\"3YH\"p_oyM;&Fael$!3mLejX5RX>Fa
q7$$\"32Q)e.drz@&Fael$!3/AN\")GU/e9Faq7$$\"3j.#[i]*z7`Fael$!3J\\(QCfQV
,\"Faq7$$\"3>pv8Uui2aFael$!3)3ErP,5hw(F*7$$\"3Nv6W0`\"4i&Fael$!3Q[b())
eu71&F*7$$\"3^lH:zphNeFael$!36Y%3k[p8s$F*7$$\"3#Ra(G8#)H\\gFael$!3,]$R
`jjG#HF*7$$\"3!)=AVq!3uC'Fael$!3u.aQeZy>CF*7$$\"33kZ`_$RDX'Fael$!3[#4V
3')Qr.#F*7$$\"35[&GzT'okmFael$!3Bvt7Y>aM<F*7$$\"3%=3$*\\7`h(oFael$!3+'
4PBsKa\\\"F*7$$\"3'yHrdisO4(Fael$!3?)[d$*R(\\%H\"F*7$$\"3'G3b/#RE&G(Fa
el$!3z^#Gk**=\\9\"F*7$$\"3%*)H%**R.&4](Fael$!3S]0u>1.%***Fael7$$\"3Hm[
p)QAvr(Fael$!3soiicH#)=()Fael7$$\"3-BEW(oHi#zFael$!3Qa<9(HZci(Fael7$$
\"3(48A#4fv:\")Fael$!3@7d,N^(Qs'Fael7$$\"3i3mG=47T$)Fael$!3$Hl_$z!G4u&
Fael7$$\"3RE69UY.K&)Fael$!3EwT<r%H\"p\\Fael7$$\"3+C`s)o7Tv)Fael$!3@JZ
\\Il+FTFael7$$\"39'>H'*Q*o]*)Fael$!37gmrydM@MFael7$$\"3=4=D'=lj;*Fael$
!3W\"o6&zq]!o#Fael7$$\"31j&>vaR<P*Fael$!3-HzZPTx**>Fael7$$\"39xrJ[h-'e
*Fael$!3l!Gasd>zI\"Fael7$$\"3K&[')393Gy*Fael$!3jV](eM#*Q$oFihl7$$\"3w
\"G'>/T1&***Fael$!3aWvXU'e1b\"F_il7$$\"3fVBmRQb@5F*$\"3g9.NVlq\"y'Fihl
7$$\"3&H_I;>Y2/\"F*$\"3uQv2.t6(G\"Fael7$$\"3))>yadWZh5F*$\"3;=?T')=lb>
Fael7$$\"3MnE]%y))G3\"F*$\"3;To5$*)GXm#Fael7$$\"3ropiD&QQ5\"F*$\"3)pK#
[q)\\IQ$Fael7$$\"39\\.Qx%3T7\"F*$\"3+iT1Q,O4TFael7$$\"396.(G![hY6F*$\"
3Kax9DC,i\\Fael7$$\"3)oIKttPo;\"F*$\"3'*)R?!32n!y&Fael7$$\"3y>%pj.I%)=
\"F*$\"3r;#R2&=8CnFael7$$\"3M\"pV&oe*z?\"F*$\"3]ta&GdRkl(Fael7$$\"3:\"
oB6\\'QH7F*$\"3QGr#fL*)>y)Fael7$$\"3)GDh**R8&\\7F*$\"3]#G'>hEZp**Fael7
$$\"37*o,!>=bq7F*$\"3%y#*pb;\\#Q6F*7$$\"3'*3m3>27\"H\"F*$\"3Is:P\\vg)H
\"F*7$$\"3@t!\\7XaEJ\"F*$\"3GiVPW*y\")\\\"F*7$$\"3q-jm%*RRL8F*$\"38FF2
_L\"Gt\"F*7$$\"3cZ!*GtJga8F*$\"3PT*=C8H[.#F*7$$\"3'G^/#Hnjv8F*$\"3yD6Z
m!RzU#F*7$$\"3g8-j#Qk\\R\"F*$\"3I+yH\")[p>HF*7$$\"3%pj!eag6<9F*$\"3))y
u%))*zB`PF*7$$\"3+%4XSxGpV\"F*$\"3%=Tz&=0h!)\\F*7$$\"3A!ow`E`!e9F*$\"3
ZR#48=`Wa(F*7$$\"3'Q-\">#oi\"o9F*$\"3!e(H[.)\\Y'**F*7$$\"3]n`+*4s#y9F*
$\"37arlrhqi9Faq7$$\"3oDS]tSq$[\"F*$\"3e141KFg^>Faq7$$\"3%Qo-![g8*[\"F
*$\"3z*[p^FE)GHFaq7$$\"3#G,__._=\\\"F*$\"3g3A:i]w0RFaq7$$\"3!=M,D-oX\\
\"F*$\"3m?*oKgd$feFaq7$$\"3S1g7;g#f\\\"F*$\"3(>j\"f'>2G\"yFaq7$$\"3+r1
v4SG(\\\"F*$\"3;bAu+i&><\"Fiy7$$\"3=.Ic1I'z\\\"F*$\"3q\">kMGCEc\"Fiy7$
$\"3QN`P.?k)\\\"F*$\"3_(>G`Z_RM#Fiy7$$\"3ynw=+5K*\\\"F*$\"3#*GLxDf\"zo
%Fiy7$$\"3)******p******\\\"F*$\"3U%Rtg*G.h5Fjin7ao7$$\"3(******\\+++]
\"F*$!3mqx$\\$z>mjF\\^l7$$\"3u'*31m6o+:F*$!3+6n%4M2In%Fiy7$$\"3_$z@rKi
8]\"F*$!3c&>/ia,lL#Fiy7$$\"3H!p#=)[V?]\"F*$!3%y=hXrdwb\"Fiy7$$\"3%oeV#
\\Ys-:F*$!3Y-z]<:Bo6Fiy7$$\"3<!Ql8(p3/:F*$!3ES!po5xzy(Faq7$$\"3rtr[$H
\\a]\"F*$!3$pr$R]\\tSeFaq7$$\"3eg2tPR<3:F*$!3W#z,3#*[L*QFaq7$$\"3XZV(>
e)*3^\"F*$!3.UH**eF^>HFaq7$$\"3?@:YqyM;:F*$!3#4y7]\"3RX>Faq7$$\"3;&p[*
erz@:F*$!3gq=Y+T/e9Faq7$$\"3]+(*\\_*z7`\"F*$!3b=uCJ&QV,\"Faq7$$\"3%eq]
gui2a\"F*$!3Q\"RS*f'4hw(F*7$$\"3Q^dHK:4i:F*$!3euqSUWFh]F*7$$\"3w\\5e*p
hNe\"F*$!3A:Sq3%p8s$F*7$$\"3>N!4H#)H\\g\"F*$!3,J:'zejG#HF*7$$\"3^eUa33
uC;F*$!3-jz0EZy>CF*7$$\"3cgCnORDX;F*$!3Wb`)y$)Qr.#F*7$$\"3=!)p7V'okm\"
F*$!3))ogKH>aM<F*7$$\"3)p%)[PJ:wo\"F*$!3u*yK'4FV&\\\"F*7$$\"3kg'RPEn$4
<F*$!35;_e*Q(\\%H\"F*7$$\"3x-98$RE&G<F*$!3)o'\\a))*=\\9\"F*7$$\"3i\\!*
*\\.&4]<F*$!3Oz+(pbIS***Fael7$$\"3kxCyRAvr<F*$!3J@+91H#)=()Fael7$$\"3K
qPdpHi#z\"F*$!3*H6GfDZci(Fael7$$\"3(3\"fn\"fv:\"=F*$!3t9Ej+^(Qs'Fael7$
$\"3_7A\\#47T$=F*$!3Cdgj^!G4u&Fael7$$\"3s)H,[Y.K&=F*$!3E/A<[%H\"p\\Fae
l7$$\"3I()3Pp7Tv=F*$!3yS=<7l+FTFael7$$\"3'Qk#QR*o]*=F*$!3TJr)RwX8U$Fae
l7$$\"3uM'e*=lj;>F*$!3#)GKGoq]!o#Fael7$$\"3]gK+bR<P>F*$!3+[sEHTx**>Fae
l7$$\"3E2t*\\h-'e>F*$!3uOJ'>d>zI\"Fael7$$\"3EDb<93Gy>F*$!3GZWX=B*Q$oFi
hl7$$\"3q-;U5k]**>F*$!3+9YDO'e1b\"F_il7$$\"3bGhdRQb@?F*$\"3'z]Sh^1<y'F
ihl7$$\"3AQvY\">Y2/#F*$\"3)HRsyH<rG\"Fael7$$\"39A>IdWZh?F*$\"3%*e9Ry=l
b>Fael7$$\"3776<%y))G3#F*$\"3337&>))GXm#Fael7$$\"3u9;@D&QQ5#F*$\"3]k,%
f&)\\IQ$Fael7$$\"38:R)oZ3T7#F*$\"3&pcM)>,O4TFael7$$\"3E_QG-[hY@F*$\"3+
$H(=-C,i\\Fael7$$\"3#e&\\mOx$o;#F*$\"37)4\\+oq1y&Fael7$$\"3g*p:c.I%)=#
F*$\"3/uUN;=8CnFael7$$\"3>3<rne*z?#F*$\"33heRJ&Rkl(Fael7$$\"36Nh?!\\'Q
HAF*$\"39D.([G*)>y)Fael7$$\"3C*>j*)R8&\\AF*$\"31iyn)fs%p**Fael7$$\"3o
\"[>z\"=bqAF*$\"3K#=lx:\\#Q6F*7$$\"3-E@#zr?6H#F*$\"31)*QaRvg)H\"F*7$$
\"3gb%)**\\Wl7BF*$\"3%[yE;$*y\")\\\"F*7$$\"3zEFL$*RRLBF*$\"3G]MINL\"Gt
\"F*7$$\"31N1(=<.YN#F*$\"3=dB^4\"H[.#F*7$$\"3Km>qFnjvBF*$\"3buX#R.RzU#
F*7$$\"3Yc.0\"Qk\\R#F*$\"3TvX-M[p>HF*7$$\"3\"H<7H0;rT#F*$\"3-2!o(>zB`P
F*7$$\"3#*ytHs(GpV#F*$\"3=>h)oP51)\\F*7$$\"3inWajK0eCF*$\"3NA]#z%GXWvF
*7$$\"3gt$=.oi\"oCF*$\"3%3XTM@\\Y'**F*7$$\"39zA4(4s#yCF*$\"3#yaoC/1FY
\"Faq7$$\"3e4#p:2/P[#F*$\"3kj;%**\\-;&>Faq7$$\"3eRh/Yg8*[#F*$\"3f9DHZd
#)GHFaq7$$\"3e/YGL?&=\\#F*$\"3f]Rl=Tw0RFaq7$$\"3epI_?!oX\\#F*$\"3y3b$*
oaNfeFaq7$$\"3I-B99g#f\\#F*$\"3D<i'>R.G\"yFaq7$$\"3-N:w2SG(\\#F*$\"3Kn
UQU`&><\"Fiy7$$\"3Q^6d/I'z\\#F*$\"3)y'*GavAEc\"Fiy7$$\"3wn2Q,?k)\\#F*$
\"3L7PfK!\\RM#Fiy7$$\"3m$Q!>)*4K*\\#F*$\"3MPKAO@!zo%Fiy7$$\"3-+++&****
**\\#F*$\"3E-bSOx>mjF\\^l7ao7$$\"39+++2+++DF*$!3o!\\_A*RGZXF\\^l7$$\"3
)><e!o6o+DF*$!3%)>n-RO*Hn%Fiy7$$\"3EWj6HBO,DF*$!3d%=^`7)\\OBFiy7$$\"3b
;X<!\\V?]#F*$!3#o%GC'>cwb\"Fiy7$$\"3$))oK7lCF]#F*$!3#Gd6YmI#o6Fiy7$$\"
3'H.\\L(p3/DF*$!3?n?[EL(zy(Faq7$$\"34x`Y&H\\a]#F*$!3cHbnHGtSeFaq7$$\"3
Nl!)pRR<3DF*$!3'Q/QL)zM$*QFaq7$$\"3g`2$Re)*3^#F*$!3#G0,XB7&>HFaq7$$\"3
6IhRsyM;DF*$!3/*p*Q%e!RX>Faq7$$\"3j1:'3;(z@DF*$!3]Y-6sR/e9Faq7$$\"3v!e
uV&*z7`#F*$!3!\\4c+ZQV,\"Faq7$$\"3)[l()yui2a#F*$!3'Qb4hI4hw(F*7$$\"3a&
QZS`\"4iDF*$!3[r&QfHu71&F*7$$\"3%G!oC,<c$e#F*$!3W&f**4Lp8s$F*7$$\"3?;$
*[C)H\\g#F*$!3X3PeSN'G#HF*7$$\"3O&HX+\"3uCEF*$!3'\\_IPp%y>CF*7$$\"3#[W
\"4QRDXEF*$!3%[hF\\\")Qr.#F*7$$\"3'e5hWkokm#F*$!35aZ_7>aM<F*7$$\"3o&Q)
*\\J:wo#F*$!3C&[GppKa\\\"F*7$$\"3g\">-\\En$4FF*$!3)3%H\")zt\\%H\"F*7$$
\"3[(H<URE&GFF*$!3&)y;m!)*=\\9\"F*7$$\"3yp')*f.&4]FF*$!3y#e*>%\\IS***F
ael7$$\"3moapSAvrFF*$!39mPlbG#)=()Fael7$$\"3MyKSqHi#z#F*$!3/wWr9skDwFa
el7$$\"3U3'HCfv:\"GF*$!3/0&\\i1vQs'Fael7$$\"3=kd:$47T$GF*$!3xt%>R-G4u&
Fael7$$\"3![[)QlM?`GF*$!3[W-<D%H\"p\\Fael7$$\"3AU#p)p7TvGF*$!3N]*[Q\\1
q7%Fael7$$\"37oB!)R*o]*GF*$!3F)fd#\\dM@MFael7$$\"3*))3#H>lj;HF*$!3)Gwa
q020o#Fael7$$\"3#[ca_&R<PHF*$!3Edl0@Tx**>Fael7$$\"3e'*G;:EgeHF*$!3f:?n
m&>zI\"Fael7$$\"3A-CE93GyHF*$!3v.Q.\"H#*Q$oFihl7$$\"37xNU5k]**HF*$!3Wr
D0I'e1b\"F_il7$$\"328**[RQb@IF*$\"3Y$[I*)[1<y'Fihl7$$\"3[`XI\">Y2/$F*$
\"3yUsm#H<rG\"Fael7$$\"3;Cg0dWZhIF*$\"3o')3Pq=lb>Fael7$$\"3)obRQy))G3$
F*$\"3U*e&zq)GXm#Fael7$$\"3xgizC&QQ5$F*$\"3#f*zRT)\\IQ$Fael7$$\"38\"[(
Qw%3T7$F*$\"3Cg\\g,,O4TFael7$$\"3]#R(p,[hYJF*$\"3i!yE#zB,i\\Fael7$$\"3
w/w*ftPo;$F*$\"33&yx?lq1y&Fael7$$\"3Sz>'[.I%)=$F*$\"3F5$p>yJTs'Fael7$$
\"3QC(zo'e*z?$F*$\"3Q0i$**[Rkl(Fael7$$\"32*e)G*['QHKF*$\"3cQN\"QB*)>y)
Fael7$$\"3gX^'zR8&\\KF*$\"3uK%fh`s%p**Fael7$$\"3[us$o\"=bqKF*$\"38S/'*
\\\"\\#Q6F*7$$\"3GVwv;27\"H$F*$\"3]GirHvg)H\"F*7$$\"3KPyu[Wl7LF*$\"3_/
#z)=*y\")\\\"F*7$$\"3(3:**>*RRLLF*$\"3JqT`=L\"Gt\"F*7$$\"37AAXqJgaLF*$
\"3ckdg'3H[.#F*7$$\"3!*=%*>EnjvLF*$\"3y**zP,!RzU#F*7$$\"3W)\\q%zV'\\R$
F*$\"3aE8v'y%p>HF*7$$\"3V3PC^g6<MF*$\"3GE&)oSyB`PF*7$$\"3hj'\\0xGpV$F*
$\"3kLG>N-h!)\\F*7$$\"39aArhK0eMF*$\"3EI1a9DXWvF*7$$\"3CAdWyE;oMF*$\"3
CK(*RB'[Y'**F*7$$\"3M!>z^4s#yMF*$\"31X*zK\"fqi9Faq7$$\"3%GRM'pSq$[$F*$
\"3mTC#yE-;&>Faq7$$\"3K&f*3Wg8*[$F*$\"3&e!eT>_#)GHFaq7$$\"3c'><8._=\\$
F*$\"3v=k:vJw0RFaq7$$\"3#yzW&=!oX\\$F*$\"3!)fUgMLNfeFaq7$$\"3W)fe@,Ef
\\$F*$\"3s3XM(e*z7yFaq7$$\"31*Rsd+%G(\\$F*$\"3S!yFS[a><\"Fiy7$$\"3Q*Hz
D+jz\\$F*$\"3u:lRF7ii:Fiy7$$\"3o*>'Q**>k)\\$F*$\"3I^)o)*eXRM#Fiy7$$\"3
c*4$>'*4K*\\$F*$\"3qoUvY$))yo%Fiy7$$\"3')*****H******\\$F*$\"3?E-(R-%G
ZXF\\^l7gn7$$\"39+++2+++NF*$!3#\\<J7?%GZXF\\^l7$$\"3Z<'[]1M+]$F*$!3.XM
k3#yTM*F]\\l7$$\"3OMs4B\"o+]$F*$!3!eXn@8pDn%F]\\l7$$\"3q^e9\"=-,]$F*$!
3$yT5!GF::JF]\\l7$$\"3.pW>Ri8+NF*$!3m'Q\"*y\\/kL#F]\\l7$$\"3r.<HbV?+NF
*$!3!y?HobHwb\"F]\\l7$$\"3QQ*)QrCF+NF*$!3o&Gmp/K#o6F]\\l7$$\"3s2Me.(3/
]$F*$!3w!4tV'zF)y(Fiy7$$\"31xyxN\\a+NF*$!3fyx'y)4BTeFiy7$$\"3J:o;+u\"3
]$F*$!3CzZv5f;%*QFiy7$$\"3b`dbk)*3,NF*$!3TqUw#pD1#HFiy7$$\"3!emrT+k:]$
F*$!38X#*\\Ww?N?Fiy7$$\"3hxvyV\"Q?]$F*$!3QJ8>(HU<c\"Fiy7$$\"3l#>hHe/J]
$F*$!3OLY**4yDD5Fiy7$$\"3^,f=\"4yT]$F*$!3ahr*[b5\"=wFaq7$$\"3'yS8s\\Y_
]$F*$!3i(=)[yRamgFaq7$$\"3)yRmj/Pi]$F*$!36t9C8D)G5&Faq7$$\"3rs%REqis]$
F*$!33b%*)zbT?Q%Faq7$$\"3*GICyVB$3NF*$!3#4))yd7)QBQFaq7$$\"3,Vz@r2Q4NF
*$!3+nA9\">MAR$Faq7$$\"3z&4w&o$o/^$F*$!3J1\"3*3leRIFaq7$$\"3\")[6#\\KE
9^$F*$!3wi'\\VViXy#Faq7$$\"3hMV(ov/D^$F*$!3./xzl2?WDFaq7$$\"3ZMF(ph(e8
NF*$!34DjLBQATBFaq7$$\"3XRmK`6j9NF*$!3!G]:37IS<#Faq7$$\"3>a&3Vyyb^$F*$
!3xU/Dp]fT?Faq7$$\"3*=)G?4cq;NF*$!3^zb10vl.>Faq7$$\"3mUUpx,m<NF*$!3h7c
l^jc+=Faq7$$\"3%4Pyxcq(=NF*$!3GEFoCk#Qp\"Faq7$$\"35%=^6X`(>NF*$!3,M\\&
p$\\M4;Faq7$$\"3\\%z*)*H=$3_$F*$!3r!)[X(e:e_\"Faq7$$\"35K&e;qe=_$F*$!3
%*Gr))Gh#RX\"Faq7$$\"3a[*=X8IH_$F*$!3gyeU72x&Q\"Faq7$$\"3\"3&*\\V/9R_$
F*$!3DcZ%G9`&G8Faq7$$\"3E)GCSKv\\_$F*$!3)*[\\mR5)=F\"Faq7$$\"3_1(Q`pxg
_$F*$!3oobmf\"*)y@\"Faq7$$\"3iE&3GJPq_$F*$!3'z-6i#eYu6Faq7$$\"3;7b'fst
!GNF*$!3G6x5aT*38\"Faq7$$\"3jy(p@WW\"HNF*$!3Q6H\"zpE\"*3\"Faq7$$\"32Ic
3H>>INF*$!3y\\$3NTC60\"Faq7$$\"3f!\\PlU07`$F*$!37Nutfmx;5Faq7$$\"3^'>h
EuIB`$F*$!3y_o\"f_W:\")*F*7$$\"3@-)[#*)=MLNF*$!3$*3aa9]!>^*F*7$$\"3()*
)f0/:UMNF*$!3Y)HYXRR8@*F*7$$\"3G7O`&z*RNNF*$!3%)=Y5\\ava*)F*7$$\"3e%H>
lKpk`$F*$!3SWy\"*>'H**o)F*7$$\"3sAj#=nvu`$F*$!3)zhZRB$\\a%)F*7$$\"3O(Q
PEfF&QNF*$!3Th%Qpd#[@#)F*7$$\"3`@Q]PgbRNF*$!3mtct.jf0!)F*7$$\"3$*ow'Rs
K1a$F*$!3gUXocSB\"z(F*7$$\"3Xv&**4qp;a$F*$!3%o?p/j%>&f(F*7$$\"3%4h)yf,
tUNF*$!3NRuBK<]/uF*7$$\"3afMWP=yVNF*$!3\\%**R.kdWA(F*7$$\"3O\\_)*>#[Za
$F*$!3=a8qY4VmqF*7$$\"39aVV.e&ea$F*$!3M@3MzJX$*oF*7$$\"3cJ[FRk%oa$F*$!
3ei0=bCiXnF*7$$\"3%p()*pjE!za$F*$!3#Qfo;O\"p%f'F*7$$\"3?XeC0O\"*[NF*$!
3%\\<R%>5HckF*7$$\"3#)************\\NF*$!35(4vY^^PJ'F*-%'COLOURG6&%$RG
BG$\"*++++\"!\")$\"\"!FhhvFghv-%*THICKNESSG6#\"\"#-F$6%7S7$F($\"3?x)ov
!30K<F*7$$!3gmmT0-C&R$F*Faiv7$$!3=LekNXegKF*Faiv7$$!3Vm;z^,:4JF*Faiv7$
$!3Cm;ai>rcHF*Faiv7$$!32LeR&G)*\\!GF*Faiv7$$!3sm\"zfGSVm#F*Faiv7$$!3r*
\\7[2(p=DF*Faiv7$$!3Sm\"zuls!oBF*Faiv7$$!3#)*\\iXHJz@#F*Faiv7$$!3?LL3[
C\\j?F*Faiv7$$!3Ym;/GEYF>F*Faiv7$$!3y***\\#p_Ku<F*Faiv7$$!3&****\\P6f0
i\"F*Faiv7$$!3))***\\2$pPs9F*Faiv7$$!3om\"zCJ8yL\"F*Faiv7$$!3!HL$3bT!y
<\"F*Faiv7$$!3-LLL5aDU5F*Faiv7$$!3;'*\\7j**zX))FaelFaiv7$$!3aJLLy`5]uF
aelFaiv7$$!3\"p*\\78s!)=fFaelFaiv7$$!3O'*\\PR#\\1Y%FaelFaiv7$$!3'oK$ek
V@RHFaelFaiv7$$!3CI$e*3A1U:FaelFaiv7$$!3T@kmTg[/NF_ilFaiv7$$\"3tN$3_iA
.`\"FaelFaiv7$$\"3m-]7B'zH*GFaelFaiv7$$\"3eNL37lokVFaelFaiv7$$\"3G,++N
P5&)eFaelFaiv7$$\"3_.+vj``stFaelFaiv7$$\"33**\\(o%>q6))FaelFaiv7$$\"3I
+]i1^'4/\"F*Faiv7$$\"3qmm;Uza%=\"F*Faiv7$$\"3n++DmK&yL\"F*Faiv7$$\"3?L
3_v1xw9F*Faiv7$$\"3W++v'4W'G;F*Faiv7$$\"3;mTN]^ar<F*Faiv7$$\"3D+vVEz\"
4#>F*Faiv7$$\"3/nm\"z6dp1#F*Faiv7$$\"3w*\\(=<m%)>AF*Faiv7$$\"3iLLLwt4n
BF*Faiv7$$\"3PL$eaa#o<DF*Faiv7$$\"3?nTN$z?qm#F*Faiv7$$\"3a++]LrC/GF*Fa
iv7$$\"3Qn;H0S_hHF*Faiv7$$\"3[LLL9V>-JF*Faiv7$$\"3W+vo.#y@D$F*Faiv7$$
\"3s+DJB>t&R$F*Faiv7$F\\hvFaiv-Fahv6&FchvFghvFghvFdhvFihv-F$6%7$7$$!3+
+++++++NF*$!\"&Fhhv7$Fgbw$\"\"&Fhhv-Fahv6&FchvFhhvFhhvFhhv-%*LINESTYLE
G6#\"\"$-F$6%7$7$$!3++++++++DF*Fibw7$FhcwF\\cwF^cwF`cw-F$6%7$7$$!3++++
++++:F*Fibw7$F_dwF\\cwF^cwF`cw-F$6%7$7$$!3++++++++]FaelFibw7$FfdwF\\cw
F^cwF`cw-F$6%7$7$$\"3++++++++]FaelFibw7$F]ewF\\cwF^cwF`cw-F$6%7$7$$\"3
++++++++:F*Fibw7$FdewF\\cwF^cwF`cw-F$6%7$7$$\"3++++++++DF*Fibw7$F[fwF
\\cwF^cwF`cw-F$6%7$7$$\"3++++++++NF*Fibw7$FbfwF\\cwF^cwF`cw-F$6&7$7$$!
3_mmmmmmmEF*Fghv7$FifwFaiv-Fahv6&Fchv$\")!\\DP\"FfhvF^gw$\")viobFfhv-F
jhv6#\"\"\"-FacwF[iv-F$6&7$7$$!3ummmmmmm;F*Fghv7$FjgwFaivF\\gwFbgwFegw
-F$6&7$7$$!3ImmmmmmmmFaelFghv7$FahwFaivF\\gwFbgwFegw-F$6&7$7$$\"3:LLLL
LLLLFaelFghv7$FhhwFaivF\\gwFbgwFegw-F$6&7$7$$\"3ELLLLLLL8F*Fghv7$F_iwF
aivF\\gwFbgwFegw-F$6&7$7$$\"3[LLLLLLLBF*Fghv7$FfiwFaivF\\gwFbgwFegw-F$
6&7$7$$\"3[LLLLLLLLF*Fghv7$F]jwFaivF\\gwFbgwFegw-F$6&7$7$$\"3.LLLLLLLV
F*Fghv7$FdjwFaivF\\gwFbgwFegw-F$6&7#F[gw-Fahv6&FchvFghvFdhvFghv-%'SYMB
OLG6#%'CIRCLEG-%&STYLEG6#%&POINTG-F$6&7#F\\hwFjjwF\\[xF`[x-F$6&7#FchwF
jjwF\\[xF`[x-F$6&7#FjhwFjjwF\\[xF`[x-F$6&7#FaiwFjjwF\\[xF`[x-F$6&7#Fhi
wFjjwF\\[xF`[x-F$6&7#F_jwFjjwF\\[xF`[x-F$6&7#FfjwFjjwF\\[xF`[x-F$6&Fij
wFjjw-F][x6#%(DIAMONDGF`[x-F$6&Ff[xFjjwF[]xF`[x-F$6&Fi[xFjjwF[]xF`[x-F
$6&F\\\\xFjjwF[]xF`[x-F$6&F_\\xFjjwF[]xF`[x-F$6&Fb\\xFjjwF[]xF`[x-F$6&
Fe\\xFjjwF[]xF`[x-F$6&Fh\\xFjjwF[]xF`[x-F$6&FijwFjjw-F][x6#%&CROSSGF`[
x-F$6&Ff[xFjjwF^^xF`[x-F$6&Fi[xFjjwF^^xF`[x-F$6&F\\\\xFjjwF^^xF`[x-F$6
&F_\\xFjjwF^^xF`[x-F$6&Fb\\xFjjwF^^xF`[x-F$6&Fe\\xFjjwF^^xF`[x-F$6&Fh
\\xFjjwF^^xF`[x-F$6&FijwF^cw-F][x6$F_[x\"#7F`[x-F$6&Ff[xF^cwFa_xF`[x-F
$6&Fi[xF^cwFa_xF`[x-F$6&F\\\\xF^cwFa_xF`[x-F$6&F_\\xF^cwFa_xF`[x-F$6&F
b\\xF^cwFa_xF`[x-F$6&Fe\\xF^cwFa_xF`[x-F$6&Fh\\xF^cwFa_xF`[x-F$6$7$7$F
ifw$!3-+++++++))FaelFhfwF^cw-F$6$7$7$FjgwFf`xFigwF^cw-F$6$7$7$FahwFf`x
F`hwF^cw-F$6$7$7$FhhwFf`xFghwF^cw-F$6$7$7$F_iwFf`xF^iwF^cw-F$6$7$7$Ffi
wFf`xFeiwF^cw-F$6$7$7$F]jwFf`xF\\jwF^cw-F$6$7$7$FdjwFf`xFcjwF^cw-%)POL
YGONSG6+7%7$$!+nmm1FFjin$!\"\"F\\cx7$$!+nmmmEFjinFghv7$$!+nmmEEFjinF[c
x7%7$$!+nmm1<FjinF[cx7$$!+nmmm;FjinFghv7$$!+nmmE;FjinF[cx7%7$$!+nmmmqF
\\^lF[cx7$$!+nmmmmF\\^lFghv7$$!+nmmmiF\\^lF[cx7%7$$\"+LLLLHF\\^lF[cx7$
$\"+LLLLLF\\^lFghv7$$\"+LLLLPF\\^lF[cx7%7$$\"+LLL$H\"FjinF[cx7$$\"+LLL
L8FjinFghv7$$\"+LLLt8FjinF[cx7%7$$\"+LLL$H#FjinF[cx7$$\"+LLLLBFjinFghv
7$$\"+LLLtBFjinF[cx7%7$$\"+LLL$H$FjinF[cx7$$F]exFjinFghv7$$\"+LLLtLFji
nF[cx7%7$$\"+LLL$H%FjinF[cx7$$\"+LLLLVFjinFghv7$$\"+LLLtVFjinF[cx-Fahv
6&FchvFdhv$\")AR!)\\FfhvFghv-F$6$7U7$$!+nmmmCFjin$!#6F\\cx7$$!+FPCoCFj
in$!+`L$\\2\"Fjin7$$!+N+&HZ#Fjin$!+B?E]5Fjin7$$!+q8r![#Fjin$!+*3vj-\"F
jin7$$!+J`S\"\\#Fjin$!+l#\\O+\"Fjin7$$!+oK'[]#Fjin$!+&\\HW#)*F\\^l7$$!
+UH(3_#Fjin$!+)y04j*F\\^l7$$!+p==RDFjin$!+9N(*e%*F\\^l7$$!+38]fDFjin$!
+\\TM6$*F\\^l7$$!+43^\"e#Fjin$!+&*eM!>*F\\^l7$$!+oK'[g#Fjin$!+np)y4*F
\\^l7$$!+//>HEFjin$!+)\\Da.*F\\^l7$$!+j&3Tl#Fjin$!+Vl%R+*F\\^l7$$!+rZA
zEFjinF^\\y7$$!+IH9/FFjin$!+*\\Da.*F\\^l7$$!+m+ZGFFjinFd[y7$$!+DD#=v#F
jinF_[y7$$!+E?$Qx#FjinFjjx7$$!+l9:%z#FjinFejx7$$!+$RgC\"GFjin$!+!z04j*
F\\^l7$$!+m+ZGGFjin$!+(\\HW#)*F\\^l7$$!+.!G>%GFjinFfix7$$!+k>i_GFjin$!
+!4vj-\"Fjin7$$!+*H$QgGFjinF\\ix7$$!+2'*3lGFjinFghx7$$!+nmmmGFjin$!+++
++6Fjin7$Fj^y$!+Zm1D6Fjin7$Fg^y$!+xzt\\6Fjin7$Fb^y$!+6\\it6Fjin7$F_^y$
!+N2N'>\"Fjin7$Fj]y$!+]qb<7Fjin7$$!+#RgC\"GFjin$!+@%4pB\"Fjin7$Fb]y$!+
\\E5a7Fjin7$F_]y$!+&el)o7Fjin7$F\\]y$!+6a'4G\"Fjin7$Fi\\y$!+.8@!H\"Fji
n7$Fd\\y$!+]uX'H\"Fjin7$Fa\\y$!+Y`g*H\"Fjin7$F\\\\yFeay7$Fg[yFbay7$Fb[
yF_ay7$F][y$!+5a'4G\"Fjin7$FhjxFi`y7$FcjxFf`y7$$!+TH(3_#FjinFc`y7$Fiix
F^`y7$FdixF[`y7$F_ix$!+5\\it6Fjin7$FjhxFe_y7$FehxFb_y7$F`hxF__yF^cw-F$
6$7U7$$!+nmmm9FjinFbhx7$$!+FPCo9FjinFghx7$$!+N+&HZ\"FjinF\\ix7$$!+q8r!
[\"FjinFaix7$$!+J`S\"\\\"FjinFfix7$$!+oK'[]\"FjinF[jx7$$!+UH(3_\"FjinF
`jx7$$!+p==R:FjinFejx7$$!+38]f:FjinFjjx7$$!+43^\"e\"FjinF_[y7$$!+oK'[g
\"FjinFd[y7$$!+//>H;FjinFi[y7$$!+j&3Tl\"FjinF^\\y7$$!+rZAz;FjinF^\\y7$
$!+IH9/<FjinFf\\y7$$!+m+ZG<FjinFd[y7$$!+DD#=v\"FjinF_[y7$$!+E?$Qx\"Fji
nFjjx7$$!+l9:%z\"FjinFejx7$$!+$RgC\"=FjinFg]y7$$!+m+ZG=FjinF\\^y7$$!+.
!G>%=FjinFfix7$$!+k>i_=FjinFd^y7$$!+*H$Qg=FjinF\\ix7$$!+2'*3l=FjinFghx
7$$!+nmmm=FjinF__y7$FfgyFb_y7$FcgyFe_y7$F`gyFh_y7$F]gyF[`y7$FjfyF^`y7$
$!+#RgC\"=FjinFc`y7$FdfyFf`y7$FafyFi`y7$F^fyF\\ay7$F[fyF_ay7$FheyFbay7
$FeeyFeay7$FbeyFeay7$F_eyFbay7$F\\eyF_ay7$FidyF[by7$FfdyFi`y7$FcdyFf`y
7$$!+TH(3_\"FjinFc`y7$F]dyF^`y7$FjcyF[`y7$FgcyFeby7$FdcyFe_y7$FacyFb_y
7$F^cyF__yF^cw-F$6$7U7$$!+nmmmYF\\^lFbhx7$$!+ksV#o%F\\^lFghx7$$!+X.]HZ
F\\^lF\\ix7$$!+&p8r![F\\^lFaix7$$!+2L09\\F\\^lFfix7$$!+yEj[]F\\^lF[jx7
$$!+7%H(3_F\\^lF`jx7$$!+)o==R&F\\^lFejx7$$!+xI,&f&F\\^lFjjx7$$!+$33^\"
eF\\^lF_[y7$$!+zEj[gF\\^lFd[y7$$!+RS!>H'F\\^lFi[y7$$!+Gc3TlF\\^lF^\\y7
$$!+1xC#z'F\\^lF^\\y7$$!+'HH9/(F\\^lFf\\y7$$!+b1q%G(F\\^lFd[y7$$!+]_A=
vF\\^lF_[y7$$!+c-KQxF\\^lFjjx7$$!+YY^TzF\\^lFejx7$$!+CRgC\")F\\^lFg]y7
$$!+d1q%G)F\\^lF\\^y7$$!+G+G>%)F\\^lFfix7$$!+R'>i_)F\\^lFd^y7$$!+!*H$Q
g)F\\^lF\\ix7$$!+qg*3l)F\\^lFghx7$$!+nmmm')F\\^lF__y7$Fd^zFb_y7$$!+*)H
$Qg)F\\^lFe_y7$F^^zFh_y7$$!+F+G>%)F\\^lF[`y7$$!+c1q%G)F\\^lF^`y7$$!+AR
gC\")F\\^lFc`y7$$!+WY^TzF\\^lFf`y7$$!+b-KQxF\\^lFi`y7$$!+[_A=vF\\^lF\\
ay7$$!+a1q%G(F\\^lF_ay7$$!+&HH9/(F\\^lFbay7$$!+0xC#z'F\\^lFeay7$$!+Fc3
TlF\\^lFeay7$$!+PS!>H'F\\^lFbay7$$!+yEj[gF\\^lF_ay7$$!+%33^\"eF\\^lF[b
y7$Fd[zFi`y7$Fa[zFf`y7$$!+5%H(3_F\\^lFc`y7$$!+wEj[]F\\^lF^`y7$$!+1L09
\\F\\^lF[`y7$$!+%p8r![F\\^lFeby7$$!+W.]HZF\\^lFe_y7$F_jyFb_y7$F\\jyF__
yF^cw-F$6$7U7$$\"+LLLL`F\\^lFbhx7$$\"+OFc<`F\\^lFghx7$$\"+b'*\\q_F\\^l
F\\ix7$$\"+0j)G>&F\\^lFaix7$$\"+$pYf3&F\\^lFfix7$$\"+AtO^\\F\\^lF[jx7$
$\"+)eq7z%F\\^lF`jx7$$\"+78=3YF\\^lFejx7$$\"+Bp)\\S%F\\^lFjjx7$$\"+<>*
[=%F\\^lF_[y7$$\"+@tO^RF\\^lFd[y7$$\"+hf43PF\\^lFi[y7$$\"+sV\"*eMF\\^l
F^\\y7$$\"+%H_x?$F\\^lF^\\y7$$\"+/2deHF\\^lFf\\y7$$\"+X$*H:FF\\^lFd[y7
$$\"+]Zx\"[#F\\^lF_[y7$$\"+W(z;E#F\\^lFjjx7$$\"+a`[e?F\\^lFejx7$$\"+wg
Rv=F\\^lFg]y7$$\"+V$*H:<F\\^lF\\^y7$$\"+s*>2e\"F\\^lFfix7$$\"+h.yt9F\\
^lFd^y7$$\"+5q;'R\"F\\^lF\\ix7$$\"+IR5\\8F\\^lFghx7$$FgexF\\^lF__y7$Fd
gzFb_y7$$\"+6q;'R\"F\\^lFe_y7$F^gzFh_y7$$\"+t*>2e\"F\\^lF[`y7$$\"+W$*H
:<F\\^lF^`y7$$\"+ygRv=F\\^lFc`y7$$\"+c`[e?F\\^lFf`y7$$\"+X(z;E#F\\^lFi
`y7$$\"+_Zx\"[#F\\^lF\\ay7$$\"+Y$*H:FF\\^lF_ay7$$\"+02deHF\\^lFbay7$$
\"+&H_x?$F\\^lFeay7$$\"+tV\"*eMF\\^lFeay7$$\"+jf43PF\\^lFbay7$$\"+AtO^
RF\\^lF_ay7$$\"+;>*[=%F\\^lF[by7$FddzFi`y7$FadzFf`y7$$\"+!fq7z%F\\^lFc
`y7$$\"+CtO^\\F\\^lF^`y7$$\"+%pYf3&F\\^lF[`y7$$\"+1j)G>&F\\^lFeby7$$\"
+c'*\\q_F\\^lFe_y7$F_czFb_y7$F\\czF__yF^cw-F$6$7U7$$\"+LLLL:FjinFbhx7$
$\"+tivJ:FjinFghx7$$\"+l*\\q_\"FjinF\\ix7$$\"+I')G>:FjinFaix7$$\"+pYf3
:FjinFfix7$$\"+Kn8&\\\"FjinF[jx7$$\"+eq7z9FjinF`jx7$$\"+J\"=3Y\"FjinFe
jx7$$\"+#p)\\S9FjinFjjx7$$\"+\">*[=9FjinF_[y7$$\"+Kn8&R\"FjinFd[y7$$\"
+'f43P\"FjinFi[y7$$\"+P9*eM\"FjinF^\\y7$$\"+H_x?8FjinF^\\y7$$\"+qq&eH
\"FjinFf\\y7$$\"+M*H:F\"FjinFd[y7$$\"+vu<[7FjinF_[y7$$\"+uz;E7FjinFjjx
7$$\"+N&[e?\"FjinFejx7$$\"+2'Rv=\"FjinFg]y7$$\"+M*H:<\"FjinF\\^y7$$\"+
(*>2e6FjinFfix7$$\"+O!yt9\"FjinFd^y7$$\"+,nhR6FjinF\\ix7$$\"+$R5\\8\"F
jinFghx7$$\"+LLLL6FjinF__y7$Fc`[lFb_y7$F``[lFe_y7$F]`[lFh_y7$Fj_[lF[`y
7$Fg_[lF^`y7$$\"+3'Rv=\"FjinFc`y7$Fa_[lFf`y7$F^_[lFi`y7$F[_[lF\\ay7$Fh
^[lF_ay7$Fe^[lFbay7$Fb^[lFeay7$F_^[lFeay7$F\\^[lFbay7$Fi][lF_ay7$Ff][l
F[by7$Fc][lFi`y7$F`][lFf`y7$$\"+fq7z9FjinFc`y7$Fj\\[lF^`y7$Fg\\[lF[`y7
$Fd\\[lFeby7$Fa\\[lFe_y7$F^\\[lFb_y7$F[\\[lF__yF^cw-F$6$7U7$$\"+LLLLDF
jinFbhx7$$\"+tivJDFjinFghx7$$\"+l*\\q_#FjinF\\ix7$$\"+I')G>DFjinFaix7$
$\"+pYf3DFjinFfix7$$\"+Kn8&\\#FjinF[jx7$$\"+eq7zCFjinF`jx7$$\"+J\"=3Y#
FjinFejx7$$\"+#p)\\SCFjinFjjx7$$\"+\">*[=CFjinF_[y7$$\"+Kn8&R#FjinFd[y
7$$\"+'f43P#FjinFi[y7$$\"+P9*eM#FjinF^\\y7$$\"+H_x?BFjinF^\\y7$$\"+qq&
eH#FjinFf\\y7$$\"+M*H:F#FjinFd[y7$$\"+vu<[AFjinF_[y7$$\"+uz;EAFjinFjjx
7$$\"+N&[e?#FjinFejx7$$\"+2'Rv=#FjinFg]y7$$\"+M*H:<#FjinF\\^y7$$\"+(*>
2e@FjinFfix7$$\"+O!yt9#FjinFd^y7$$\"+,nhR@FjinF\\ix7$$\"+$R5\\8#FjinFg
hx7$$\"+LLLL@FjinF__y7$Fag[lFb_y7$F^g[lFe_y7$F[g[lFh_y7$Fhf[lF[`y7$Fef
[lF^`y7$$\"+3'Rv=#FjinFc`y7$F_f[lFf`y7$F\\f[lFi`y7$Fie[lF\\ay7$Ffe[lF_
ay7$Fce[lFbay7$F`e[lFeay7$F]e[lFeay7$Fjd[lFbay7$Fgd[lF_ay7$Fdd[lF[by7$
Fad[lFi`y7$F^d[lFf`y7$$\"+fq7zCFjinFc`y7$Fhc[lF^`y7$Fec[lF[`y7$Fbc[lFe
by7$F_c[lFe_y7$F\\c[lFb_y7$Fib[lF__yF^cw-F$6$7U7$$\"+LLLLNFjinFbhx7$$
\"+tivJNFjinFghx7$$\"+l*\\q_$FjinF\\ix7$$\"+I')G>NFjinFaix7$$\"+pYf3NF
jinFfix7$$\"+Kn8&\\$FjinF[jx7$$\"+eq7zMFjinF`jx7$$\"+J\"=3Y$FjinFejx7$
$\"+#p)\\SMFjinFjjx7$$\"+\">*[=MFjinF_[y7$$\"+Kn8&R$FjinFd[y7$$\"+'f43
P$FjinFi[y7$$\"+P9*eM$FjinF^\\y7$$\"+H_x?LFjinF^\\y7$$\"+qq&eH$FjinFf
\\y7$$\"+M*H:F$FjinFd[y7$$\"+vu<[KFjinF_[y7$$\"+uz;EKFjinFjjx7$$\"+N&[
e?$FjinFejx7$$\"+2'Rv=$FjinFg]y7$$\"+M*H:<$FjinF\\^y7$$\"+(*>2eJFjinFf
ix7$$\"+O!yt9$FjinFd^y7$$\"+,nhRJFjinF\\ix7$$\"+$R5\\8$FjinFghx7$$\"+L
LLLJFjinF__y7$F_^\\lFb_y7$F\\^\\lFe_y7$Fi]\\lFh_y7$Ff]\\lF[`y7$Fc]\\lF
^`y7$$\"+3'Rv=$FjinFc`y7$F]]\\lFf`y7$Fj\\\\lFi`y7$Fg\\\\lF\\ay7$Fd\\\\
lF_ay7$Fa\\\\lFbay7$F^\\\\lFeay7$F[\\\\lFeay7$Fh[\\lFbay7$Fe[\\lF_ay7$
Fb[\\lF[by7$F_[\\lFi`y7$F\\[\\lFf`y7$$\"+fq7zMFjinFc`y7$Ffj[lF^`y7$Fcj
[lF[`y7$F`j[lFeby7$F]j[lFe_y7$Fji[lFb_y7$Fgi[lF__yF^cw-%%TEXTG6&7$$Ffh
v!\"#F\\cwQ\"y6\"-%&COLORG6&Fchv$FdgwFf`\\lF\\a\\lF\\a\\l-%%FONTG6$%*H
ELVETICAG\"#5-Fb`\\l6&7$$\"#QF\\cx$Ff`\\lF\\cxQ\"qFh`\\lFi`\\l-F^a\\l6
$F][x\"#6-Fb`\\l6&7$F^cxFbhxQ&-8p/3Fh`\\lFi`\\l-F^a\\l6$F][xFaa\\l-Fb`
\\l6&7$FhcxFbhxQ&-5p/3Fh`\\lFi`\\lF`b\\l-Fb`\\l6&7$FbdxFbhxQ&-2p/3Fh`
\\lFi`\\lF`b\\l-Fb`\\l6&7$FfexFbhxQ%4p/3Fh`\\lFi`\\lF`b\\l-Fb`\\l6&7$F
`fxFbhxQ%7p/3Fh`\\lFi`\\lF`b\\l-Fb`\\l6&7$FjfxFbhxQ&10p/3Fh`\\lFi`\\lF
`b\\l-Fb`\\l6&7$F\\exFbhxQ$p/3Fh`\\lFi`\\lF`b\\l-%*AXESTICKSG6$71/$!#N
F\\cx%&-7p/2G/!\"$%$-3pG/$!#DF\\cx%&-5p/2G/Ff`\\l%$-2pG/$FiyF\\cx%&-3p
/2G/F\\cx%#-pG/$FjbwF\\cx%%-p/2G/Fhhv%\"0G/$F]cwF\\cx%$p/2G/Fdgw%\"pG/
$\"#:F\\cx%%3p/2G/F\\iv%#2pG/$\"#DF\\cx%%5p/2G/Fccw%#3pG/$\"#NF\\cx%%7
p/2GF]cw-F^a\\l6$F][x\"\"*-%+AXESLABELSG6%%!GF`g\\l-F^a\\l6#%(DEFAULTG
-%%VIEWG6$;$!$b$Ff`\\lFea\\l;FibwF\\cw" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17
" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "C
urve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve
 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36
" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "C
urve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve
 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55
" "Curve 56" "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "C
urve 62" "Curve 63" "Curve 64" "Curve 65" "Curve 66" "Curve 67" "Curve
 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72" "Curve 73" "Curve 74
" "Curve 75" }}{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 12 "solution set" }{TEXT 
-1 17 " of the equation " }{XPPEDIT 18 0 "tan*theta = sqrt(3);" "6#/*&
%$tanG\"\"\"%&thetaGF&-%%sqrtG6#\"\"$" }{TEXT -1 5 " is: " }}{PARA 
256 "" 0 "" {TEXT -1 2 " \{" }{XPPEDIT 18 0 "Pi/3+k*Pi" "6#,&*&%#PiG\"
\"\"\"\"$!\"\"F&*&%\"kGF&F%F&F&" }{TEXT -1 3 " | " }{TEXT 287 1 "k" }
{TEXT -1 17 " is an integer\}. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 154 "_EnvAllSolutions := true:\ntan(theta)=sqrt(3);\nsolve(%,theta
):\nsubs(op(indets(%))=k,%);\nsol := %:\n_EnvAllSolutions := '_EnvAllS
olutions':\nseq(sol,k=-4..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$t
anG6#%&thetaG*$\"\"$#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&
*&#\"\"\"\"\"$F&%#PiGF&F&*&F(F&%\"kGF&F&" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6+,$*(\"#6\"\"\"\"\"$!\"\"%#PiGF&F(,$*(\"\")F&F'F(F)F&F(,$*(\"\"&F
&F'F(F)F&F(,$*(\"\"#F&F'F(F)F&F(,$*&F'F(F)F&F&,$*(\"\"%F&F'F(F)F&F&,$*
(\"\"(F&F'F(F)F&F&,$*(\"#5F&F'F(F)F&F&,$*(\"#8F&F'F(F)F&F&" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "
" {TEXT 278 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 
42 "Find the general solution of the equation " }{XPPEDIT 18 0 "tan*th
eta=2" "6#/*&%$tanG\"\"\"%&thetaGF&\"\"#" }{TEXT -1 81 ", giving the s
olution in terms of an appropriate inverse trigonometric function. " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 8 "Solution
" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 29 "One solution of the \+
equation " }{XPPEDIT 18 0 "tan*theta = 2;" "6#/*&%$tanG\"\"\"%&thetaGF
&\"\"#" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "theta = arctan*2;" "6#/%&th
etaG*&%'arctanG\"\"\"\"\"#F'" }{TEXT -1 1 " " }{TEXT 294 1 "~" }{TEXT 
-1 59 " 1.107148718, and this is the only solution in the interval" }
{XPPEDIT 18 0 "``(-Pi/2,Pi/2)" "6#-%!G6$,$*&%#PiG\"\"\"\"\"#!\"\"F+*&F
(F)F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 51 "Since the tangent function is periodic with period
 " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 20 ", the real numbers: " }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*2" "6#*&%'arc
tanG\"\"\"\"\"#F%" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "arctan*2+Pi" "6
#,&*&%'arctanG\"\"\"\"\"#F&F&%#PiGF&" }{TEXT -1 5 ",    " }{XPPEDIT 
18 0 "arctan*2+2*Pi" "6#,&*&%'arctanG\"\"\"\"\"#F&F&*&F'F&%#PiGF&F&" }
{TEXT -1 5 ",    " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT 
-1 8 ", etc., " }}{PARA 0 "" 0 "" {TEXT -1 40 "are also solutions, as \+
are the numbers: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
arctan*2-Pi" "6#,&*&%'arctanG\"\"\"\"\"#F&F&%#PiG!\"\"" }{TEXT -1 5 ",
    " }{XPPEDIT 18 0 "arctan*2-2*Pi" "6#,&*&%'arctanG\"\"\"\"\"#F&F&*&
F'F&%#PiGF&!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "arctan*2-3*Pi" "
6#,&*&%'arctanG\"\"\"\"\"#F&F&*&\"\"$F&%#PiGF&!\"\"" }{TEXT -1 5 ",   \+
 " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT -1 7 ", etc. " }}
{PARA 0 "" 0 "" {TEXT -1 38 "In general, any number of the form:   " }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta = arctan*2+k*P
i;" "6#/%&thetaG,&*&%'arctanG\"\"\"\"\"#F(F(*&%\"kGF(%#PiGF(F(" }
{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 281 9 "______
___" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 6 "where " }{TEXT 280 1 "k" }{TEXT -1 25 " is an integer, t
hat is, " }{XPPEDIT 18 0 "k=` . . . `,-4,-3,-2,-1,0,1,2,3,4,` . . . `
" "6-/%\"kG%(~.~.~.~G,$\"\"%!\"\",$\"\"$F(,$\"\"#F(,$\"\"\"F(\"\"!F.F,
F*F'F%" }{TEXT -1 32 ", is a solution of the equation " }{XPPEDIT 18 
0 "tan*theta = 2;" "6#/*&%$tanG\"\"\"%&thetaGF&\"\"#" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 75 "The following picture shows the point
s of intersection P and Q of the line " }{XPPEDIT 18 0 "y = 2*x;" "6#/
%\"yG*&\"\"#\"\"\"%\"xGF'" }{TEXT -1 22 " with the unit circle " }
{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 127 "All the solutions are a
ssociated with either P or Q, more precisely, the \"wrapping function
\" W, which associates a real number " }{XPPEDIT 18 0 "theta" "6#%&the
taG" }{TEXT -1 16 " with the point " }{XPPEDIT 18 0 "W(theta)=``(cos*t
heta,sin*theta)" "6#/-%\"WG6#%&thetaG-%!G6$*&%$cosG\"\"\"F'F-*&%$sinGF
-F'F-" }{TEXT -1 40 ", maps each solution to either P or Q.  " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 282 234 234 {PLOTDATA 2 "6
0-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4hRPij!**!#=$\"3Ikwb#=y_O\"
F/7$$\"3E8J#))4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\"3-N5')yke[#*F/$\"3gLj&[K5J
!QF/7$$\"3?goz=42`')F/$\"3j9NXR4U7]F/7$$\"3Sb](G._U!zF/$\"3t_H-qceDhF/
7$$\"3_\\$R'oTd#3(F/$\"3!GO#*3qU&fqF/7$$\"3?H$>jubk6'F/$\"3!\\@@&\\!>8
\"zF/7$$\"3VGuIc:x5]F/$\"3-$>p(Qh-a')F/7$$\"37C3nW'R,#QF/$\"3aK+16bcT#
*F/7$$\"3$oY@iF'QDDF/$\"3s&Q\"[M\"oen*F/7$$\"3OB^hAo8X8F/$\"33)fVPO<\"
4**F/7$$!3+qB/u(p5(f!#@$\"2%HhJ<#)******!#<7$$!33)\\T#fB[i8F/$\"3ap%>w
GZn!**F/7$$!3)e:d.:#=YEF/$\"3PA>=\\D`V'*F/7$$!3%*yV1J*3Jx$F/$\"3IzF#e`
m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'yI2&oN')F/7$$!3[igNw%*[RgF/$\"3!G;)RQ+B
qzF/7$$!3/XUwYjJ*3(F/$\"3AfvOg?x_qF/7$$!3&=e_b?/U!zF/$\"3u3<J%QZc7'F/7
$$!3w!G5)RnIf')F/$\"3D%H$\\4/k,]F/7$$!3/`k8ti$4B*F/$\"3In'[*>HvXQF/7$$
!3!p!RS4Nij'*F/$\"3o$p9m#Q%=d#F/7$$!3#p(pT>+.2**F/$\"31V?00]Ug8F/7$$!3
/gKG4>&*****F/$\"3vCA\\O%485$!#?7$$!3__Is=!Q%3**F/$!3q#4*>c=8]8F/7$$!3
)>b`sYlSn*F/$!3UNbFGIGKDF/7$$!37_J.8AEj#*F/$!3q4&=j`Bsw$F/7$$!3%\\F)4K
h=u')F/$!3kENX')3zv\\F/7$$!3EB*z7xng%zF/$!37W(H!pUCrgF/7$$!3XD84Pfc5rF
/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[gF/$!3)4j2yeGL'zF/7$$!3g@UD=%)o!*\\F/$!
3I_Mh6Mil')F/7$$!3o+1s\"[\"ysPF/$!3#\\IN,%***4E*F/7$$!30yCH\"el'3EF/$!
3=V>(fp[Pl*F/7$$!3g67Cvnc\"H\"F/$!3;$RoI*>C;**F/7$$!3EzyOHMQdIFas$!3W1
O#>E`*****F/7$$\"3GIAj`Ks(G\"F/$!3lYZ3X=u;**F/7$$\"3EKRqX8/bDF/$!3U$[o
%H'z!o'*F/7$$\"3@%)yb&Q)zNQF/$!36.2<#>x]B*F/7$$\"3O7w4w0I.]F/$!3wHgb5w
Me')F/7$$\"3@\\#)[*R]$4hF/$!3/a*[=H2o\"zF/7$$\"33/wY'Q+$*4(F/$!3)4d)eg
?sUqF/7$$\"3[f=s$Ry,!zF/$!3D1$H<bQ38'F/7$$\"3%*R,@;vLu')F/$!3e>)zD(p_v
\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2ndo*fQF/7$$\"31mGAp))oa'*F/$!3%)f]nRQ=0E
F/7$$\"3@zb'f`bp!**F/$!3/up91t'4O\"F/7$F($\"36YKhSr8/#)!#F-%'COLOURG6&
%$RGBG$\"*++++\"!\")F+F+-%*THICKNESSG6#\"\"#-F$6%7$7$$!3)******R&f8sWF
/$!3$******z!>FW*)F/7$$\"3)******R&f8sWF/$\"3$******z!>FW*)F/-Fjz6&F\\
[l$\")#)eqkF_[l$\"))eqk\"F_[lFe\\lF`[l-F$6&7$F\\\\lFg[l-Fjz6&F\\[lF+F]
[lF][l-%'SYMBOLG6$%&CROSSG\"#5-%&STYLEG6#%&POINTG-F$6&Fi\\lFj\\l-F]]l6
$%(DIAMONDGF`]lFa]l-F$6&Fi\\lFj\\l-F]]l6$%'CIRCLEGF`]lFa]l-F$6&Fi\\l-F
jz6&F\\[lF*F*F*-F]]l6$F^^l\"#7Fa]l-%%TEXTG6%7$$\"#b!\"#F(Q\"P6\"-%&COL
ORG6&F\\[l$F)F\\_lFb_lFb_l-Fg^l6%7$$!#bF\\_l$!\"\"F*Q\"QF^_lF__l-Fg^l6
%7$$\"$D\"F\\_l$!\"'F\\_lQ\"xF^_lF__l-Fg^l6%7$F``lF^`lQ\"yF^_lF__l-%(S
CALINGG6#%,CONSTRAINEDG-%*AXESTICKSG6$F*F*-%+AXESLABELSG6%Q!F^_lFaal-%
%FONTG6#%(DEFAULTG-%%VIEWG6$;$!#6Fi_l$\"#8Fi_l;FjalF^`l" 1 2 0 1 10 0 
2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+
3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+
10" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 
12 "solution set" }{TEXT -1 17 " of the equation " }{XPPEDIT 18 0 "tan
*theta = 2;" "6#/*&%$tanG\"\"\"%&thetaGF&\"\"#" }{TEXT -1 5 " is: " }}
{PARA 256 "" 0 "" {TEXT -1 2 " \{" }{XPPEDIT 18 0 "arctan*2+k*Pi;" "6#
,&*&%'arctanG\"\"\"\"\"#F&F&*&%\"kGF&%#PiGF&F&" }{TEXT -1 3 " | " }
{TEXT 282 1 "k" }{TEXT -1 17 " is an integer\}. " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{TEXT 299 16 "________________" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 5 ": If " }{XPPEDIT 18 0 
"theta" "6#%&thetaG" }{TEXT -1 46 " is an angle in degrees, then the s
olution of " }{XPPEDIT 18 0 "tan*theta=2" "6#/*&%$tanG\"\"\"%&thetaGF&
\"\"#" }{TEXT -1 9 " between " }{XPPEDIT 18 0 "-90^o" "6#,$)\"#!*%\"oG
!\"\"" }{TEXT -1 7 " and + " }{XPPEDIT 18 0 "90^o" "6#)\"#!*%\"oG" }
{TEXT -1 4 " is " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 1 " " 
}{TEXT 295 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1.107148718*`.`*``(18
0^o/Pi);" "6#*(-%&FloatG6$\"+=([r5\"!\"*\"\"\"%\".GF)-%!G6#*&)\"$!=%\"
oGF)%#PiG!\"\"F)" }{TEXT -1 1 " " }{TEXT 296 1 "~" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "63.43494882^o" "6#)-%&FloatG6$\"+#)[\\Vj!\")%\"oG" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 59 "The general solution ca
n be given in the approximate form: " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 1 " " }{TEXT 297 1 "~
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "63.43494882^o+k*180^o;" "6#,&)-%&Flo
atG6$\"+#)[\\Vj!\")%\"oG\"\"\"*&%\"kGF+)\"$!=F*F+F+" }{TEXT -1 2 ", " 
}}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 298 1 "k" }{TEXT -1 16 " \+
is an integer. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "_EnvAllSolu
tions := true:\ntan(theta)=2;\nsolve(%,theta):\nsubs(op(indets(%))=k,%
);\nsol := %:\n_EnvAllSolutions := '_EnvAllSolutions':\nseq(sol,k=-2..
3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$tanG6#%&thetaG\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'arctanG6#\"\"#\"\"\"*&%#PiGF(%\"k
GF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(,&-%'arctanG6#\"\"#\"\"\"*&F'
F(%#PiGF(!\"\",&F$F(F*F+F$,&F$F(F*F(,&F$F(*&F'F(F*F(F(,&F$F(*&\"\"$F(F
*F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 72 "The previous two examples illustrate the \+
fact that, for any real number " }{TEXT 288 1 "a" }{TEXT -1 15 ", the \+
equation " }{XPPEDIT 18 0 "tan*theta=a" "6#/*&%$tanG\"\"\"%&thetaGF&%
\"aG" }{TEXT -1 9 " has the " }{TEXT 259 16 "general solution" }{TEXT 
-1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta=arc
tan*a+k*Pi" "6#/%&thetaG,&*&%'arctanG\"\"\"%\"aGF(F(*&%\"kGF(%#PiGF(F(
" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 289 11 "_
__________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{TEXT 290 1 "k" }{TEXT -1 50 " is an integer, that is, it has the solu
tion set: " }}{PARA 256 "" 0 "" {TEXT -1 2 " \{" }{XPPEDIT 18 0 "arcta
n*a+k*Pi" "6#,&*&%'arctanG\"\"\"%\"aGF&F&*&%\"kGF&%#PiGF&F&" }{TEXT 
-1 3 " | " }{TEXT 380 1 "k" }{TEXT -1 17 " is an integer\}. " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{TEXT 383 16 "________________" }{TEXT -1 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "N
ote" }{TEXT -1 56 ": The previous result is true for a general real nu
mber " }{TEXT 378 1 "a" }{TEXT -1 20 " (no matter whether " }{TEXT 
377 1 "a" }{TEXT -1 79 " is positive or negative). However, it may be \+
useful to note that the equation " }{XPPEDIT 18 0 "tan*theta=-a" "6#/*
&%$tanG\"\"\"%&thetaGF&,$%\"aG!\"\"" }{TEXT -1 27 " has the general so
lution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta=-ar
ctan*a+k*Pi" "6#/%&thetaG,&*&%'arctanG\"\"\"%\"aGF(!\"\"*&%\"kGF(%#PiG
F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 
376 1 "k" }{TEXT -1 16 " is an integer. " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "The genera
l solution of the equation " }{XPPEDIT 18 0 "cos*theta = a;" "6#/*&%$c
osG\"\"\"%&thetaGF&%\"aG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 \+
" }}{PARA 0 "" 0 "" {TEXT 310 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 
"" 0 "" {TEXT -1 42 "Find the general solution of the equation " }
{XPPEDIT 18 0 "cos*theta = sqrt(3)/2;" "6#/*&%$cosG\"\"\"%&thetaGF&*&-
%%sqrtG6#\"\"$F&\"\"#!\"\"" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 311 8 "Solution" }{TEXT -1 2 ": " }}
{PARA 0 "" 0 "" {TEXT -1 29 "One solution of the equation " }{XPPEDIT 
18 0 "cos*theta = sqrt(3)/2;" "6#/*&%$cosG\"\"\"%&thetaGF&*&-%%sqrtG6#
\"\"$F&\"\"#!\"\"" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "theta = arccos(s
qrt(3)/2);" "6#/%&thetaG-%'arccosG6#*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"
" }{XPPEDIT 18 0 "`` = Pi/6;" "6#/%!G*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "Another solution is given by \+
" }{XPPEDIT 18 0 "theta=-arccos(sqrt(3)/2)" "6#/%&thetaG,$-%'arccosG6#
*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F0" }{XPPEDIT 18 0 "``=-Pi/6" "6#/%!G
,$*&%#PiG\"\"\"\"\"'!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 4 "The " }{TEXT 259 13 "two solutions" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "theta=Pi/6" "6#/%&thetaG*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 2 ", \+
" }{XPPEDIT 18 0 "theta=-Pi/6" "6#/%&thetaG,$*&%#PiG\"\"\"\"\"'!\"\"F*
" }{TEXT -1 41 " are the only solutions in the interval [" }{XPPEDIT 
18 0 "-Pi,Pi;" "6$,$%#PiG!\"\"F$" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" 
{TEXT -1 14 "We may write: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "theta=``" "6#/%&thetaG%!G" }{TEXT 315 1 "+" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "Pi/6" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 2 ",
 " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "-Pi<=theta" "6#
1,$%#PiG!\"\"%&thetaG" }{XPPEDIT 18 0 "``<Pi" "6#2%!G%#PiG" }{TEXT -1 
2 ". " }}{PARA 0 "" 0 "" {TEXT -1 50 "Since the cosine function is per
iodic with period " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }
{TEXT -1 15 ", the numbers: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 1 " " }{TEXT 
316 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi/6" "6#*&%#PiG\"\"\"\"\"'!
\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "4*Pi" "6#*&\"\"%\"\"\"%#PiGF
%" }{TEXT -1 1 " " }{TEXT 317 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi
/6" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "6*
Pi" "6#*&\"\"'\"\"\"%#PiGF%" }{TEXT -1 1 " " }{TEXT 318 1 "+" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "Pi/6" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 
5 ",    " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT -1 8 ", etc
., " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "11*Pi/6" "6#*(\"#6\"\"\"%#PiGF%\"\"'!\"
\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "13*Pi/6" "6#*(\"#8\"\"\"%#PiGF
%\"\"'!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "23*Pi/6" "6#*(\"#B\"
\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "25*Pi/6" "6
#*(\"#D\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "35
*Pi/6" "6#*(\"#N\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 5 ",    " }
{XPPEDIT 18 0 "37*Pi/6" "6#*(\"#P\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 5 
",    " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT -1 8 ", etc.,
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "are \+
also solutions, as are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "-2*Pi;" "6#,$*&\"\"#\"\"\"%#PiGF&!\"\"" }{TEXT -1 1 " \+
" }{TEXT 319 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi/6" "6#*&%#PiG\"
\"\"\"\"'!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "-4*Pi;" "6#,$*&\"
\"%\"\"\"%#PiGF&!\"\"" }{TEXT -1 1 " " }{TEXT 320 1 "+" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Pi/6" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 5 ",    \+
" }{XPPEDIT 18 0 "-6*Pi;" "6#,$*&\"\"'\"\"\"%#PiGF&!\"\"" }{TEXT -1 1 
" " }{TEXT 321 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi/6" "6#*&%#PiG
\"\"\"\"\"'!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "` . . . `" "6#%(
~.~.~.~G" }{TEXT -1 7 ", etc.," }}{PARA 0 "" 0 "" {TEXT -1 8 "that is,
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-11*Pi/6;" "6#,$*
(\"#6\"\"\"%#PiGF&\"\"'!\"\"F)" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "-1
3*Pi/6;" "6#,$*(\"#8\"\"\"%#PiGF&\"\"'!\"\"F)" }{TEXT -1 5 ",    " }
{XPPEDIT 18 0 "-23*Pi/6;" "6#,$*(\"#B\"\"\"%#PiGF&\"\"'!\"\"F)" }
{TEXT -1 5 ",    " }{XPPEDIT 18 0 "-25*Pi/6;" "6#,$*(\"#D\"\"\"%#PiGF&
\"\"'!\"\"F)" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "-35*Pi/6;" "6#,$*(\"
#N\"\"\"%#PiGF&\"\"'!\"\"F)" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "-37*P
i/6;" "6#,$*(\"#P\"\"\"%#PiGF&\"\"'!\"\"F)" }{TEXT -1 5 ",    " }
{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT -1 6 ", etc." }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "In general, \+
any number of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "theta=2*k*Pi" "6#/%&thetaG*(\"\"#\"\"\"%\"kGF'%#PiGF'" 
}{TEXT -1 1 " " }{TEXT 322 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi/6
" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 3 ",  " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{TEXT 313 7 "_______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 312 1 "k
" }{TEXT -1 25 " is an integer, that is, " }{XPPEDIT 18 0 "k=` . . . `
,-4,-3,-2,-1,0,1,2,3,4,` . . . `" "6-/%\"kG%(~.~.~.~G,$\"\"%!\"\",$\"
\"$F(,$\"\"#F(,$\"\"\"F(\"\"!F.F,F*F'F%" }{TEXT -1 32 ", is a solution
 of the equation " }{XPPEDIT 18 0 "cos*theta = sqrt(3)/2;" "6#/*&%$cos
G\"\"\"%&thetaGF&*&-%%sqrtG6#\"\"$F&\"\"#!\"\"" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 75 "The following picture shows the points of
 intersection P and Q of the line " }{XPPEDIT 18 0 "x=sqrt(3)/2" "6#/%
\"xG*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 22 " with the unit ci
rcle " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F
(" }{TEXT -1 129 ". All the solutions are associated with either P or \+
Q, more precisely, the \"wrapping function\" W, which associates a rea
l number " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 16 " with the
 point " }{XPPEDIT 18 0 "W(theta)=``(cos*theta,sin*theta)" "6#/-%\"WG6
#%&thetaG-%!G6$*&%$cosG\"\"\"F'F-*&%$sinGF-F'F-" }{TEXT -1 39 ", maps \+
each solution to either P or Q. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{GLPLOT2D 297 262 262 {PLOTDATA 2 "62-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*
7$$\"3w\"4hRPij!**!#=$\"3Ikwb#=y_O\"F/7$$\"3E8J#))4-Qn*F/$\"3%[#\\ff*)
GLDF/7$$\"3-N5')yke[#*F/$\"3gLj&[K5J!QF/7$$\"3?goz=42`')F/$\"3j9NXR4U7
]F/7$$\"3Sb](G._U!zF/$\"3t_H-qceDhF/7$$\"3_\\$R'oTd#3(F/$\"3!GO#*3qU&f
qF/7$$\"3?H$>jubk6'F/$\"3!\\@@&\\!>8\"zF/7$$\"3VGuIc:x5]F/$\"3-$>p(Qh-
a')F/7$$\"37C3nW'R,#QF/$\"3aK+16bcT#*F/7$$\"3$oY@iF'QDDF/$\"3s&Q\"[M\"
oen*F/7$$\"3OB^hAo8X8F/$\"33)fVPO<\"4**F/7$$!3+qB/u(p5(f!#@$\"2%HhJ<#)
******!#<7$$!33)\\T#fB[i8F/$\"3ap%>wGZn!**F/7$$!3)e:d.:#=YEF/$\"3PA>=
\\D`V'*F/7$$!3%*yV1J*3Jx$F/$\"3IzF#e`m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'yI
2&oN')F/7$$!3[igNw%*[RgF/$\"3!G;)RQ+BqzF/7$$!3/XUwYjJ*3(F/$\"3AfvOg?x_
qF/7$$!3&=e_b?/U!zF/$\"3u3<J%QZc7'F/7$$!3w!G5)RnIf')F/$\"3D%H$\\4/k,]F
/7$$!3/`k8ti$4B*F/$\"3In'[*>HvXQF/7$$!3!p!RS4Nij'*F/$\"3o$p9m#Q%=d#F/7
$$!3#p(pT>+.2**F/$\"31V?00]Ug8F/7$$!3/gKG4>&*****F/$\"3vCA\\O%485$!#?7
$$!3__Is=!Q%3**F/$!3q#4*>c=8]8F/7$$!3)>b`sYlSn*F/$!3UNbFGIGKDF/7$$!37_
J.8AEj#*F/$!3q4&=j`Bsw$F/7$$!3%\\F)4Kh=u')F/$!3kENX')3zv\\F/7$$!3EB*z7
xng%zF/$!37W(H!pUCrgF/7$$!3XD84Pfc5rF/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[gF
/$!3)4j2yeGL'zF/7$$!3g@UD=%)o!*\\F/$!3I_Mh6Mil')F/7$$!3o+1s\"[\"ysPF/$
!3#\\IN,%***4E*F/7$$!30yCH\"el'3EF/$!3=V>(fp[Pl*F/7$$!3g67Cvnc\"H\"F/$
!3;$RoI*>C;**F/7$$!3EzyOHMQdIFas$!3W1O#>E`*****F/7$$\"3GIAj`Ks(G\"F/$!
3lYZ3X=u;**F/7$$\"3EKRqX8/bDF/$!3U$[o%H'z!o'*F/7$$\"3@%)yb&Q)zNQF/$!36
.2<#>x]B*F/7$$\"3O7w4w0I.]F/$!3wHgb5wMe')F/7$$\"3@\\#)[*R]$4hF/$!3/a*[
=H2o\"zF/7$$\"33/wY'Q+$*4(F/$!3)4d)eg?sUqF/7$$\"3[f=s$Ry,!zF/$!3D1$H<b
Q38'F/7$$\"3%*R,@;vLu')F/$!3e>)zD(p_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2ndo
*fQF/7$$\"31mGAp))oa'*F/$!3%)f]nRQ=0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4O
\"F/7$F($\"36YKhSr8/#)!#F-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%*THICKN
ESSG6#\"\"#-F$6&7$7$F+F+7$$\"3a+++SSDg')F/$\"3++++++++]F/-Fjz6&F\\[l$
\")#)eqkF_[l$\"))eqk\"F_[lFa\\lF`[l-%*LINESTYLEG6#F)-F$6&7$Fg[l7$Fi[l$
!3++++++++]F/F]\\lF`[lFc\\l-F$6&7$Fi\\lFh[l-Fjz6&F\\[lF*F*F*-Fa[lFe\\l
-Fd\\lFb[l-F$6&7$Fh[lFi\\l-Fjz6&F\\[lF+F][lF][l-%'SYMBOLG6$%&CROSSG\"#
5-%&STYLEG6#%&POINTG-F$6&Fe]lFf]l-Fi]l6$%(DIAMONDGF\\^lF]^l-F$6&Fe]lFf
]l-Fi]l6$%'CIRCLEGF\\^lF]^l-F$6&Fe]lF_]l-Fi]l6$Fj^l\"#7F]^l-%%TEXTG6%7
$F($\"#`!\"#Q\"P6\"-%&COLORG6&F\\[l$F)Ff_lF\\`lF\\`l-Fa_l6%7$F($!#`Ff_
lQ\"QFh_lFi_l-Fa_l6%7$$\"$D\"Ff_l$!\"'Ff_lQ\"xFh_lFi_l-Fa_l6%7$Fh`lFf`
lQ\"yFh_lFi_l-%(SCALINGG6#%,CONSTRAINEDG-%*AXESTICKSG6$F*F*-%+AXESLABE
LSG6%Q!Fh_lFial-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!#6!\"\"$\"#8Fdbl;FbblF
f`l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1
" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8
" "Curve 9" "Curve 10" "Curve 11" "Curve 12" }}{TEXT -1 3 "   " }}
{PARA 0 "" 0 "" {TEXT -1 42 "The following picture shows the graphs of
 " }{XPPEDIT 18 0 "y=cos*theta" "6#/%\"yG*&%$cosG\"\"\"%&thetaGF'" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=sqrt(3)/2" "6#/%\"yG*&-%%sqrtG6#
\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "
The solutions of the equation " }{XPPEDIT 18 0 "cos*theta=sqrt(3)/2" "
6#/*&%$cosG\"\"\"%&thetaGF&*&-%%sqrtG6#\"\"$F&\"\"#!\"\"" }{TEXT -1 
18 " are given by the " }{TEXT 389 1 "x" }{TEXT -1 62 " coordinates of
 the points of intersection of the two graphs. " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{GLPLOT2D 889 218 218 {PLOTDATA 2 "6eo-%'CURVESG6%7[v7
$$!3#)************\\D!#<$!3w'H-/lWVc\"!#=7$$!3jmTN^+J6DF*$!3W19`K)4Cb$
!#>7$$!3WL$3F5?EZ#F*$\"3k\\Bp6`/\"f)F37$$!3z*\\iS:IRV#F*$\"3ae.e8Lxg?F
-7$$!3gmmT0-C&R#F*$\"3*>*))QRQ.KKF-7$$!3n\\7`qB\"zK#F*$\"3=PSC!)evY^F-
7$$!3=LekNXegAF*$\"3l\\c=UH2KoF-7$$!3-](=PMn[=#F*$\"3s#)*e8&4Og$)F-7$$
!3Vm;z^,:4@F*$\"3$p?H,.=yT*F-7$$!3Em\"zWgS52#F*$\"3kevlSS)>v*F-7$$!3cm
m;d5$H.#F*$\"3q;!*HH?`Y**F-7$$!3q;/^$GwQ,#F*$\"3Y=Ho\"[*\\!***F-7$$!3S
mT&)4:#[*>F*$\"3;8b6vmn)***F-7$$!3a;z>Onwv>F*$\"3wJ(y`JM5(**F-7$$!3Ym;
ai>rc>F*$\"3JoWFH9n2**F-7$$!3))\\(oR7b3)=F*$\"3j'p7By?wI*F-7$$!32LeR&G
)*\\!=F*$\"3)oYo4?m9=)F-7$$!3o*\\(o&GpYt\"F*$\"35TVShpVAnF-7$$!3]m\"zf
GSVm\"F*$\"34CX(*4KdO\\F-7$$!3v*\\(=$[Hzi\"F*$\"3uRc!e$zp6RF-7$$!3ALeR
!o=:f\"F*$\"3**R*zmY&pNGF-7$$!3omTgxy5b:F*$\"3\\c8F8&HEs\"F-7$$!3%**\\
7[2(p=:F*$\"3U.xQ$=#[qeF37$$!3]m\"z/(4/\"[\"F*$!3UZu'ovLE&fF37$$!3GLe9
m[QV9F*$!3_K7[:SDp<F-7$$!3%)*\\7=wGdS\"F*$!3mfxUWH^=HF-7$$!3Sm\"zuls!o
8F*$!3OOc$37vp-%F-7$$!3k***\\nJP0L\"F*$!3/s\\GDt(e2&F-7$$!35L3-w>+$H\"
F*$!31S=il'zU0'F-7$$!3em;HNmYb7F*$!3;NiA=Hf[pF-7$$!3#)*\\iXHJz@\"F*$!3
!=P\\%*y&RYxF-7$$!3_;HKr=rS6F*$!3%4ngbtD(Q!*F-7$$!3)HL$3[C\\j5F*$!3q2*
=W[A<!)*F-7$$!3#[7y0s)[Y5F*$!3+'RMxVQN*)*F-7$$!3T;H2$*\\[H5F*$!3)4u#oK
#Hr&**F-7$$!3,3xcl7[75F*$!3g#*)R=X8B***F-7$$!3C(*\\i!QvZ&**F-$!3-sYc=2
**)***F-7$$!3O9Hd0\"QZy*F-$!3Y/@O$)>9x**F-7$$!3NI3_I3q9'*F-$!3I\"\\H#y
&Ho#**F-7$$!3[Z(oabjYW*F-$!3/--$)Qq>[)*F-7$$!3gkmT!GEYF*F-$!39tC+&po9u
*F-7$$!3?J$ek[R*3&)F-$!3UuSee+yA*)F-7$$!3\"y***\\#p_Ku(F-$!3w@<8VgF!f(
F-7$$!3y)**\\PIP)etF-$!3o)z1LqL1v'F-7$$!3k)****\\\">UupF-$!3X\"[Wo9aE
\"eF-7$$!3])**\\i_1+f'F-$!3c-m8C5+!z%F-7$$!3Z****\\P6f0iF-$!3=t'RYVrvp
$F-7$$!3s)*****zc8NeF-$!3_gSwQ$fOf#F-7$$!33****\\A-okaF-$!3Qc^A]jla9F-
7$$!3U*****\\wCU4&F-$!3ce%fd+E(fHF37$$!3o)***\\2$pPs%F-$\"3e&H_$z]:n')
F37$$!3s:H#=EgtQ%F-$\"3$*o,\"\\#Q!G\">F-7$$!3wKe9;7&40%F-$\"3j*[SE*oaP
HF-7$$!3z\\(o/<UXr$F-$\"3!z&o&yX4&HRF-7$$!3%om\"zCJ8yLF-$\"3'G%zcp?ix[
F-7$$!3U#3-8B5\"yHF-$\"3f7kev\"[L$fF-7$$!3+)\\7yL(3yDF-$\"3o=3ER6\\&*o
F-7$$!3e8HKWW1y@F-$\"3S$yamlv)[xF-7$$!3;HL$3bT!y<F-$\"3+&**yb!=/![)F-7
$$!3oHL3FyH+6F-$\"3J!zcxq!\\3%*F-7$$!36-LLL5aDUF3$\"3#e\"GY`u,7**F-7$$
!3c%)G2e$)faAF3$\"3Kdx9Je#\\(**F-7$$!3.qY7GobOG!#?$\"3-rhuXHg****F-7$$
\"3c]zW#*pG(o\"F3$\"39A$yj@af)**F-7$$\"37o$3xmH#eOF3$\"38'[G]&>.M**F-7
$$\"3@.#H#=]6+wF3$\"3mC&=My3jr*F-7$$\"3%Q+vo.+U:\"F-$\"3Y&GX[8o(\\$*F-
7$$\"39O3FHt/_=F-$\"3)\\$R=R1aa$)F-7$$\"3[omm@Y*)\\DF-$\"3;FS)>ql$fpF-
7$$\"38_(oH;>F$HF-$\"3C;m7>z_ZgF-7$$\"3yN3F/Pa:LF-$\"3;8[+*)=K[]F-7$$
\"3U>HdX#o$)p$F-$\"3I<>$p2$=wRF-7$$\"33.](oy#>\"3%F-$\"3-'yNMg+m%GF-7$
$\"3C.DJ!GKdW%F-$\"3j$3$fWs\\K<F-7$$\"3P.+vt<F5[F-$\"37*3$y)Hfp&fF37$$
\"3].v=n7\"[<&F-$!3#4a%)Qs(4*[&F37$$\"3k.]ig2NRbF-$!3Ez2JYSK'o\"F-7$$
\"3-@HKz%4(>fF-$!3A,&pAx<$\\GF-7$$\"3RQ3-)>o+I'F-$!3WyItMbnrRF-7$$\"3y
b(=n\"pU!o'F-$!3V2kyj1RP]F-7$$\"39tmTNcygqF-$!3\\`?u1VEJgF-7$$\"3Wr\"H
Krh$fxF-$!3FT.Ecu7BwF-7$$\"3wp;/\"zPzX)F-$!3[J%\\%=xH\\))F-7$$\"3F'3x
\"G@pM))F-$!3!*pt@3KLP$*F-7$$\"3x-DJlkW6#*F-$!3Czi$y)=r%p*F-7$$\"3-6-)
QjB)*R*F-$!34\"G\\(p'oF#)*F-7$$\"3G>zW-3?)e*F-$!3d9!Gj(HV;**F-7$$\"3_F
c,rzdw(*F-$!3+)4*egoPv**F-7$$\"3yNLeR^&\\'**F-$!337#=A%RR****F-7$$\"33
Y'Gvgig,\"F*$!3%\\cc'[0F()**F-7$$\"3ee*)4,(Hc.\"F*$!3)ysU<F>u$**F-7$$
\"33r#pYz'>b5F*$!3oh7^H%G+&)*F-7$$\"3e$eR#))Qwu5F*$!3L#RY$\\\"Gas*F-7$
$\"3d3-Qv!)*Q6\"F*$!3+TZ)z\"3im$*F-7$$\"3eL3_iA.`6F*$!314$)RvQTm))F-7$
$\"3#pm;C6l6A\"F*$!3=nQgy\"\\<o(F-7$$\"3E+DJizH*G\"F*$!3Y^?H9E[YhF-7$$
\"3qek`M14E8F*$!3'e')[LxSd>&F-7$$\"3#pTgnI$)GO\"F*$!33:@lR&ec<%F-7$$\"
37vV)*yfn*R\"F*$!3k,n<X&\\)*4$F-7$$\"3cL$37lokV\"F*$!3w'H'=I6n#)>F-7$$
\"3K]7y\"3zWZ\"F*$!30flA'zX!4!)F37$$\"3%o;aB^*[7:F*$\"3E02$GK*oARF37$$
\"3Q$3FH%**\\]:F*$\"3OhHy:b&)z:F-7$$\"37++]t.^)e\"F*$\"3'o4+N&3%\\u#F-
7$$\"3>](=U;'pD;F*$\"3ARN]&*4.ZQF-7$$\"3C+v$\\&>)Gm\"F*$\"3Qxr$GQym*[F
-7$$\"3I]ilXx1+<F*$\"3E=#RY@u&zeF-7$$\"3N+]PONDP<F*$\"3#H:Txk>By'F-7$$
\"38]7`l=@4=F*$\"3cdo[MK'oD)F-7$$\"3!**\\(o%>q6)=F*$\"381\\&)p_B6$*F-7
$$\"3,v=nA*=6#>F*$\"3C,YG'p8Xp*F-7$$\"35]il]w1h>F*$\"3A\"oN&H]HD**F-7$
$\"3yP%[Y,U5)>F*$\"3A:;'fIpA)**F-7$$\"3VD1kyj,,?F*$\"3mwhFA!\\*****F-7
$$\"3'G\"GjU2*4-#F*$\"3sR\\T$ek#y**F-7$$\"3I+]i1^'4/#F*$\"3&>nu([8I<**
F-7$$\"3]LeRClv7@F*$\"3spLIs47z$*F-7$$\"3qmm;Uza%=#F*$\"3W=Y;.B'eO)F-7
$$\"3oL$3Ug+7E#F*$\"3nJ$3R7Fz\"oF-7$$\"3n++DmK&yL#F*$\"3KxEL%GEl([F-7$
$\"3e3xc=EesBF*$\"3MbJMcm)o*QF-7$$\"3$pT&)3(>J2CF*$\"3g/T&zY/4(GF-7$$
\"3GDJ?B8/UCF*$\"3p'pPe!4y5=F-7$$\"3?L3_v1xwCF*$\"3ATp`F<B\"H(F37$$\"3
+D\"G3.RZ^#F*$!3$RUH#G'[(GYF37$$\"3#oTNhQ2Fb#F*$!3!p\"*yy#\\H[;F-7$$\"
3i3FWTdn!f#F*$!3Mp2&[(3H5GF-7$$\"3W++v'4W'GEF*$!3PiJ9R%\\B$RF-7$$\"3H$
3_Ni%4+FF*$!3A1q7<sD!)eF-7$$\"3;mTN]^arFF*$!3d)[6F\"Q7LvF-7$$\"3?LeRQ:
BYGF*$!3My\")*oF%pb))F-7$$\"3D+vVEz\"4#HF*$!3a@>eCH'Hp*F-7$$\"3K'Rs`#G
<RHF*$!3>c'=&p*pz\")*F-7$$\"3%>H2VsFu&HF*$!3#R4<=&\\p5**F-7$$\"3e(=UKi
#ovHF*$!3)HV#G')H$3(**F-7$$\"3k$3x@_PR*HF*$!3)yt?bL'=)***F-7$$\"3rz>6@
C>7IF*$!3m166g]m#***F-7$$\"3Mvo/?tWIIF*$!3[%yJVJ(Ga**F-7$$\"3'4x\")*=A
q[IF*$!3S@yY'GzJ))*F-7$$\"3/nm\"z6dp1$F*$!3w!e'*Qyu&z(*F-7$$\"3S$3_v'=
SVJF*$!3Ar\\2C/D-!*F-7$$\"3w*\\(=<m%)>KF*$!3>VXfkI?3xF-7$$\"3AeR(pIfmD
$F*$!3>?i;;')f@pF-7$$\"3o;/w'*>Z$H$F*$!3&Q7W&*p?D/'F-7$$\"3:voa'o%GILF
*$!3?/bN6Rr#3&F-7$$\"3iLLLwt4nLF*$!3;@Ld'R,]0%F-7$$\"3b$e9'oOu/MF*$!3k
3P:#>*4[HF-7$$\"3\\Le*3'**QUMF*$!3nX?64\"3+!=F-7$$\"3W$3xJDO+[$F*$!3;(
ox[6&oniF37$$\"3PL$eaa#o<NF*$\"3k)y;QyyAb&F37$$\"3#=HKu5<]b$F*$\"3T8i@
V/#)><F-7$$\"3G]iSp;N#f$F*$\"3#)[%37l!ygGF-7$$\"3u3-QJioHOF*$\"3IfOPm.
ViRF-7$$\"3?nTN$z?qm$F*$\"3UX1;D;j4]F-7$$\"3(Q3FM'RjNPF*$\"3H8huvE%[u'
F-7$$\"3a++]LrC/QF*$\"3gI#[\"zU(y;)F-7$$\"3'R$eRpb)G)QF*$\"3s3(op*QvI$
*F-7$$\"3Qn;H0S_hRF*$\"3#>hR#))R.F**F-7$$\"3wv=#Rz2\"zRF*$\"3))p'*)e4o
%y**F-7$$\"3o$3_De\"p'*RF*$\"3+/%p)f)f%****F-7$$\"3h\"H#=r`F9SF*$\"3%f
TN%Q_%**)**F-7$$\"3)**\\7)f\"f=.%F*$\"3MQk#)eK&*\\**F-7$$\"3u;H2Pn-nSF
*$\"3#)3d%R'y6z(*F-7$$\"3[LLL9V>-TF*$\"33H$ef#o.*[*F-7$$\"3S</,fi=xTF*
$\"3eEn/hQJ!\\)F-7$$\"3W+vo.#y@D%F*$\"3\"4:i:6:D-(F-7$$\"3_]PfLm1)G%F*
$\"3)\\=jQ,]p<'F-7$$\"3f++]j]&RK%F*$\"3omLs9w%HD&F-7$$\"3m]iS$\\V)fVF*
$\"3/L)[TDTAE%F-7$$\"3s+DJB>t&R%F*$\"3]2!f[I6u@$F-7$$\"31vV[U*)HMWF*$
\"3EXDn%yM%\\?F-7$$\"3G]ilhf'GZ%F*$\"3*H[WHX*39&)F37$$\"3\\D\"G3)HV6XF
*$!37p]2CY+\"f$F37$$\"3#)************\\XF*$!3E%H-/lWVc\"F--%'COLOURG6&
%$RGBG$\"*++++\"!\")$\"\"!FhbnFgbn-%*THICKNESSG6#\"\"#-F$6%7S7$F($\"3'
fQWy.a-m)F-7$F?Facn7$FIFacn7$FSFacn7$F[pFacn7$FepFacn7$F_qFacn7$FcrFac
n7$FgsFacn7$F[uFacn7$FeuFacn7$F]xFacn7$FgxFacn7$F[zFacn7$F_[lFacn7$Fc
\\lFacn7$Fg]lFacn7$Fa^lFacn7$F``lFacn7$Fj`lFacn7$F^blFacn7$FbclFacn7$F
fdlFacn7$F`elFacn7$F^glFacn7$F\\ilFacn7$FfilFacn7$FjjlFacn7$F^\\mFacn7
$Fb]mFacn7$F\\^mFacn7$Fj_mFacn7$Fd`mFacn7$F^amFacn7$FbbmFacn7$FfcmFacn
7$F`dmFacn7$FjdmFacn7$FbgmFacn7$F\\hmFacn7$F`imFacn7$FdjmFacn7$Fh[nFac
n7$Fb\\nFacn7$F\\]nFacn7$Fj^nFacn7$Fd_nFacn7$Fh`nFacn7$F\\bnFacn-Fabn6
&FcbnFgbnFgbnFdbnFibn-F$6%7$7$$!3ELLLLLLL=F*Fgbn7$FifnFacn-Fabn6&Fcbn$
\")!\\DP\"FfbnF^gn$\")viobFfbn-%*LINESTYLEGF[cn-F$6%7$7$$\"3emmmmmmm;F
-Fgbn7$FhgnFacnF\\gnFbgn-F$6%7$7$$\"3_mmmmmmm@F*Fgbn7$F_hnFacnF\\gnFbg
n-F$6%7$7$$\"3'pmmmmmm;%F*Fgbn7$FfhnFacnF\\gnFbgn-F$6%7$7$$!3_mmmmmmm@
F*Fgbn7$F]inFacnF\\gnFbgn-F$6%7$7$$!3emmmmmmm;F-Fgbn7$FdinFacnF\\gnFbg
n-F$6%7$7$$\"3ELLLLLLL=F*Fgbn7$F[jnFacnF\\gnFbgn-F$6%7$7$$\"3[LLLLLLLQ
F*Fgbn7$FbjnFacnF\\gnFbgn-F$6&7#F[gn-Fabn6&FcbnFgbnFdbnFgbn-%'SYMBOLG6
#%'CIRCLEG-%&STYLEG6#%&POINTG-F$6&7#FjgnFhjnFjjnF^[o-F$6&7#FahnFhjnFjj
nF^[o-F$6&7#FhhnFhjnFjjnF^[o-F$6&7#F_inFhjnFjjnF^[o-F$6&7#FfinFhjnFjjn
F^[o-F$6&7#F]jnFhjnFjjnF^[o-F$6&7#FdjnFhjnFjjnF^[o-F$6&FgjnFhjn-F[[o6#
%(DIAMONDGF^[o-F$6&Fd[oFhjnFi\\oF^[o-F$6&Fg[oFhjnFi\\oF^[o-F$6&Fj[oFhj
nFi\\oF^[oFg\\oF\\]oF^]oF`]o-F$6&FgjnFhjn-F[[o6#%&CROSSGF^[o-F$6&Fd[oF
hjnFd]oF^[o-F$6&Fg[oFhjnFd]oF^[o-F$6&Fj[oFhjnFd]oF^[o-F$6&F]\\oFhjnFd]
oF^[o-F$6&F`\\oFhjnFd]oF^[o-F$6&Fc\\oFhjnFd]oF^[o-F$6&Ff\\oFhjnFd]oF^[
o-F$6&Fgjn-Fabn6&FcbnFhbnFhbnFhbn-F[[o6$F][o\"#7F^[o-F$6&Fd[oFg^oFi^oF
^[o-F$6&Fg[oFg^oFi^oF^[o-F$6&Fj[oFg^oFi^oF^[o-F$6&F]\\oFg^oFi^oF^[o-F$
6&F`\\oFg^oFi^oF^[o-F$6&Fc\\oFg^oFi^oF^[o-F$6&Ff\\oFg^oFi^oF^[o-F$6$7$
7$Fifn$!33+++++++^F-FhfnFg^o-F$6$7$7$FhgnF^`oFggnFg^o-F$6$7$7$F_hnF^`o
F^hnFg^o-F$6$7$7$FfhnF^`oFehnFg^o-F$6$7$7$F]inF^`oF\\inFg^o-F$6$7$7$Fd
inF^`oFcinFg^o-F$6$7$7$F[jnF^`oFjinFg^o-F$6$7$7$FbjnF^`oFajnFg^o-%)POL
YGONSG6+7%7$$!+LLLt=!\"*$!\"\"Febo7$$!+LLLL=FcboFgbn7$$!+LLL$z\"FcboFd
bo7%7$$\"+nmmm7!#5Fdbo7$$\"+nmmm;F`coFgbn7$$\"+nmmm?F`coFdbo7%7$$\"+nm
mE@FcboFdbo7$$\"+nmmm@FcboFgbn7$$\"+nmm1AFcboFdbo7%7$$\"+nmmETFcboFdbo
7$$\"+nmmmTFcboFgbn7$$\"+nmm1UFcboFdbo7%7$$!+nmm1AFcboFdbo7$$!+nmmm@Fc
boFgbn7$$!+nmmE@FcboFdbo7%7$$!+nmmm?F`coFdbo7$$!+nmmm;F`coFgbn7$$!+nmm
m7F`coFdbo7%7$$\"+LLL$z\"FcboFdbo7$$\"+LLLL=FcboFgbn7$$\"+LLLt=FcboFdb
o7%7$$\"+LLL$z$FcboFdbo7$$\"+LLLLQFcboFgbn7$$\"+LLLtQFcboFdbo-Fabn6&Fc
bnFdbn$\")AR!)\\FfbnFgbn-F$6$7U7$$!+LLLj:Fcbo$!\"(Febo7$$!+$R5\\c\"Fcb
o$!+LNL\\nF`co7$$!+,nhp:Fcbo$!+D-i-lF`co7$$!+O!ytd\"Fcbo$!+&*3vjiF`co7
$$!+(*>2)e\"Fcbo$!+_E\\OgF`co7$$!+M*H:g\"Fcbo$!+&\\HW#eF`co7$$!+3'Rvh
\"Fcbo$!+)y04j&F`co7$$!+N&[ej\"Fcbo$!+9N(*eaF`co7$$!+uz;c;Fcbo$!+\\TM6
`F`co7$$!+vu<y;Fcbo$!+&*eM!>&F`co7$$!+M*H:q\"Fcbo$!+np)y4&F`co7$$!+qq&
es\"Fcbo$!+)\\Da.&F`co7$$!+H_x]<Fcbo$!+Vl%R+&F`co7$$!+P9*ex\"FcboFi[p7
$$!+'f43!=Fcbo$!+*\\Da.&F`co7$$!+Kn8D=FcboF_[p7$$!+\">*[[=FcboFjjo7$$!
+#p)\\q=FcboFejo7$$!+J\"=3*=FcboF`jo7$$!+fq74>Fcbo$!+!z04j&F`co7$$!+Kn
8D>Fcbo$!+(\\HW#eF`co7$$!+pYfQ>Fcbo$!+aE\\OgF`co7$$!+I')G\\>Fcbo$!+'*3
vjiF`co7$$!+l*\\q&>Fcbo$!+F-i-lF`co7$$!+tivh>Fcbo$!+MNL\\nF`co7$$!+LLL
j>Fcbo$!+,+++qF`co7$Fi^p$!+okm]sF`co7$Fd^p$!+v(zt\\(F`co7$F_^p$!+0\"\\
it(F`co7$Fj]p$!+[t]jzF`co7$Fe]p$!+/0dv\")F`co7$$!+eq74>Fcbo$!+6U4p$)F`
co7$F]]p$!+([E5a)F`co7$Fj\\p$!+_el)o)F`co7$Fg\\p$!+1Tl4))F`co7$Fd\\p$!
+LI6-*)F`co7$F_\\p$!+-Xdk*)F`co7$F\\\\p$!+dM0'**)F`co7$Fg[pFfap7$Fb[p$
!+,Xdk*)F`co7$F][p$!+KI6-*)F`co7$Fhjo$!+0Tl4))F`co7$Fcjo$!+^el)o)F`co7
$F^jo$!+'[E5a)F`co7$$!+2'Rvh\"Fcbo$!+4U4p$)F`co7$Fdio$!+-0dv\")F`co7$F
_io$!+Yt]jzF`co7$Fjho$!+.\"\\it(F`co7$Feho$!+s(zt\\(F`co7$F`ho$!+lkm]s
F`co7$F[ho$!+)*******pF`coFg^o-F$6$7U7$$\"+nmmmVF`coF]ho7$$\"+qg*3N%F`
coFbho7$$\"+*)H$QI%F`coFgho7$$\"+R'>iA%F`coF\\io7$$\"+F+G>TF`coFaio7$$
\"+c1q%)RF`coFfio7$$\"+ARgCQF`coF[jo7$$\"+YY^TOF`coF`jo7$$\"+d-KQMF`co
Fejo7$$\"+^_A=KF`coFjjo7$$\"+b1q%)HF`coF_[p7$$\"+&HH9u#F`coFd[p7$$\"+1
xC#\\#F`coFi[p7$$\"+Gc3TAF`coFi[p7$$\"+QS!>*>F`coFa\\p7$$\"+zEj[<F`coF
_[p7$$\"+%33^^\"F`coFjjo7$$\"+yI,&H\"F`coFejo7$$\"+)o==4\"F`coF`jo7$$
\"*5%H(3*F`coFb]p7$$\"*xEj[(F`coFg]p7$$\"*1L09'F`coF\\^p7$$\"*&p8r]F`c
oFa^p7$$\"*W.]H%F`coFf^p7$$\"*ksV#QF`coF[_p7$$\"*nmmm$F`coF`_p7$F[ipFc
_p7$$\"*X.]H%F`coFf_p7$FehpFi_p7$$\"*2L09'F`coF\\`p7$$\"*yEj[(F`coF_`p
7$$\"*7%H(3*F`coFd`p7$$\"+!p==4\"F`coFg`p7$$\"+zI,&H\"F`coFj`p7$$\"+'3
3^^\"F`coF]ap7$$\"+!oK'[<F`coF`ap7$$\"+RS!>*>F`coFcap7$$\"+Hc3TAF`coFf
ap7$$\"+2xC#\\#F`coFfap7$$\"+(HH9u#F`coFjap7$$\"+c1q%)HF`coF]bp7$$\"+]
_A=KF`coF`bp7$F[fpFcbp7$FhepFfbp7$$\"+CRgCQF`coF[cp7$$\"+e1q%)RF`coF^c
p7$$\"+G+G>TF`coFacp7$$\"+S'>iA%F`coFdcp7$$\"+!*H$QI%F`coFgcp7$FfdpFjc
p7$FcdpF]dpFg^o-F$6$7U7$$\"+nmmOCFcboF]ho7$$\"+2'*3NCFcboFbho7$$\"+*H$
QICFcboFgho7$$\"+k>iACFcboF\\io7$$\"+.!G>T#FcboFaio7$$\"+m+Z)R#FcboFfi
o7$$\"+#RgCQ#FcboF[jo7$$\"+l9:kBFcboF`jo7$$\"+E?$QM#FcboFejo7$$\"+DD#=
K#FcboFjjo7$$\"+m+Z)H#FcboF_[p7$$\"+IH9uAFcboFd[p7$$\"+rZA\\AFcboFi[p7
$$\"+j&3TA#FcboFi[p7$$\"+//>*>#FcboFa\\p7$$\"+oK'[<#FcboF_[p7$$\"+43^^
@FcboFjjo7$$\"+38]H@FcboFejo7$$\"+p==4@FcboF`jo7$$\"+TH(34#FcboFb]p7$$
\"+oK'[2#FcboFg]p7$$\"+J`Sh?FcboF\\^p7$$\"+q8r]?FcboFa^p7$$\"+N+&H/#Fc
boFf^p7$$\"+FPCQ?FcboF[_p7$$\"+nmmO?FcboF`_p7$F[bqFc_p7$FhaqFf_p7$Feaq
Fi_p7$FbaqF\\`p7$F_aqF_`p7$$\"+UH(34#FcboFd`p7$Fi`qFg`p7$Ff`qFj`p7$Fc`
qF]ap7$F``qF`ap7$F]`qFcap7$Fj_qFfap7$Fg_qFfap7$Fd_qFjap7$Fa_qF]bp7$F^_
qF`bp7$F[_qFcbp7$Fh^qFfbp7$$\"+$RgCQ#FcboF[cp7$Fb^qF^cp7$F_^qFacp7$F\\
^qFdcp7$Fi]qFgcp7$Ff]qFjcp7$Fc]qF]dpFg^o-F$6$7U7$$\"+nmmOWFcboF]ho7$$
\"+2'*3NWFcboFbho7$$\"+*H$QIWFcboFgho7$$\"+k>iAWFcboF\\io7$$\"+.!G>T%F
cboFaio7$$\"+m+Z)R%FcboFfio7$$\"+#RgCQ%FcboF[jo7$$\"+l9:kVFcboF`jo7$$
\"+E?$QM%FcboFejo7$$\"+DD#=K%FcboFjjo7$$\"+m+Z)H%FcboF_[p7$$\"+IH9uUFc
boFd[p7$$\"+rZA\\UFcboFi[p7$$\"+j&3TA%FcboFi[p7$$\"+//>*>%FcboFa\\p7$$
\"+oK'[<%FcboF_[p7$$\"+43^^TFcboFjjo7$$\"+38]HTFcboFejo7$$\"+p==4TFcbo
F`jo7$$\"+TH(34%FcboFb]p7$$\"+oK'[2%FcboFg]p7$$\"+J`ShSFcboF\\^p7$$\"+
q8r]SFcboFa^p7$$\"+N+&H/%FcboFf^p7$$\"+FPCQSFcboF[_p7$$\"+nmmOSFcboF`_
p7$FihqFc_p7$FfhqFf_p7$FchqFi_p7$F`hqF\\`p7$F]hqF_`p7$$\"+UH(34%FcboFd
`p7$FggqFg`p7$FdgqFj`p7$FagqF]ap7$F^gqF`ap7$F[gqFcap7$FhfqFfap7$FefqFf
ap7$FbfqFjap7$F_fqF]bp7$F\\fqF`bp7$FieqFcbp7$FfeqFfbp7$$\"+$RgCQ%FcboF
[cp7$F`eqF^cp7$F]eqFacp7$FjdqFdcp7$FgdqFgcp7$FddqFjcp7$FadqF]dpFg^o-F$
6$7U7$$!+nmmO?FcboF]ho7$$!+FPCQ?FcboFbho7$$!+N+&H/#FcboFgho7$$!+q8r]?F
cboF\\io7$$!+J`Sh?FcboFaio7$$!+oK'[2#FcboFfio7$$!+UH(34#FcboF[jo7$$!+p
==4@FcboF`jo7$$!+38]H@FcboFejo7$$!+43^^@FcboFjjo7$$!+oK'[<#FcboF_[p7$$
!+//>*>#FcboFd[p7$$!+j&3TA#FcboFi[p7$$!+rZA\\AFcboFi[p7$$!+IH9uAFcboFa
\\p7$$!+m+Z)H#FcboF_[p7$$!+DD#=K#FcboFjjo7$$!+E?$QM#FcboFejo7$$!+l9:kB
FcboF`jo7$$!+$RgCQ#FcboFb]p7$$!+m+Z)R#FcboFg]p7$$!+.!G>T#FcboF\\^p7$$!
+k>iACFcboFa^p7$$!+*H$QICFcboFf^p7$$!+2'*3NCFcboF[_p7$$!+nmmOCFcboF`_p
7$Fg_rFc_p7$Fd_rFf_p7$Fa_rFi_p7$F^_rF\\`p7$F[_rF_`p7$$!+#RgCQ#FcboFd`p
7$Fe^rFg`p7$Fb^rFj`p7$F_^rF]ap7$F\\^rF`ap7$Fi]rFcap7$Ff]rFfap7$Fc]rFfa
p7$F`]rFjap7$F]]rF]bp7$Fj\\rF`bp7$Fg\\rFcbp7$Fd\\rFfbp7$$!+TH(34#FcboF
[cp7$F^\\rF^cp7$F[\\rFacp7$Fh[rFdcp7$Fe[rFgcp7$Fb[rFjcp7$F_[rF]dpFg^o-
F$6$7U7$$!*nmmm$F`coF]ho7$$!*ksV#QF`coFbho7$$!*X.]H%F`coFgho7$$!*&p8r]
F`coF\\io7$$!*2L09'F`coFaio7$$!*yEj[(F`coFfio7$$!*7%H(3*F`coF[jo7$$!+)
o==4\"F`coF`jo7$$!+xI,&H\"F`coFejo7$$!+$33^^\"F`coFjjo7$$!+zEj[<F`coF_
[p7$$!+RS!>*>F`coFd[p7$$!+Gc3TAF`coFi[p7$$!+1xC#\\#F`coFi[p7$$!+'HH9u#
F`coFa\\p7$$!+b1q%)HF`coF_[p7$$!+]_A=KF`coFjjo7$$!+c-KQMF`coFejo7$$!+Y
Y^TOF`coF`jo7$$!+CRgCQF`coFb]p7$$!+d1q%)RF`coFg]p7$$!+G+G>TF`coF\\^p7$
$!+R'>iA%F`coFa^p7$$!+!*H$QI%F`coFf^p7$$!+qg*3N%F`coF[_p7$$!+nmmmVF`co
F`_p7$FefrFc_p7$$!+*)H$QI%F`coFf_p7$F_frFi_p7$$!+F+G>TF`coF\\`p7$$!+c1
q%)RF`coF_`p7$$!+ARgCQF`coFd`p7$$!+WY^TOF`coFg`p7$$!+b-KQMF`coFj`p7$$!
+[_A=KF`coF]ap7$$!+a1q%)HF`coF`ap7$$!+&HH9u#F`coFcap7$$!+0xC#\\#F`coFf
ap7$$!+Fc3TAF`coFfap7$$!+PS!>*>F`coFjap7$$!+yEj[<F`coF]bp7$$!+%33^^\"F
`coF`bp7$FecrFcbp7$FbcrFfbp7$$!*5%H(3*F`coF[cp7$$!*wEj[(F`coF^cp7$$!*1
L09'F`coFacp7$$!*%p8r]F`coFdcp7$$!*W.]H%F`coFgcp7$F`brFjcp7$F]brF]dpFg
^o-F$6$7U7$$\"+LLLj>FcboF]ho7$$\"+tivh>FcboFbho7$$\"+l*\\q&>FcboFgho7$
$\"+I')G\\>FcboF\\io7$$\"+pYfQ>FcboFaio7$$\"+Kn8D>FcboFfio7$$\"+eq74>F
cboF[jo7$$\"+J\"=3*=FcboF`jo7$$\"+#p)\\q=FcboFejo7$$\"+\">*[[=FcboFjjo
7$$\"+Kn8D=FcboF_[p7$$\"+'f43!=FcboFd[p7$$\"+P9*ex\"FcboFi[p7$$\"+H_x]
<FcboFi[p7$$\"+qq&es\"FcboFa\\p7$$\"+M*H:q\"FcboF_[p7$$\"+vu<y;FcboFjj
o7$$\"+uz;c;FcboFejo7$$\"+N&[ej\"FcboF`jo7$$\"+2'Rvh\"FcboFb]p7$$\"+M*
H:g\"FcboFg]p7$$\"+(*>2)e\"FcboF\\^p7$$\"+O!ytd\"FcboFa^p7$$\"+,nhp:Fc
boFf^p7$$\"+$R5\\c\"FcboF[_p7$$\"+LLLj:FcboF`_p7$Fe_sFc_p7$Fb_sFf_p7$F
__sFi_p7$F\\_sF\\`p7$Fi^sF_`p7$$\"+3'Rvh\"FcboFd`p7$Fc^sFg`p7$F`^sFj`p
7$F]^sF]ap7$Fj]sF`ap7$Fg]sFcap7$Fd]sFfap7$Fa]sFfap7$F^]sFjap7$F[]sF]bp
7$Fh\\sF`bp7$Fe\\sFcbp7$Fb\\sFfbp7$$\"+fq74>FcboF[cp7$F\\\\sF^cp7$Fi[s
Facp7$Ff[sFdcp7$Fc[sFgcp7$F`[sFjcp7$F][sF]dpFg^o-F$6$7U7$$\"+LLLjRFcbo
F]ho7$$\"+tivhRFcboFbho7$$\"+l*\\q&RFcboFgho7$$\"+I')G\\RFcboF\\io7$$
\"+pYfQRFcboFaio7$$\"+Kn8DRFcboFfio7$$\"+eq74RFcboF[jo7$$\"+J\"=3*QFcb
oF`jo7$$\"+#p)\\qQFcboFejo7$$\"+\">*[[QFcboFjjo7$$\"+Kn8DQFcboF_[p7$$
\"+'f43!QFcboFd[p7$$\"+P9*ex$FcboFi[p7$$\"+H_x]PFcboFi[p7$$\"+qq&es$Fc
boFa\\p7$$\"+M*H:q$FcboF_[p7$$\"+vu<yOFcboFjjo7$$\"+uz;cOFcboFejo7$$\"
+N&[ej$FcboF`jo7$$\"+2'Rvh$FcboFb]p7$$\"+M*H:g$FcboFg]p7$$\"+(*>2)e$Fc
boF\\^p7$$\"+O!ytd$FcboFa^p7$$\"+,nhpNFcboFf^p7$$\"+$R5\\c$FcboF[_p7$$
\"+LLLjNFcboF`_p7$FcfsFc_p7$F`fsFf_p7$F]fsFi_p7$FjesF\\`p7$FgesF_`p7$$
\"+3'Rvh$FcboFd`p7$FaesFg`p7$F^esFj`p7$F[esF]ap7$FhdsF`ap7$FedsFcap7$F
bdsFfap7$F_dsFfap7$F\\dsFjap7$FicsF]bp7$FfcsF`bp7$FccsFcbp7$F`csFfbp7$
$\"+fq74RFcboF[cp7$FjbsF^cp7$FgbsFacp7$FdbsFdcp7$FabsFgcp7$F^bsFjcp7$F
[bsF]dpFg^o-%%TEXTG6&7$$Ffbn!\"#$\"$H\"FjhsQ\"y6\"-%&COLORG6&Fcbn$\"\"
\"FjhsFbisFbis-%%FONTG6$%*HELVETICAG\"#5-Ffhs6&7$$\"#ZFebo$!#8FjhsQ\"q
F^isF_is-Feis6$F[[o\"#6-Ffhs6&7$$!+LLLj<FcboF]hoQ'-11p/6F^isF_is-Feis6
$F[[oFhis-Ffhs6&7$$\"+nmmOAFcboF]hoQ&13p/6F^isF_isFjjs-Ffhs6&7$$\"+nmm
OUFcboF]hoQ&25p/6F^isF_isFjjs-Ffhs6&7$$!+nmmOAFcboF]hoQ'-13p/6F^isF_is
Fjjs-Ffhs6&7$$\"+LLLj<FcboF]hoQ&11p/6F^isF_isFjjs-Ffhs6&7$$\"+LLLjPFcb
oF]hoQ&23p/6F^isF_isFjjs-Ffhs6&7$$\"+nmmmBF`coF]hoQ$p/6F^isF_isFjjs-Ff
hs6&7$$!+nmmmBF`coF]hoQ%-p/6F^isF_isFjjs-%*AXESTICKSG6$71/$!#DFebo%&-5
p/2G/Fjhs%$-2pG/$!#:Febo%&-3p/2G/Febo%#-pG/$!\"&Febo%%-p/2G/Fhbn%\"0G/
$\"\"&Febo%$p/2G/Fcis%\"pG/$\"#:Febo%%3p/2G/F\\cn%#2pG/$\"#DFebo%%5p/2
G/\"\"$%#3pG/$\"#NFebo%%7p/2G/\"\"%%#4pG/$\"#XFebo%%9p/2GF]`t-Feis6$F[
[o\"\"*-%+AXESLABELSG6%%!GF`at-Feis6#%(DEFAULTG-%%VIEWG6$;$!$b#FjhsF\\
js;$!$K\"FjhsF[is" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu
rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+
19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" 
"Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Cur
ve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 3
8" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "
Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curv
e 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57
" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "Curve 63" "C
urve 64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve 69" }}
{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "The " }{TEXT 259 12 "solution set" }{TEXT -1 17 " of the e
quation " }{XPPEDIT 18 0 "cos*theta = sqrt(3)/2;" "6#/*&%$cosG\"\"\"%&
thetaGF&*&-%%sqrtG6#\"\"$F&\"\"#!\"\"" }{TEXT -1 5 " is: " }}{PARA 
256 "" 0 "" {TEXT -1 2 " \{" }{XPPEDIT 18 0 "2*k*Pi;" "6#*(\"\"#\"\"\"
%\"kGF%%#PiGF%" }{TEXT -1 1 " " }{TEXT 324 1 "+" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Pi/6" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 2 " |" }
{TEXT 323 2 " k" }{TEXT -1 17 " is an integer\}. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT 
307 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 42 "Find \+
the general solution of the equation " }{XPPEDIT 18 0 "cos*theta = 2/3
;" "6#/*&%$cosG\"\"\"%&thetaGF&*&\"\"#F&\"\"$!\"\"" }{TEXT -1 81 ", gi
ving the solution in terms of an appropriate inverse trigonometric fun
ction. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 308 
8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 29 "One solut
ion of the equation " }{XPPEDIT 18 0 "cos*theta = 2/3;" "6#/*&%$cosG\"
\"\"%&thetaGF&*&\"\"#F&\"\"$!\"\"" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "
theta = arccos(2/3);" "6#/%&thetaG-%'arccosG6#*&\"\"#\"\"\"\"\"$!\"\"
" }{TEXT -1 1 " " }{TEXT 335 1 "~" }{TEXT -1 15 " 0.8410686706. " }}
{PARA 0 "" 0 "" {TEXT -1 29 "Another solution is given by " }{XPPEDIT 
18 0 "theta = -arccos(2/3);" "6#/%&thetaG,$-%'arccosG6#*&\"\"#\"\"\"\"
\"$!\"\"F-" }{TEXT -1 1 " " }{TEXT 336 1 "~" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "-0" "6#,$\"\"!!\"\"" }{TEXT -1 13 ".8410686706. " }}
{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 13 "two solutions" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "theta = arccos(2/3);" "6#/%&thetaG-%'ar
ccosG6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta
 = -arccos(2/3);" "6#/%&thetaG,$-%'arccosG6#*&\"\"#\"\"\"\"\"$!\"\"F-
" }{TEXT -1 40 " are the only solution in the interval [" }{XPPEDIT 
18 0 "-Pi,Pi;" "6$,$%#PiG!\"\"F$" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" 
{TEXT -1 14 "We may write: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "theta=``" "6#/%&thetaG%!G" }{TEXT 327 1 "+" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "arccos(2/3);" "6#-%'arccosG6#*&\"\"#\"\"\"\"\"$!
\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 
18 0 "-Pi<=theta" "6#1,$%#PiG!\"\"%&thetaG" }{XPPEDIT 18 0 "``<Pi" "6#
2%!G%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 50 "Since the c
osine function is periodic with period " }{XPPEDIT 18 0 "2*Pi;" "6#*&
\"\"#\"\"\"%#PiGF%" }{TEXT -1 15 ", the numbers: " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }
{TEXT -1 1 " " }{TEXT 328 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arccos
(2/3);" "6#-%'arccosG6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 5 ",    " }
{XPPEDIT 18 0 "4*Pi" "6#*&\"\"%\"\"\"%#PiGF%" }{TEXT -1 1 " " }{TEXT 
329 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arccos(2/3);" "6#-%'arccosG6
#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "6*Pi" "6
#*&\"\"'\"\"\"%#PiGF%" }{TEXT -1 1 " " }{TEXT 330 1 "+" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "arccos(2/3);" "6#-%'arccosG6#*&\"\"#\"\"\"\"\"$!\"\"
" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }
{TEXT -1 8 ", etc., " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 28 "are also solutions, as are: " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "-2*Pi;" "6#,$*&\"\"#\"\"\"%#PiGF&!\"\"
" }{TEXT -1 1 " " }{TEXT 331 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arc
cos(2/3);" "6#-%'arccosG6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 5 ",    \+
" }{XPPEDIT 18 0 "-4*Pi;" "6#,$*&\"\"%\"\"\"%#PiGF&!\"\"" }{TEXT -1 1 
" " }{TEXT 332 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arccos(2/3);" "6#
-%'arccosG6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 
0 "-6*Pi;" "6#,$*&\"\"'\"\"\"%#PiGF&!\"\"" }{TEXT -1 1 " " }{TEXT 333 
1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcccos(2/3);" "6#-%(arcccosG6#*
&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "` . . . `
" "6#%(~.~.~.~G" }{TEXT -1 7 ", etc. " }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 36 "In general, any number of the form:
 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta=2*k*Pi" "6
#/%&thetaG*(\"\"#\"\"\"%\"kGF'%#PiGF'" }{TEXT -1 1 " " }{TEXT 334 1 "+
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arccos(2/3);" "6#-%'arccosG6#*&\"\"#
\"\"\"\"\"$!\"\"" }{TEXT -1 3 ",  " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 326 12 "____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 325 1 "k" }
{TEXT -1 25 " is an integer, that is, " }{XPPEDIT 18 0 "k=` . . . `,-4
,-3,-2,-1,0,1,2,3,4,` . . . `" "6-/%\"kG%(~.~.~.~G,$\"\"%!\"\",$\"\"$F
(,$\"\"#F(,$\"\"\"F(\"\"!F.F,F*F'F%" }{TEXT -1 32 ", is a solution of \+
the equation " }{XPPEDIT 18 0 "cos*theta = 2/3;" "6#/*&%$cosG\"\"\"%&t
hetaGF&*&\"\"#F&\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 75 "The following picture shows the points of intersection P and Q \+
of the line " }{XPPEDIT 18 0 "x = 2/3;" "6#/%\"xG*&\"\"#\"\"\"\"\"$!\"
\"" }{TEXT -1 22 " with the unit circle " }{XPPEDIT 18 0 "x^2+y^2=1" "
6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 129 ". All the solutio
ns are associated with either P or Q, more precisely, the \"wrapping f
unction\" W, which associates a real number " }{XPPEDIT 18 0 "theta" "
6#%&thetaG" }{TEXT -1 16 " with the point " }{XPPEDIT 18 0 "W(theta)=`
`(cos*theta,sin*theta)" "6#/-%\"WG6#%&thetaG-%!G6$*&%$cosG\"\"\"F'F-*&
%$sinGF-F'F-" }{TEXT -1 39 ", maps each solution to either P or Q. " }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 278 249 249 {PLOTDATA 2 "
62-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4hRPij!**!#=$\"3Ikwb#=y_O
\"F/7$$\"3E8J#))4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\"3-N5')yke[#*F/$\"3gLj&[K
5J!QF/7$$\"3?goz=42`')F/$\"3j9NXR4U7]F/7$$\"3Sb](G._U!zF/$\"3t_H-qceDh
F/7$$\"3_\\$R'oTd#3(F/$\"3!GO#*3qU&fqF/7$$\"3?H$>jubk6'F/$\"3!\\@@&\\!
>8\"zF/7$$\"3VGuIc:x5]F/$\"3-$>p(Qh-a')F/7$$\"37C3nW'R,#QF/$\"3aK+16bc
T#*F/7$$\"3$oY@iF'QDDF/$\"3s&Q\"[M\"oen*F/7$$\"3OB^hAo8X8F/$\"33)fVPO<
\"4**F/7$$!3+qB/u(p5(f!#@$\"2%HhJ<#)******!#<7$$!33)\\T#fB[i8F/$\"3ap%
>wGZn!**F/7$$!3)e:d.:#=YEF/$\"3PA>=\\D`V'*F/7$$!3%*yV1J*3Jx$F/$\"3IzF#
e`m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'yI2&oN')F/7$$!3[igNw%*[RgF/$\"3!G;)RQ
+BqzF/7$$!3/XUwYjJ*3(F/$\"3AfvOg?x_qF/7$$!3&=e_b?/U!zF/$\"3u3<J%QZc7'F
/7$$!3w!G5)RnIf')F/$\"3D%H$\\4/k,]F/7$$!3/`k8ti$4B*F/$\"3In'[*>HvXQF/7
$$!3!p!RS4Nij'*F/$\"3o$p9m#Q%=d#F/7$$!3#p(pT>+.2**F/$\"31V?00]Ug8F/7$$
!3/gKG4>&*****F/$\"3vCA\\O%485$!#?7$$!3__Is=!Q%3**F/$!3q#4*>c=8]8F/7$$
!3)>b`sYlSn*F/$!3UNbFGIGKDF/7$$!37_J.8AEj#*F/$!3q4&=j`Bsw$F/7$$!3%\\F)
4Kh=u')F/$!3kENX')3zv\\F/7$$!3EB*z7xng%zF/$!37W(H!pUCrgF/7$$!3XD84Pfc5
rF/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[gF/$!3)4j2yeGL'zF/7$$!3g@UD=%)o!*\\F/
$!3I_Mh6Mil')F/7$$!3o+1s\"[\"ysPF/$!3#\\IN,%***4E*F/7$$!30yCH\"el'3EF/
$!3=V>(fp[Pl*F/7$$!3g67Cvnc\"H\"F/$!3;$RoI*>C;**F/7$$!3EzyOHMQdIFas$!3
W1O#>E`*****F/7$$\"3GIAj`Ks(G\"F/$!3lYZ3X=u;**F/7$$\"3EKRqX8/bDF/$!3U$
[o%H'z!o'*F/7$$\"3@%)yb&Q)zNQF/$!36.2<#>x]B*F/7$$\"3O7w4w0I.]F/$!3wHgb
5wMe')F/7$$\"3@\\#)[*R]$4hF/$!3/a*[=H2o\"zF/7$$\"33/wY'Q+$*4(F/$!3)4d)
eg?sUqF/7$$\"3[f=s$Ry,!zF/$!3D1$H<bQ38'F/7$$\"3%*R,@;vLu')F/$!3e>)zD(p
_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2ndo*fQF/7$$\"31mGAp))oa'*F/$!3%)f]nRQ=
0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4O\"F/7$F($\"36YKhSr8/#)!#F-%'COLOURG
6&%$RGBG$\"*++++\"!\")F+F+-%*THICKNESSG6#\"\"#-F$6&7$7$F+F+7$$\"3m****
*pmmmm'F/$\"3[+++B*fNX(F/-Fjz6&F\\[l$\")#)eqkF_[l$\"))eqk\"F_[lFa\\lF`
[l-%*LINESTYLEG6#F)-F$6&7$Fg[l7$Fi[l$!3[+++B*fNX(F/F]\\lF`[lFc\\l-F$6&
7$Fi\\lFh[l-Fjz6&F\\[lF*F*F*-Fa[lFe\\l-Fd\\lFb[l-F$6&7$Fh[lFi\\l-Fjz6&
F\\[lF+F][lF][l-%'SYMBOLG6$%&CROSSG\"#5-%&STYLEG6#%&POINTG-F$6&Fe]lFf]
l-Fi]l6$%(DIAMONDGF\\^lF]^l-F$6&Fe]lFf]l-Fi]l6$%'CIRCLEGF\\^lF]^l-F$6&
Fe]lF_]l-Fi]l6$Fj^l\"#7F]^l-%%TEXTG6%7$$\"#\")!\"#$\"#zFf_lQ\"P6\"-%&C
OLORG6&F\\[l$F)Ff_lF^`lF^`l-Fa_l6%7$Fd_l$!#zFf_lQ\"QFj_lF[`l-Fa_l6%7$$
\"$D\"Ff_l$!\"'Ff_lQ\"xFj_lF[`l-Fa_l6%7$Fj`lFh`lQ\"yFj_lF[`l-%(SCALING
G6#%,CONSTRAINEDG-%*AXESTICKSG6$F*F*-%+AXESLABELSG6%Q!Fj_lF[bl-%%FONTG
6#%(DEFAULTG-%%VIEWG6$;$!#6!\"\"$\"#8Ffbl;FdblFh`l" 1 2 0 1 10 0 2 9 
1 4 1 1.000000 130.000000 349.000000 0 0 "Curve 1" "Curve 2" "Curve 3
" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 1
0" "Curve 11" "Curve 12" }}{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT 
-1 4 "The " }{TEXT 259 12 "solution set" }{TEXT -1 17 " of the equatio
n " }{XPPEDIT 18 0 "cos*theta = 2/3;" "6#/*&%$cosG\"\"\"%&thetaGF&*&\"
\"#F&\"\"$!\"\"" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 3 " \+
\{ " }{XPPEDIT 18 0 "2*k*Pi;" "6#*(\"\"#\"\"\"%\"kGF%%#PiGF%" }{TEXT 
-1 1 " " }{TEXT 337 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arccos(2/3);
" "6#-%'arccosG6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 3 " | " }{TEXT 
309 1 "k" }{TEXT -1 17 " is an integer\}. " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{TEXT 338 18 "__________________" }{TEXT -1 1 " " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 
5 ": If " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 47 " is an ang
le in degrees, then the solutions of " }{XPPEDIT 18 0 "cos*theta = 2/3
;" "6#/*&%$cosG\"\"\"%&thetaGF&*&\"\"#F&\"\"$!\"\"" }{TEXT -1 9 " betw
een " }{XPPEDIT 18 0 "-90^o" "6#,$)\"#!*%\"oG!\"\"" }{TEXT -1 7 " and \+
+ " }{XPPEDIT 18 0 "90^o" "6#)\"#!*%\"oG" }{TEXT -1 5 " are " }
{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 1 " " }{TEXT 344 1 "~" }
{TEXT -1 2 "  " }{TEXT 348 1 "+" }{TEXT -1 2 " 0" }{XPPEDIT 18 0 ".841
0686706*`.`*``(180^o/Pi)" "6#*(-%&FloatG6$\"+1no5%)!#5\"\"\"%\".GF)-%!
G6#*&)\"$!=%\"oGF)%#PiG!\"\"F)" }{TEXT -1 1 " " }{TEXT 345 1 "~" }
{TEXT -1 1 " " }{TEXT 347 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "48.189
68510^o" "6#)-%&FloatG6$\"+5&o*=[!\")%\"oG" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 59 "The general solution can be given in the approx
imate form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta
" "6#%&thetaG" }{TEXT -1 1 " " }{TEXT 349 1 "~" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "k*360^o;" "6#*&%\"kG\"\"\")\"$g$%\"oGF%" }{TEXT -1 1 " \+
" }{TEXT 350 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "48.18968510^o" "6#)
-%&FloatG6$\"+5&o*=[!\")%\"oG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" 
{TEXT -1 6 "where " }{TEXT 346 1 "k" }{TEXT -1 16 " is an integer. " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 
0 "" {TEXT -1 72 "The previous two examples illustrate the fact that, \+
for any real number " }{TEXT 314 1 "a" }{TEXT -1 6 " with " }{XPPEDIT 
18 0 "-1<a" "6#2,$\"\"\"!\"\"%\"aG" }{XPPEDIT 18 0 "``<1" "6#2%!G\"\"
\"" }{TEXT -1 15 ", the equation " }{XPPEDIT 18 0 "cos*theta = a;" "6#
/*&%$cosG\"\"\"%&thetaGF&%\"aG" }{TEXT -1 9 " has the " }{TEXT 259 16 
"general solution" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "theta = 2*k*Pi;" "6#/%&thetaG*(\"\"#\"\"\"%\"kGF'%#P
iGF'" }{TEXT -1 1 " " }{TEXT 381 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 
"arccos*a;" "6#*&%'arccosG\"\"\"%\"aGF%" }{TEXT -1 2 ", " }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{TEXT 384 11 "___________" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 382 1 "k" }{TEXT -1 49 " is
 an integer, that is, it has the solution set:" }}{PARA 256 "" 0 "" 
{TEXT -1 3 " \{ " }{XPPEDIT 18 0 "2*k*Pi;" "6#*(\"\"#\"\"\"%\"kGF%%#Pi
GF%" }{TEXT -1 1 " " }{TEXT 340 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
arccos*a;" "6#*&%'arccosG\"\"\"%\"aGF%" }{TEXT -1 3 " | " }{TEXT 339 
1 "k" }{TEXT -1 17 " is an integer\}. " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{TEXT 341 17 "_________________" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 45 "
: It may be useful to note that the equation " }{XPPEDIT 18 0 "cos*the
ta = -a;" "6#/*&%$cosG\"\"\"%&thetaGF&,$%\"aG!\"\"" }{TEXT -1 27 " has
 the general solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "theta = (2*k-1)*Pi;" "6#/%&thetaG*&,&*&\"\"#\"\"\"%\"kGF)F)F)!\"
\"F)%#PiGF)" }{TEXT -1 1 " " }{TEXT 385 1 "+" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "arccos*a;" "6#*&%'arccosG\"\"\"%\"aGF%" }{TEXT -1 2 ", \+
" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 379 1 "k" }{TEXT -1 16 
" is an integer. " }}{PARA 0 "" 0 "" {TEXT -1 22 "We have the followin
g " }{TEXT 259 13 "special cases" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 37 "The general solution of the equation " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta=1" "6#/*&%$cosG\"\"\"%&thet
aGF&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "is" }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta=2*k*Pi" "6#/%&thetaG*(\"
\"#\"\"\"%\"kGF'%#PiGF'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 
6 "where " }{TEXT 342 1 "k" }{TEXT -1 58 " is an integer, while the ge
neral solution of the equation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "cos*theta=-1" "6#/*&%$cosG\"\"\"%&thetaGF&,$F&!\"\"" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "is " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "theta=(2*k-1)*Pi" "6#/%&thetaG*&,&*&\"
\"#\"\"\"%\"kGF)F)F)!\"\"F)%#PiGF)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "
" {TEXT -1 6 "where " }{TEXT 343 1 "k" }{TEXT -1 15 " is an integer." 
}}{PARA 0 "" 0 "" {TEXT -1 43 "Also, the general solution of the equat
ion " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta = 0
;" "6#/*&%$cosG\"\"\"%&thetaGF&\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 3 "is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
theta = ``(2*k-1);" "6#/%&thetaG-%!G6#,&*&\"\"#\"\"\"%\"kGF+F+F+!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 354 1 "k
" }{TEXT -1 15 " is an integer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 37 "The general solution of the equation " }{XPPEDIT 18 0 "
sin*theta = a;" "6#/*&%$sinG\"\"\"%&thetaGF&%\"aG" }{TEXT -1 1 " " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT 273 8 "Question" }
{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 42 "Find the general soluti
on of the equation " }{XPPEDIT 18 0 "sin*theta = 1/2;" "6#/*&%$sinG\"
\"\"%&thetaGF&*&F&F&\"\"#!\"\"" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 8 "Solution" }{TEXT -1 2 ": \+
" }}{PARA 0 "" 0 "" {TEXT -1 29 "One solution of the equation " }
{XPPEDIT 18 0 "sin*theta = 1/2;" "6#/*&%$sinG\"\"\"%&thetaGF&*&F&F&\"
\"#!\"\"" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "theta = arcsin(1/2);" "6#
/%&thetaG-%'arcsinG6#*&\"\"\"F)\"\"#!\"\"" }{XPPEDIT 18 0 "`` = Pi/6;
" "6#/%!G*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Since " }{XPPEDIT 18 0 "sin(Pi-theta)=sin*theta" "6#/-%$si
nG6#,&%#PiG\"\"\"%&thetaG!\"\"*&F%F)F*F)" }{TEXT -1 85 " (supplementar
y angles have the same sine), we see that another solution is given by
 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta = Pi-arcsi
n(1/2);" "6#/%&thetaG,&%#PiG\"\"\"-%'arcsinG6#*&F'F'\"\"#!\"\"F-" }
{XPPEDIT 18 0 "`` = 5*Pi/6;" "6#/%!G*(\"\"&\"\"\"%#PiGF'\"\"'!\"\"" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 13 "two
 solutions" }{TEXT -1 1 " " }{XPPEDIT 18 0 "theta = Pi/6;" "6#/%&theta
G*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta = 5*P
i/6;" "6#/%&thetaG*(\"\"&\"\"\"%#PiGF'\"\"'!\"\"" }{TEXT -1 41 " are t
he only solutions in the interval [" }{XPPEDIT 18 0 "0,2*Pi;" "6$\"\"!
*&\"\"#\"\"\"%#PiGF&" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 48 
"Since the sine function is periodic with period " }{XPPEDIT 18 0 "2*P
i;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 15 ", the numbers: " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }
{TEXT -1 5 ",    " }{XPPEDIT 18 0 "Pi/6-4*Pi;" "6#,&*&%#PiG\"\"\"\"\"'
!\"\"F&*&\"\"%F&F%F&F(" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "Pi/6-2*Pi;
" "6#,&*&%#PiG\"\"\"\"\"'!\"\"F&*&\"\"#F&F%F&F(" }{TEXT -1 5 ",    " }
{XPPEDIT 18 0 "Pi/6;" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 5 ",    " 
}{XPPEDIT 18 0 "Pi/6+2*Pi;" "6#,&*&%#PiG\"\"\"\"\"'!\"\"F&*&\"\"#F&F%F
&F&" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "Pi/6+4*Pi;" "6#,&*&%#PiG\"\"
\"\"\"'!\"\"F&*&\"\"%F&F%F&F&" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "Pi/
6+6*Pi;" "6#,&*&%#PiG\"\"\"\"\"'!\"\"F&*&F'F&F%F&F&" }{TEXT -1 5 ",   \+
 " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT -1 8 ", etc., " }}
{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT -1 5 ",    " }
{XPPEDIT 18 0 "-23*Pi/6;" "6#,$*(\"#B\"\"\"%#PiGF&\"\"'!\"\"F)" }
{TEXT -1 5 ",    " }{XPPEDIT 18 0 "-11*Pi/6;" "6#,$*(\"#6\"\"\"%#PiGF&
\"\"'!\"\"F)" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "Pi/6;" "6#*&%#PiG\"
\"\"\"\"'!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "13*Pi/6;" "6#*(\"#
8\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "25*Pi/6;
" "6#*(\"#D\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 
0 "37*Pi/6;" "6#*(\"#P\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 5 ",    " }
{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT -1 8 ", etc., " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "are solut
ions, as are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` .
 . . `" "6#%(~.~.~.~G" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "5*Pi/6-4*Pi
;" "6#,&*(\"\"&\"\"\"%#PiGF&\"\"'!\"\"F&*&\"\"%F&F'F&F)" }{TEXT -1 5 "
,    " }{XPPEDIT 18 0 "5*Pi/6-2*Pi;" "6#,&*(\"\"&\"\"\"%#PiGF&\"\"'!\"
\"F&*&\"\"#F&F'F&F)" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "5*Pi/6;" "6#*
(\"\"&\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "5*P
i/6+2*Pi;" "6#,&*(\"\"&\"\"\"%#PiGF&\"\"'!\"\"F&*&\"\"#F&F'F&F&" }
{TEXT -1 5 ",    " }{XPPEDIT 18 0 "5*Pi/6+4*Pi;" "6#,&*(\"\"&\"\"\"%#P
iGF&\"\"'!\"\"F&*&\"\"%F&F'F&F&" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "5
*Pi/6+6*Pi;" "6#,&*(\"\"&\"\"\"%#PiGF&\"\"'!\"\"F&*&F(F&F'F&F&" }
{TEXT -1 5 ",    " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT 
-1 8 ", etc., " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }
{TEXT -1 5 ",    " }{XPPEDIT 18 0 "-19*Pi/6;" "6#,$*(\"#>\"\"\"%#PiGF&
\"\"'!\"\"F)" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "-7*Pi/6;" "6#,$*(\"
\"(\"\"\"%#PiGF&\"\"'!\"\"F)" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "5*Pi
/6;" "6#*(\"\"&\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 
18 0 "17*Pi/6;" "6#*(\"#<\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 5 ",    " 
}{XPPEDIT 18 0 "29*Pi/6;" "6#*(\"#H\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 
5 ",    " }{XPPEDIT 18 0 "41*Pi/6;" "6#*(\"#T\"\"\"%#PiGF%\"\"'!\"\"" 
}{TEXT -1 5 ",    " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT 
-1 7 ", etc. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 36 "In general, any number of the form: " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta = Pi/6+2*k*Pi;" "6#/%&thetaG,&*
&%#PiG\"\"\"\"\"'!\"\"F(*(\"\"#F(%\"kGF(F'F(F(" }{TEXT -1 6 "  or  " }
{XPPEDIT 18 0 "theta = 5*Pi/6+2*k*Pi;" "6#/%&thetaG,&*(\"\"&\"\"\"%#Pi
GF(\"\"'!\"\"F(*(\"\"#F(%\"kGF(F)F(F(" }{XPPEDIT 18 0 "`` = (2*k+1)*Pi
-Pi/6;" "6#/%!G,&*&,&*&\"\"#\"\"\"%\"kGF*F*F*F*F*%#PiGF*F**&F,F*\"\"'!
\"\"F/" }{TEXT -1 3 ",  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 
276 28 "____________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 275 1 "k
" }{TEXT -1 25 " is an integer, that is, " }{XPPEDIT 18 0 "k=` . . . `
,-4,-3,-2,-1,0,1,2,3,4,` . . . `" "6-/%\"kG%(~.~.~.~G,$\"\"%!\"\",$\"
\"$F(,$\"\"#F(,$\"\"\"F(\"\"!F.F,F*F'F%" }{TEXT -1 32 ", is a solution
 of the equation " }{XPPEDIT 18 0 "sin*theta = 1/2;" "6#/*&%$sinG\"\"
\"%&thetaGF&*&F&F&\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 75 "The following picture shows the points of intersection P \+
and Q of the line " }{XPPEDIT 18 0 "y = 1/2;" "6#/%\"yG*&\"\"\"F&\"\"#
!\"\"" }{TEXT -1 22 " with the unit circle " }{XPPEDIT 18 0 "x^2+y^2=1
" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 129 ". All the solu
tions are associated with either P or Q, more precisely, the \"wrappin
g function\" W, which associates a real number " }{XPPEDIT 18 0 "theta
" "6#%&thetaG" }{TEXT -1 16 " with the point " }{XPPEDIT 18 0 "W(theta
)=``(cos*theta,sin*theta)" "6#/-%\"WG6#%&thetaG-%!G6$*&%$cosG\"\"\"F'F
-*&%$sinGF-F'F-" }{TEXT -1 39 ", maps each solution to either P or Q. \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 253 247 247 {PLOTDATA 
2 "62-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4hRPij!**!#=$\"3Ikwb#=y
_O\"F/7$$\"3E8J#))4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\"3-N5')yke[#*F/$\"3gLj&
[K5J!QF/7$$\"3?goz=42`')F/$\"3j9NXR4U7]F/7$$\"3Sb](G._U!zF/$\"3t_H-qce
DhF/7$$\"3_\\$R'oTd#3(F/$\"3!GO#*3qU&fqF/7$$\"3?H$>jubk6'F/$\"3!\\@@&
\\!>8\"zF/7$$\"3VGuIc:x5]F/$\"3-$>p(Qh-a')F/7$$\"37C3nW'R,#QF/$\"3aK+1
6bcT#*F/7$$\"3$oY@iF'QDDF/$\"3s&Q\"[M\"oen*F/7$$\"3OB^hAo8X8F/$\"33)fV
PO<\"4**F/7$$!3+qB/u(p5(f!#@$\"2%HhJ<#)******!#<7$$!33)\\T#fB[i8F/$\"3
ap%>wGZn!**F/7$$!3)e:d.:#=YEF/$\"3PA>=\\D`V'*F/7$$!3%*yV1J*3Jx$F/$\"3I
zF#e`m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'yI2&oN')F/7$$!3[igNw%*[RgF/$\"3!G;
)RQ+BqzF/7$$!3/XUwYjJ*3(F/$\"3AfvOg?x_qF/7$$!3&=e_b?/U!zF/$\"3u3<J%QZc
7'F/7$$!3w!G5)RnIf')F/$\"3D%H$\\4/k,]F/7$$!3/`k8ti$4B*F/$\"3In'[*>HvXQ
F/7$$!3!p!RS4Nij'*F/$\"3o$p9m#Q%=d#F/7$$!3#p(pT>+.2**F/$\"31V?00]Ug8F/
7$$!3/gKG4>&*****F/$\"3vCA\\O%485$!#?7$$!3__Is=!Q%3**F/$!3q#4*>c=8]8F/
7$$!3)>b`sYlSn*F/$!3UNbFGIGKDF/7$$!37_J.8AEj#*F/$!3q4&=j`Bsw$F/7$$!3%
\\F)4Kh=u')F/$!3kENX')3zv\\F/7$$!3EB*z7xng%zF/$!37W(H!pUCrgF/7$$!3XD84
Pfc5rF/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[gF/$!3)4j2yeGL'zF/7$$!3g@UD=%)o!*
\\F/$!3I_Mh6Mil')F/7$$!3o+1s\"[\"ysPF/$!3#\\IN,%***4E*F/7$$!30yCH\"el'
3EF/$!3=V>(fp[Pl*F/7$$!3g67Cvnc\"H\"F/$!3;$RoI*>C;**F/7$$!3EzyOHMQdIFa
s$!3W1O#>E`*****F/7$$\"3GIAj`Ks(G\"F/$!3lYZ3X=u;**F/7$$\"3EKRqX8/bDF/$
!3U$[o%H'z!o'*F/7$$\"3@%)yb&Q)zNQF/$!36.2<#>x]B*F/7$$\"3O7w4w0I.]F/$!3
wHgb5wMe')F/7$$\"3@\\#)[*R]$4hF/$!3/a*[=H2o\"zF/7$$\"33/wY'Q+$*4(F/$!3
)4d)eg?sUqF/7$$\"3[f=s$Ry,!zF/$!3D1$H<bQ38'F/7$$\"3%*R,@;vLu')F/$!3e>)
zD(p_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2ndo*fQF/7$$\"31mGAp))oa'*F/$!3%)f]
nRQ=0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4O\"F/7$F($\"36YKhSr8/#)!#F-%'COL
OURG6&%$RGBG$\"*++++\"!\")F+F+-%*THICKNESSG6#\"\"#-F$6&7$7$F+F+7$$\"3a
+++SSDg')F/$\"3++++++++]F/-Fjz6&F\\[l$\")#)eqkF_[l$\"))eqk\"F_[lFa\\lF
`[l-%*LINESTYLEG6#F)-F$6&7$Fg[l7$$!3a+++SSDg')F/F[\\lF]\\lF`[lFc\\l-F$
6&7$Fi\\lFh[l-Fjz6&F\\[lF*F*F*-Fa[lFe\\l-Fd\\lFb[l-F$6&7$Fh[lFi\\l-Fjz
6&F\\[lF+F][lF][l-%'SYMBOLG6$%&CROSSG\"#5-%&STYLEG6#%&POINTG-F$6&Fe]lF
f]l-Fi]l6$%(DIAMONDGF\\^lF]^l-F$6&Fe]lFf]l-Fi]l6$%'CIRCLEGF\\^lF]^l-F$
6&Fe]lF_]l-Fi]l6$Fj^l\"#7F]^l-%%TEXTG6%7$F($\"#`!\"#Q\"P6\"-%&COLORG6&
F\\[l$F)Ff_lF\\`lF\\`l-Fa_l6%7$$!\"\"F*Fd_lQ\"QFh_lFi_l-Fa_l6%7$$\"$D
\"Ff_l$!\"'Ff_lQ\"xFh_lFi_l-Fa_l6%7$Fh`lFf`lQ\"yFh_lFi_l-%(SCALINGG6#%
,CONSTRAINEDG-%*AXESTICKSG6$F*F*-%+AXESLABELSG6%Q!Fh_lFial-%%FONTG6#%(
DEFAULTG-%%VIEWG6$;$!#6Fa`l$\"#8Fa`l;FbblFf`l" 1 2 0 1 10 0 2 9 1 4 1 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" "Curve 12" }}{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 42 "The \+
following picture shows the graphs of " }{XPPEDIT 18 0 "y=sin*theta" "
6#/%\"yG*&%$sinG\"\"\"%&thetaGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "
y=1/2" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 30 "The solutions of the equation " }{XPPEDIT 18 0 "sin*the
ta=1/2" "6#/*&%$sinG\"\"\"%&thetaGF&*&F&F&\"\"#!\"\"" }{TEXT -1 18 " a
re given by the " }{TEXT 390 1 "x" }{TEXT -1 62 " coordinates of the p
oints of intersection of the two graphs. " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{GLPLOT2D 899 228 228 {PLOTDATA 2 "6eo-%'CURVESG6%7[v7$$!3#)
************\\?!#<$!3K(H-/lWVc\"!#=7$$!3jmTN^+J6?F*$!3q79`K)4Cb$!#>7$$
!3AL$3F5?E(>F*$\"3!HV#p6`/\"f)F37$$!3z*\\iS:IR$>F*$\"3)zN!e8Lxg?F-7$$!
3gmmT0-C&*=F*$\"3)3*))QRQ.KKF-7$$!3*)\\7`qB\"z#=F*$\"3UHSC!)evY^F-7$$!
3=LekNXeg<F*$\"3a[c=UH2KoF-7$$!3!)\\(=PMn[o\"F*$\"3;()*e8&4Og$)F-7$$!3
Vm;z^,:4;F*$\"3$p?H,.=yT*F-7$$!3]m\"zWgS5d\"F*$\"3UcvlSS)>v*F-7$$!3cmm
;d5$H`\"F*$\"3q;!*HH?`Y**F-7$$!3[;/^$GwQ^\"F*$\"3Y=Ho\"[*\\!***F-7$$!3
SmT&)4:#[\\\"F*$\"3;8b6vmn)***F-7$$!3a;z>Onwv9F*$\"3wJ(y`JM5(**F-7$$!3
Ym;ai>rc9F*$\"3JoWFH9n2**F-7$$!3))\\(oR7b3Q\"F*$\"3j'p7By?wI*F-7$$!32L
eR&G)*\\I\"F*$\"3)oYo4?m9=)F-7$$!3o*\\(o&GpYB\"F*$\"3xPVShpVAnF-7$$!3]
m\"zfGSV;\"F*$\"3lCX(*4KdO\\F-7$$!3v*\\(=$[Hz7\"F*$\"3HSc!e$zp6RF-7$$!
3ALeR!o=:4\"F*$\"3aS*zmY&pNGF-7$$!3omTgxy5b5F*$\"3xh8F8&HEs\"F-7$$!3%*
*\\7[2(p=5F*$\"3ckwQ$=#[qeF37$$!3'\\m\"z/(4/\")*F-$!3'=WnovLE&fF37$$!3
nJ$e9m[QV*F-$!3oJ7[:SDp<F-7$$!3Q)*\\7=wGd!*F-$!3*HwFW%H^=HF-7$$!3)Rm\"
zuls!o)F-$!3!ejN37vp-%F-7$$!3a(***\\nJP0$)F-$!3gn\\GDt(e2&F-7$$!35J$3-
w>+$zF-$!3GU=il'zU0'F-7$$!3mkm\"HNmYb(F-$!3QPiA=Hf[pF-7$$!3A)*\\iXHJzr
F-$!3!=P\\%*y&RYxF-7$$!3/k\"HKr=rS'F-$!30s1cNdsQ!*F-7$$!3')HL$3[C\\j&F
-$!3q2*=W[A<!)*F-7$$!3)pC\"y0s)[Y&F-$!37(RMxVQN*)*F-7$$!33k\"H2$*\\[H&
F-$!3)4u#oK#Hr&**F-7$$!36!3xcl7[7&F-$!3g#*)R=X8B***F-7$$!3C(*\\i!QvZ&
\\F-$!3-sYc=2**)***F-7$$!3O9Hd0\"QZy%F-$!3Y/@O$)>9x**F-7$$!3\"4$3_I3q9
YF-$!3T#\\H#y&Ho#**F-7$$!3[Z(oabjYW%F-$!3/--$)Qq>[)*F-7$$!3gkmT!GEYF%F
-$!39tC+&po9u*F-7$$!3?J$ek[R*3NF-$!3`vSee+yA*)F-7$$!3\"y***\\#p_Ku#F-$
!3w@<8VgF!f(F-7$$!3C)**\\PIP)eBF-$!3e(z1LqL1v'F-7$$!3k)****\\\">Uu>F-$
!3X\"[Wo9aE\"eF-7$$!31***\\i_1+f\"F-$!3A/m8C5+!z%F-7$$!3Z****\\P6f07F-
$!3Iu'RYVrvp$F-7$$!3q#******zc8N)F3$!3=iSwQ$fOf#F-7$$!3u!****\\A-ok%F3
$!3Qc^A]jla9F-7$$!3z())****\\wCU*!#?$!3'HWfd+E(fHF37$$\"3=8++DpIiFF3$
\"33$G_$z]:n')F37$$\"3\"G%3x\"Q(REhF3$\"3#y;5\\#Q!G\">F-7$$\"3Ws;aQy[!
\\*F3$\"3_)[SE*oaPHF-7$$\"3?]7`HyX&G\"F-$\"3zco&yX4&HRF-7$$\"3<L$3_(o'
=i\"F-$\"3wTzcp?ix[F-7$$\"3f<zpo(*)=-#F-$\"3[6kev\"[L$fF-7$$\"3,-v=iE
\">U#F-$\"3e<3ER6\\&*oF-7$$\"3U'3xcbN>#GF-$\"3G#yamlv)[xF-7$$\"3%3nm\"
\\%e>A$F-$\"3+&**yb!=/![)F-7$$\"3Kqm\"H<-(**QF-$\"3J!zcxq!\\3%*F-7$$\"
3!)pmm'*eWxXF-$\"3#e\"GY`u,7**F-7$$\"3a6F>k,auZF-$\"3Kdx9Je#\\(**F-7$$
\"3I`(=<VM;(\\F-$\"3-rhuXHg****F-7$$\"3g&zW#*pG(o^F-$\"39A$yj@af)**F-7
$$\"3#o$3xmH#eO&F-$\"38'[G]&>.M**F-7$$\"3K?H#=]6+w&F-$\"3mC&=My3jr*F-7
$$\"3%Q+vo.+U:'F-$\"3Y&GX[8o(\\$*F-7$$\"39O3FHt/_oF-$\"3)\\$R=R1aa$)F-
7$$\"3[omm@Y*)\\vF-$\"3EGS)>ql$fpF-7$$\"38_(oH;>F$zF-$\"3O<m7>z_ZgF-7$
$\"3yN3F/Pa:$)F-$\"3Q:[+*)=K[]F-7$$\"3U>HdX#o$)p)F-$\"3k:>$p2$=wRF-7$$
\"33.](oy#>\"3*F-$\"3#\\yNMg+m%GF-7$$\"3C.DJ!GKdW*F-$\"3S'3$fWs\\K<F-7
$$\"3P.+vt<F5)*F-$\"3O&4$y)Hfp&fF37$$\"3C](=n7\"[<5F*$!3G!\\%)Qs(4*[&F
37$$\"3O+D1w]$R0\"F*$!3#4y5j/Cjo\"F-7$$\"35#HKz%4(>4\"F*$!3o+&pAx<$\\G
F-7$$\"3%Q3-)>o+I6F*$!3)y2LZ`v;(RF-7$$\"3ev=n\"pU!o6F*$!3V2kyj1RP]F-7$
$\"3Jn;aj&yg?\"F*$!3\\`?u1VEJgF-7$$\"39<HKrh$fF\"F*$!3FT.Ecu7BwF-7$$\"
3)p;/\"zPzX8F*$!3eK%\\%=xH\\))F-7$$\"3i3x\"G@pMQ\"F*$!3zot@3KLP$*F-7$$
\"3G]7`YY9@9F*$!3Czi$y)=r%p*F-7$$\"35@!)QjB)*R9F*$!3)*z#\\(p'oF#)*F-7$
$\"3#>zW-3?)e9F*$!3o:!Gj(HV;**F-7$$\"3wi:5(zdwZ\"F*$!3+)4*egoPv**F-7$$
\"3eL$eR^&\\'\\\"F*$!337#=A%RR****F-7$$\"33Y'Gvgig^\"F*$!3%\\cc'[0F()*
*F-7$$\"3ee*)4,(Hc`\"F*$!3)ysU<F>u$**F-7$$\"33r#pYz'>b:F*$!3yi7^H%G+&)
*F-7$$\"3e$eR#))Qwu:F*$!3A\"RY$\\\"Gas*F-7$$\"3d3-Qv!)*Qh\"F*$!35UZ)z
\"3im$*F-7$$\"3eL3_iA.`;F*$!314$)RvQTm))F-7$$\"3#pm;C6l6s\"F*$!3'\\'Qg
y\"\\<o(F-7$$\"3E+DJizH*y\"F*$!3Y^?H9E[YhF-7$$\"3qek`M14E=F*$!3'e')[Lx
Sd>&F-7$$\"3#pTgnI$)G'=F*$!3k:@lR&ec<%F-7$$\"37vV)*yfn**=F*$!3?-n<X&\\
)*4$F-7$$\"3cL$37lok$>F*$!3I(H'=I6n#)>F-7$$\"35]7y\"3zW(>F*$!3![lEizX!
4!)F37$$\"3imTN7&*[7?F*$\"3-*pIGK*oARF37$$\"3Q$3FH%**\\]?F*$\"3!3'Hy:b
&)z:F-7$$\"37++]t.^)3#F*$\"3J'4+N&3%\\u#F-7$$\"3>](=U;'pD@F*$\"3mQN]&*
4.ZQF-7$$\"3C+v$\\&>)G;#F*$\"3Qxr$GQym*[F-7$$\"3I]ilXx1+AF*$\"3E=#RY@u
&zeF-7$$\"3N+]PONDPAF*$\"3#H:Txk>By'F-7$$\"38]7`l=@4BF*$\"3cdo[MK'oD)F
-7$$\"3!**\\(o%>q6Q#F*$\"3-0\\&)p_B6$*F-7$$\"3yu=nA*=6U#F*$\"3-*f%G'p8
Xp*F-7$$\"35]il]w1hCF*$\"3A\"oN&H]HD**F-7$$\"3aP%[Y,U5[#F*$\"3A:;'fIpA
)**F-7$$\"3VD1kyj,,DF*$\"3mwhFA!\\*****F-7$$\"3'G\"GjU2*4_#F*$\"3sR\\T
$ek#y**F-7$$\"3I+]i1^'4a#F*$\"3&>nu([8I<**F-7$$\"3]LeRClv7EF*$\"3;uLIs
47z$*F-7$$\"3qmm;Uza%o#F*$\"3)RhkJIieO)F-7$$\"3oL$3Ug+7w#F*$\"3nJ$3R7F
z\"oF-7$$\"3n++DmK&y$GF*$\"3cpEL%GEl([F-7$$\"3e3xc=EesGF*$\"3MbJMcm)o*
QF-7$$\"3$pT&)3(>J2HF*$\"3[8T&zY/4(GF-7$$\"3GDJ?B8/UHF*$\"3O)oPe!4y5=F
-7$$\"3?L3_v1xwHF*$\"3wYp`F<B\"H(F37$$\"3+D\"G3.RZ,$F*$!3p<%H#G'[(GYF3
7$$\"3#oTNhQ2F0$F*$!3O;*yy#\\H[;F-7$$\"3i3FWTdn!4$F*$!3yo2&[(3H5GF-7$$
\"3W++v'4W'GJF*$!3#=;V\"R%\\B$RF-7$$\"3H$3_Ni%4+KF*$!3A1q7<sD!)eF-7$$
\"3;mTN]^arKF*$!3d)[6F\"Q7LvF-7$$\"3?LeRQ:BYLF*$!3Cx\")*oF%pb))F-7$$\"
3D+vVEz\"4U$F*$!3a@>eCH'Hp*F-7$$\"3K'Rs`#G<RMF*$!3>c'=&p*pz\")*F-7$$\"
3%>H2VsFuX$F*$!3#R4<=&\\p5**F-7$$\"3e(=UKi#ovMF*$!3)HV#G')H$3(**F-7$$
\"3k$3x@_PR\\$F*$!3)yt?bL'=)***F-7$$\"3rz>6@C>7NF*$!3m166g]m#***F-7$$
\"3Mvo/?tWINF*$!3[%yJVJ(Ga**F-7$$\"3'4x\")*=Aq[NF*$!3S@yY'GzJ))*F-7$$
\"3/nm\"z6dpc$F*$!3'=e'*Qyu&z(*F-7$$\"3S$3_v'=SVOF*$!3Ls\\2C/D-!*F-7$$
\"3w*\\(=<m%)>PF*$!3IWXfkI?3xF-7$$\"3AeR(pIfmv$F*$!3>?i;;')f@pF-7$$\"3
o;/w'*>Z$z$F*$!3&Q7W&*p?D/'F-7$$\"3:voa'o%GIQF*$!3?/bN6Rr#3&F-7$$\"3iL
LLwt4nQF*$!3;@Ld'R,]0%F-7$$\"3b$e9'oOu/RF*$!3?4P:#>*4[HF-7$$\"3\\Le*3'
**QURF*$!3^Y?64\"3+!=F-7$$\"3W$3xJDO+)RF*$!3q#px[6&oniF37$$\"3PL$eaa#o
<SF*$\"3S#y;QyyAb&F37$$\"3#=HKu5<]0%F*$\"3e7i@V/#)><F-7$$\"3G]iSp;N#4%
F*$\"3E[%37l!ygGF-7$$\"3u3-QJioHTF*$\"3ueOPm.ViRF-7$$\"3?nTN$z?q;%F*$
\"3UX1;D;j4]F-7$$\"3(Q3FM'RjNUF*$\"3H8huvE%[u'F-7$$\"3a++]LrC/VF*$\"3]
H#[\"zU(y;)F-7$$\"3_LeRpb)GQ%F*$\"3s3(op*QvI$*F-7$$\"3Qn;H0S_hWF*$\"3#
>hR#))R.F**F-7$$\"3'[(=#Rz2\"zWF*$\"3mn'*)e4o%y**F-7$$\"3C$3_De\"p'\\%
F*$\"3)GSp)f)f%****F-7$$\"3h\"H#=r`F9XF*$\"3%fTN%Q_%**)**F-7$$\"3)**\\
7)f\"f=`%F*$\"3MQk#)eK&*\\**F-7$$\"3u;H2Pn-nXF*$\"3#)3d%R'y6z(*F-7$$\"
3[LLL9V>-YF*$\"33H$ef#o.*[*F-7$$\"3S</,fi=xYF*$\"3oFn/hQJ!\\)F-7$$\"3W
+vo.#y@v%F*$\"3\"4:i:6:D-(F-7$$\"3_]PfLm1)y%F*$\"35'=jQ,]p<'F-7$$\"3f+
+]j]&R#[F*$\"3!yOBZhZHD&F-7$$\"3m]iS$\\V)f[F*$\"3gL)[TDTAE%F-7$$\"3s+D
JB>t&*[F*$\"313!f[I6u@$F-7$$\"31vV[U*)HM\\F*$\"3!easYyM%\\?F-7$$\"3G]i
lhf'G(\\F*$\"3b)[WHX*39&)F37$$\"3\\D\"G3)HV6]F*$!3)G1vSi/5f$F37$$\"3#)
************\\]F*$!3q$H-/lWVc\"F--%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"
!FhbnFgbn-%*THICKNESSG6#\"\"#-F$6%7S7$F($\"3++++++++]F-7$F?Facn7$FIFac
n7$FSFacn7$F[pFacn7$FepFacn7$F_qFacn7$FcrFacn7$FgsFacn7$F[uFacn7$FeuFa
cn7$F]xFacn7$FgxFacn7$F[zFacn7$F`[lFacn7$Fd\\lFacn7$Fh]lFacn7$Fb^lFacn
7$F``lFacn7$Fj`lFacn7$F^blFacn7$FbclFacn7$FfdlFacn7$F`elFacn7$F^glFacn
7$F\\ilFacn7$FfilFacn7$FjjlFacn7$F^\\mFacn7$Fb]mFacn7$F\\^mFacn7$Fj_mF
acn7$Fd`mFacn7$F^amFacn7$FbbmFacn7$FfcmFacn7$F`dmFacn7$FjdmFacn7$FbgmF
acn7$F\\hmFacn7$F`imFacn7$FdjmFacn7$Fh[nFacn7$Fb\\nFacn7$F\\]nFacn7$Fj
^nFacn7$Fd_nFacn7$Fh`nFacn7$F\\bnFacn-Fabn6&FcbnFgbnFgbnFdbnFibn-F$6%7
$7$$!3ELLLLLLL=F*Fgbn7$FifnFacn-Fabn6&Fcbn$\")!\\DP\"FfbnF^gn$\")viobF
fbn-%*LINESTYLEGF[cn-F$6%7$7$$\"3emmmmmmm;F-Fgbn7$FhgnFacnF\\gnFbgn-F$
6%7$7$$\"3_mmmmmmm@F*Fgbn7$F_hnFacnF\\gnFbgn-F$6%7$7$$\"3'pmmmmmm;%F*F
gbn7$FfhnFacnF\\gnFbgn-F$6%7$7$$!3ummmmmmm6F*Fgbn7$F]inFacnF\\gnFbgn-F
$6%7$7$$\"3qLLLLLLL$)F-Fgbn7$FdinFacnF\\gnFbgn-F$6%7$7$$\"3[LLLLLLLGF*
Fgbn7$F[jnFacnF\\gnFbgn-F$6%7$7$$\"3.LLLLLLL[F*Fgbn7$FbjnFacnF\\gnFbgn
-F$6&7#F[gn-Fabn6&FcbnFgbnFdbnFgbn-%'SYMBOLG6$%'CIRCLEG\"#5-%&STYLEG6#
%&POINTG-F$6&7#FjgnFhjnFjjnF_[o-F$6&7#FahnFhjnFjjnF_[o-F$6&7#FhhnFhjnF
jjnF_[o-F$6&7#F_inFhjnFjjnF_[o-F$6&7#FfinFhjnFjjnF_[o-F$6&7#F]jnFhjnFj
jnF_[o-F$6&7#FdjnFhjnFjjnF_[o-F$6&FgjnFhjn-F[[o6$%(DIAMONDGF^[oF_[o-F$
6&Fe[oFhjnFj\\oF_[o-F$6&Fh[oFhjnFj\\oF_[o-F$6&F[\\oFhjnFj\\oF_[o-F$6&F
^\\oFhjnFj\\oF_[o-F$6&Fa\\oFhjnFj\\oF_[o-F$6&Fd\\oFhjnFj\\oF_[o-F$6&Fg
\\oFhjnFj\\oF_[o-F$6&FgjnFhjn-F[[o6$%&CROSSG\"\")F_[o-F$6&Fe[oFhjnF]^o
F_[o-F$6&Fh[oFhjnF]^oF_[o-F$6&F[\\oFhjnF]^oF_[o-F$6&F^\\oFhjnF]^oF_[o-
F$6&Fa\\oFhjnF]^oF_[o-F$6&Fd\\oFhjnF]^oF_[o-F$6&Fg\\oFhjnF]^oF_[o-F$6&
Fgjn-Fabn6&FcbnFhbnFhbnFhbn-F[[o6$F][o\"#7F_[o-F$6&Fe[oFa_oFc_oF_[o-F$
6&Fh[oFa_oFc_oF_[o-F$6&F[\\oFa_oFc_oF_[o-F$6&F^\\oFa_oFc_oF_[o-F$6&Fa
\\oFa_oFc_oF_[o-F$6&Fd\\oFa_oFc_oF_[o-F$6&Fg\\oFa_oFc_oF_[o-F$6$7$7$Fi
fn$!33+++++++^F-FhfnFa_o-F$6$7$7$FhgnFh`oFggnFa_o-F$6$7$7$F_hnFh`oF^hn
Fa_o-F$6$7$7$FfhnFh`oFehnFa_o-F$6$7$7$F]inFh`oF\\inFa_o-F$6$7$7$FdinFh
`oFcinFa_o-F$6$7$7$F[jnFh`oFjinFa_o-F$6$7$7$FbjnFh`oFajnFa_o-%)POLYGON
SG6+7%7$$!+LLLt=!\"*$!\"\"F_co7$$!+LLLL=F]coFgbn7$$!+LLL$z\"F]coF^co7%
7$$\"+nmmm7!#5F^co7$$\"+nmmm;FjcoFgbn7$$\"+nmmm?FjcoF^co7%7$$\"+nmmE@F
]coF^co7$$\"+nmmm@F]coFgbn7$$\"+nmm1AF]coF^co7%7$$\"+nmmETF]coF^co7$$
\"+nmmmTF]coFgbn7$$\"+nmm1UF]coF^co7%7$$!+nmm17F]coF^co7$$!+nmmm6F]coF
gbn7$$!+nmmE6F]coF^co7%7$$\"+LLLLzFjcoF^co7$$\"+LLLL$)FjcoFgbn7$$\"+LL
LL()FjcoF^co7%7$$\"+LLL$z#F]coF^co7$$\"+LLLLGF]coFgbn7$$\"+LLLtGF]coF^
co7%7$$\"+LLL$z%F]coF^co7$$\"+LLLL[F]coFgbn7$$\"+LLLt[F]coF^co-Fabn6&F
cbnFdbn$\")AR!)\\FfbnFgbn-F$6$7U7$$!+LLLj:F]co$!\"(F_co7$$!+$R5\\c\"F]
co$!+LNL\\nFjco7$$!+,nhp:F]co$!+D-i-lFjco7$$!+O!ytd\"F]co$!+&*3vjiFjco
7$$!+(*>2)e\"F]co$!+_E\\OgFjco7$$!+M*H:g\"F]co$!+&\\HW#eFjco7$$!+3'Rvh
\"F]co$!+)y04j&Fjco7$$!+N&[ej\"F]co$!+9N(*eaFjco7$$!+uz;c;F]co$!+\\TM6
`Fjco7$$!+vu<y;F]co$!+&*eM!>&Fjco7$$!+M*H:q\"F]co$!+np)y4&Fjco7$$!+qq&
es\"F]co$!+)\\Da.&Fjco7$$!+H_x]<F]co$!+Vl%R+&Fjco7$$!+P9*ex\"F]coFc\\p
7$$!+'f43!=F]co$!+*\\Da.&Fjco7$$!+Kn8D=F]coFi[p7$$!+\">*[[=F]coFd[p7$$
!+#p)\\q=F]coF_[p7$$!+J\"=3*=F]coFjjo7$$!+fq74>F]co$!+!z04j&Fjco7$$!+K
n8D>F]co$!+(\\HW#eFjco7$$!+pYfQ>F]co$!+aE\\OgFjco7$$!+I')G\\>F]co$!+'*
3vjiFjco7$$!+l*\\q&>F]co$!+F-i-lFjco7$$!+tivh>F]co$!+MNL\\nFjco7$$!+LL
Lj>F]co$!+,+++qFjco7$Fc_p$!+okm]sFjco7$F^_p$!+v(zt\\(Fjco7$Fi^p$!+0\"
\\it(Fjco7$Fd^p$!+[t]jzFjco7$F_^p$!+/0dv\")Fjco7$$!+eq74>F]co$!+6U4p$)
Fjco7$Fg]p$!+([E5a)Fjco7$Fd]p$!+_el)o)Fjco7$Fa]p$!+1Tl4))Fjco7$F^]p$!+
LI6-*)Fjco7$Fi\\p$!+-Xdk*)Fjco7$Ff\\p$!+dM0'**)Fjco7$Fa\\pF`bp7$F\\\\p
$!+,Xdk*)Fjco7$Fg[p$!+KI6-*)Fjco7$Fb[p$!+0Tl4))Fjco7$F][p$!+^el)o)Fjco
7$Fhjo$!+'[E5a)Fjco7$$!+2'Rvh\"F]co$!+4U4p$)Fjco7$F^jo$!+-0dv\")Fjco7$
Fiio$!+Yt]jzFjco7$Fdio$!+.\"\\it(Fjco7$F_io$!+s(zt\\(Fjco7$Fjho$!+lkm]
sFjco7$Feho$!+)*******pFjcoFa_o-F$6$7U7$$\"+nmmmVFjcoFgho7$$\"+qg*3N%F
jcoF\\io7$$\"+*)H$QI%FjcoFaio7$$\"+R'>iA%FjcoFfio7$$\"+F+G>TFjcoF[jo7$
$\"+c1q%)RFjcoF`jo7$$\"+ARgCQFjcoFejo7$$\"+YY^TOFjcoFjjo7$$\"+d-KQMFjc
oF_[p7$$\"+^_A=KFjcoFd[p7$$\"+b1q%)HFjcoFi[p7$$\"+&HH9u#FjcoF^\\p7$$\"
+1xC#\\#FjcoFc\\p7$$\"+Gc3TAFjcoFc\\p7$$\"+QS!>*>FjcoF[]p7$$\"+zEj[<Fj
coFi[p7$$\"+%33^^\"FjcoFd[p7$$\"+yI,&H\"FjcoF_[p7$$\"+)o==4\"FjcoFjjo7
$$\"*5%H(3*FjcoF\\^p7$$\"*xEj[(FjcoFa^p7$$\"*1L09'FjcoFf^p7$$\"*&p8r]F
jcoF[_p7$$\"*W.]H%FjcoF`_p7$$\"*ksV#QFjcoFe_p7$$\"*nmmm$FjcoFj_p7$Feip
F]`p7$$\"*X.]H%FjcoF``p7$F_ipFc`p7$$\"*2L09'FjcoFf`p7$$\"*yEj[(FjcoFi`
p7$$\"*7%H(3*FjcoF^ap7$$\"+!p==4\"FjcoFaap7$$\"+zI,&H\"FjcoFdap7$$\"+'
33^^\"FjcoFgap7$$\"+!oK'[<FjcoFjap7$$\"+RS!>*>FjcoF]bp7$$\"+Hc3TAFjcoF
`bp7$$\"+2xC#\\#FjcoF`bp7$$\"+(HH9u#FjcoFdbp7$$\"+c1q%)HFjcoFgbp7$$\"+
]_A=KFjcoFjbp7$FefpF]cp7$FbfpF`cp7$$\"+CRgCQFjcoFecp7$$\"+e1q%)RFjcoFh
cp7$$\"+G+G>TFjcoF[dp7$$\"+S'>iA%FjcoF^dp7$$\"+!*H$QI%FjcoFadp7$F`epFd
dp7$F]epFgdpFa_o-F$6$7U7$$\"+nmmOCF]coFgho7$$\"+2'*3NCF]coF\\io7$$\"+*
H$QICF]coFaio7$$\"+k>iACF]coFfio7$$\"+.!G>T#F]coF[jo7$$\"+m+Z)R#F]coF`
jo7$$\"+#RgCQ#F]coFejo7$$\"+l9:kBF]coFjjo7$$\"+E?$QM#F]coF_[p7$$\"+DD#
=K#F]coFd[p7$$\"+m+Z)H#F]coFi[p7$$\"+IH9uAF]coF^\\p7$$\"+rZA\\AF]coFc
\\p7$$\"+j&3TA#F]coFc\\p7$$\"+//>*>#F]coF[]p7$$\"+oK'[<#F]coFi[p7$$\"+
43^^@F]coFd[p7$$\"+38]H@F]coF_[p7$$\"+p==4@F]coFjjo7$$\"+TH(34#F]coF\\
^p7$$\"+oK'[2#F]coFa^p7$$\"+J`Sh?F]coFf^p7$$\"+q8r]?F]coF[_p7$$\"+N+&H
/#F]coF`_p7$$\"+FPCQ?F]coFe_p7$$\"+nmmO?F]coFj_p7$FebqF]`p7$FbbqF``p7$
F_bqFc`p7$F\\bqFf`p7$FiaqFi`p7$$\"+UH(34#F]coF^ap7$FcaqFaap7$F`aqFdap7
$F]aqFgap7$Fj`qFjap7$Fg`qF]bp7$Fd`qF`bp7$Fa`qF`bp7$F^`qFdbp7$F[`qFgbp7
$Fh_qFjbp7$Fe_qF]cp7$Fb_qF`cp7$$\"+$RgCQ#F]coFecp7$F\\_qFhcp7$Fi^qF[dp
7$Ff^qF^dp7$Fc^qFadp7$F`^qFddp7$F]^qFgdpFa_o-F$6$7U7$$\"+nmmOWF]coFgho
7$$\"+2'*3NWF]coF\\io7$$\"+*H$QIWF]coFaio7$$\"+k>iAWF]coFfio7$$\"+.!G>
T%F]coF[jo7$$\"+m+Z)R%F]coF`jo7$$\"+#RgCQ%F]coFejo7$$\"+l9:kVF]coFjjo7
$$\"+E?$QM%F]coF_[p7$$\"+DD#=K%F]coFd[p7$$\"+m+Z)H%F]coFi[p7$$\"+IH9uU
F]coF^\\p7$$\"+rZA\\UF]coFc\\p7$$\"+j&3TA%F]coFc\\p7$$\"+//>*>%F]coF[]
p7$$\"+oK'[<%F]coFi[p7$$\"+43^^TF]coFd[p7$$\"+38]HTF]coF_[p7$$\"+p==4T
F]coFjjo7$$\"+TH(34%F]coF\\^p7$$\"+oK'[2%F]coFa^p7$$\"+J`ShSF]coFf^p7$
$\"+q8r]SF]coF[_p7$$\"+N+&H/%F]coF`_p7$$\"+FPCQSF]coFe_p7$$\"+nmmOSF]c
oFj_p7$FciqF]`p7$F`iqF``p7$F]iqFc`p7$FjhqFf`p7$FghqFi`p7$$\"+UH(34%F]c
oF^ap7$FahqFaap7$F^hqFdap7$F[hqFgap7$FhgqFjap7$FegqF]bp7$FbgqF`bp7$F_g
qF`bp7$F\\gqFdbp7$FifqFgbp7$FffqFjbp7$FcfqF]cp7$F`fqF`cp7$$\"+$RgCQ%F]
coFecp7$FjeqFhcp7$FgeqF[dp7$FdeqF^dp7$FaeqFadp7$F^eqFddp7$F[eqFgdpFa_o
-F$6$7U7$$!+nmmO5F]coFgho7$$!+FPCQ5F]coF\\io7$$!+N+&H/\"F]coFaio7$$!+q
8r]5F]coFfio7$$!+J`Sh5F]coF[jo7$$!+oK'[2\"F]coF`jo7$$!+UH(34\"F]coFejo
7$$!+p==46F]coFjjo7$$!+38]H6F]coF_[p7$$!+43^^6F]coFd[p7$$!+oK'[<\"F]co
Fi[p7$$!+//>*>\"F]coF^\\p7$$!+j&3TA\"F]coFc\\p7$$!+rZA\\7F]coFc\\p7$$!
+IH9u7F]coF[]p7$$!+m+Z)H\"F]coFi[p7$$!+DD#=K\"F]coFd[p7$$!+E?$QM\"F]co
F_[p7$$!+l9:k8F]coFjjo7$$!+$RgCQ\"F]coF\\^p7$$!+m+Z)R\"F]coFa^p7$$!+.!
G>T\"F]coFf^p7$$!+k>iA9F]coF[_p7$$!+*H$QI9F]coF`_p7$$!+2'*3N9F]coFe_p7
$$!+nmmO9F]coFj_p7$Fa`rF]`p7$F^`rF``p7$F[`rFc`p7$Fh_rFf`p7$Fe_rFi`p7$$
!+#RgCQ\"F]coF^ap7$F__rFaap7$F\\_rFdap7$Fi^rFgap7$Ff^rFjap7$Fc^rF]bp7$
F`^rF`bp7$F]^rF`bp7$Fj]rFdbp7$Fg]rFgbp7$Fd]rFjbp7$Fa]rF]cp7$F^]rF`cp7$
$!+TH(34\"F]coFecp7$Fh\\rFhcp7$Fe\\rF[dp7$Fb\\rF^dp7$F_\\rFadp7$F\\\\r
Fddp7$Fi[rFgdpFa_o-F$6$7U7$$\"+LLLL'*FjcoFgho7$$\"+OFc<'*FjcoF\\io7$$
\"+b'*\\q&*FjcoFaio7$$\"+0j)G\\*FjcoFfio7$$\"+$pYfQ*FjcoF[jo7$$\"+AtO^
#*FjcoF`jo7$$\"+)eq74*FjcoFejo7$$\"+78=3*)FjcoFjjo7$$\"+Bp)\\q)FjcoF_[
p7$$\"+<>*[[)FjcoFd[p7$$\"+@tO^#)FjcoFi[p7$$\"+hf43!)FjcoF^\\p7$$\"+sV
\"*exFjcoFc\\p7$$\"+%H_x](FjcoFc\\p7$$\"+/2desFjcoF[]p7$$\"+X$*H:qFjco
Fi[p7$$\"+]Zx\"y'FjcoFd[p7$$\"+W(z;c'FjcoF_[p7$$\"+a`[ejFjcoFjjo7$$\"+
wgRvhFjcoF\\^p7$$\"+V$*H:gFjcoFa^p7$$\"+s*>2)eFjcoFf^p7$$\"+h.ytdFjcoF
[_p7$$\"+5q;'p&FjcoF`_p7$$\"+IR5\\cFjcoFe_p7$$\"+LLLLcFjcoFj_p7$F_grF]
`p7$$\"+6q;'p&FjcoF``p7$FifrFc`p7$$\"+t*>2)eFjcoFf`p7$$\"+W$*H:gFjcoFi
`p7$$\"+ygRvhFjcoF^ap7$$\"+c`[ejFjcoFaap7$$\"+X(z;c'FjcoFdap7$$\"+_Zx
\"y'FjcoFgap7$$\"+Y$*H:qFjcoFjap7$$\"+02desFjcoF]bp7$$\"+&H_x](FjcoF`b
p7$$\"+tV\"*exFjcoF`bp7$$\"+jf43!)FjcoFdbp7$$\"+AtO^#)FjcoFgbp7$$\"+;>
*[[)FjcoFjbp7$F_drF]cp7$F\\drF`cp7$$\"+!fq74*FjcoFecp7$$\"+CtO^#*FjcoF
hcp7$$\"+%pYfQ*FjcoF[dp7$$\"+1j)G\\*FjcoF^dp7$$\"+c'*\\q&*FjcoFadp7$Fj
brFddp7$FgbrFgdpFa_o-F$6$7U7$$\"+LLLjHF]coFgho7$$\"+tivhHF]coF\\io7$$
\"+l*\\q&HF]coFaio7$$\"+I')G\\HF]coFfio7$$\"+pYfQHF]coF[jo7$$\"+Kn8DHF
]coF`jo7$$\"+eq74HF]coFejo7$$\"+J\"=3*GF]coFjjo7$$\"+#p)\\qGF]coF_[p7$
$\"+\">*[[GF]coFd[p7$$\"+Kn8DGF]coFi[p7$$\"+'f43!GF]coF^\\p7$$\"+P9*ex
#F]coFc\\p7$$\"+H_x]FF]coFc\\p7$$\"+qq&es#F]coF[]p7$$\"+M*H:q#F]coFi[p
7$$\"+vu<yEF]coFd[p7$$\"+uz;cEF]coF_[p7$$\"+N&[ej#F]coFjjo7$$\"+2'Rvh#
F]coF\\^p7$$\"+M*H:g#F]coFa^p7$$\"+(*>2)e#F]coFf^p7$$\"+O!ytd#F]coF[_p
7$$\"+,nhpDF]coF`_p7$$\"+$R5\\c#F]coFe_p7$$\"+LLLjDF]coFj_p7$F_`sF]`p7
$F\\`sF``p7$Fi_sFc`p7$Ff_sFf`p7$Fc_sFi`p7$$\"+3'Rvh#F]coF^ap7$F]_sFaap
7$Fj^sFdap7$Fg^sFgap7$Fd^sFjap7$Fa^sF]bp7$F^^sF`bp7$F[^sF`bp7$Fh]sFdbp
7$Fe]sFgbp7$Fb]sFjbp7$F_]sF]cp7$F\\]sF`cp7$$\"+fq74HF]coFecp7$Ff\\sFhc
p7$Fc\\sF[dp7$F`\\sF^dp7$F]\\sFadp7$Fj[sFddp7$Fg[sFgdpFa_o-F$6$7U7$$\"
+LLLj\\F]coFgho7$$\"+tivh\\F]coF\\io7$$\"+l*\\q&\\F]coFaio7$$\"+I')G\\
\\F]coFfio7$$\"+pYfQ\\F]coF[jo7$$\"+Kn8D\\F]coF`jo7$$\"+eq74\\F]coFejo
7$$\"+J\"=3*[F]coFjjo7$$\"+#p)\\q[F]coF_[p7$$\"+\">*[[[F]coFd[p7$$\"+K
n8D[F]coFi[p7$$\"+'f43![F]coF^\\p7$$\"+P9*ex%F]coFc\\p7$$\"+H_x]ZF]coF
c\\p7$$\"+qq&es%F]coF[]p7$$\"+M*H:q%F]coFi[p7$$\"+vu<yYF]coFd[p7$$\"+u
z;cYF]coF_[p7$$\"+N&[ej%F]coFjjo7$$\"+2'Rvh%F]coF\\^p7$$\"+M*H:g%F]coF
a^p7$$\"+(*>2)e%F]coFf^p7$$\"+O!ytd%F]coF[_p7$$\"+,nhpXF]coF`_p7$$\"+$
R5\\c%F]coFe_p7$$\"+LLLjXF]coFj_p7$F]gsF]`p7$FjfsF``p7$FgfsFc`p7$FdfsF
f`p7$FafsFi`p7$$\"+3'Rvh%F]coF^ap7$F[fsFaap7$FhesFdap7$FeesFgap7$FbesF
jap7$F_esF]bp7$F\\esF`bp7$FidsF`bp7$FfdsFdbp7$FcdsFgbp7$F`dsFjbp7$F]ds
F]cp7$FjcsF`cp7$$\"+fq74\\F]coFecp7$FdcsFhcp7$FacsF[dp7$F^csF^dp7$F[cs
Fadp7$FhbsFddp7$FebsFgdpFa_o-%%TEXTG6&7$$Ffbn!\"#$\"$H\"FdisQ\"y6\"-%&
COLORG6&Fcbn$\"\"\"FdisF\\jsF\\js-%%FONTG6$%*HELVETICAGF^[o-F`is6&7$$
\"#_F_co$!#8FdisQ\"qFhisFiis-F_js6$F[[o\"#6-F`is6&7$$!+LLLj<F]coFghoQ'
-11p/6FhisFiis-F_js6$F[[oF^[o-F`is6&7$$\"+nmmOAF]coFghoQ&13p/6FhisFiis
Fc[t-F`is6&7$$\"+nmmOUF]coFghoQ&25p/6FhisFiisFc[t-F`is6&7$$!+nmmO7F]co
FghoQ&-7p/6FhisFiisFc[t-F`is6&7$$\"+LLLjFF]coFghoQ&17p/6FhisFiisFc[t-F
`is6&7$$\"+LLLjZF]coFghoQ&29p/6FhisFiisFc[t-F`is6&7$$\"+nmmmBFjcoFghoQ
$p/6FhisFiisFc[t-F`is6&7$$\"+LLLLwFjcoFghoQ%5p/6FhisFiisFc[t-%*AXESTIC
KSG6$71/Fdis%$-2pG/$!#:F_co%&-3p/2G/F_co%#-pG/$!\"&F_co%%-p/2G/Fhbn%\"
0G/$\"\"&F_co%$p/2G/F]js%\"pG/$\"#:F_co%%3p/2G/F\\cn%#2pG/$\"#DF_co%%5
p/2G/\"\"$%#3pG/$\"#NF_co%%7p/2G/\"\"%%#4pG/$\"#XF_co%%9p/2G/Fc_t%#5pG
Fb`t-F_js6$F[[o\"\"*-%+AXESLABELSG6%%!GFgat-F_js6#%(DEFAULTG-%%VIEWG6$
;$!$0#FdisFejs;$!$K\"FdisFeis" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur
ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1
8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "
Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curv
e 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37
" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "C
urve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve
 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56
" "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "C
urve 63" "Curve 64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve
 69" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 
12 "solution set" }{TEXT -1 17 " of the equation " }{XPPEDIT 18 0 "sin
*theta = 1/2;" "6#/*&%$sinG\"\"\"%&thetaGF&*&F&F&\"\"#!\"\"" }{TEXT 
-1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 2 " \{" }{XPPEDIT 18 0 "Pi/6
+2*k*Pi;" "6#,&*&%#PiG\"\"\"\"\"'!\"\"F&*(\"\"#F&%\"kGF&F%F&F&" }
{TEXT -1 2 " |" }{TEXT 291 2 " k" }{TEXT -1 15 " is an integer\}" }
{XPPEDIT 18 0 "``union``" "6#-%&unionG6$%!GF&" }{TEXT -1 1 "\{" }
{XPPEDIT 18 0 "5*Pi/6+2*k*Pi;" "6#,&*(\"\"&\"\"\"%#PiGF&\"\"'!\"\"F&*(
\"\"#F&%\"kGF&F'F&F&" }{TEXT -1 1 "|" }{TEXT 351 2 " k" }{TEXT -1 17 "
 is an integer\}. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "E
xample 2 " }}{PARA 0 "" 0 "" {TEXT 271 8 "Question" }{TEXT -1 2 ": " }
}{PARA 0 "" 0 "" {TEXT -1 42 "Find the general solution of the equatio
n " }{XPPEDIT 18 0 "sin*theta = 3/5;" "6#/*&%$sinG\"\"\"%&thetaGF&*&\"
\"$F&\"\"&!\"\"" }{TEXT -1 81 ", giving the solution in terms of an ap
propriate inverse trigonometric function. " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 8 "Solution" }{TEXT -1 2 ": " }}
{PARA 0 "" 0 "" {TEXT -1 29 "One solution of the equation " }{XPPEDIT 
18 0 "sin*theta = 3/5;" "6#/*&%$sinG\"\"\"%&thetaGF&*&\"\"$F&\"\"&!\"
\"" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "theta = arcsin(3/5);" "6#/%&the
taG-%'arcsinG6#*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }{TEXT 293 1 "
~" }{TEXT -1 15 " 0.6435011088. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since \+
" }{XPPEDIT 18 0 "sin(Pi-theta)=sin*theta" "6#/-%$sinG6#,&%#PiG\"\"\"%
&thetaG!\"\"*&F%F)F*F)" }{TEXT -1 85 ", (supplementary angles have the
 same sine) we see that another solution is given by " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta = Pi-arcsin(3/5);" "6#/%&the
taG,&%#PiG\"\"\"-%'arcsinG6#*&\"\"$F'\"\"&!\"\"F." }{TEXT -1 1 " " }
{TEXT 355 1 "~" }{TEXT -1 14 " 2.498091545. " }}{PARA 0 "" 0 "" {TEXT 
-1 4 "The " }{TEXT 259 13 "two solutions" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "theta = arcsin(3/5);" "6#/%&thetaG-%'arcsinG6#*&\"\"$\"\"\"\"\"&
!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta = Pi-arcsin(3/5);" "6#/%
&thetaG,&%#PiG\"\"\"-%'arcsinG6#*&\"\"$F'\"\"&!\"\"F." }{TEXT -1 41 " \+
are the only solutions in the interval [" }{XPPEDIT 18 0 "0,2*Pi;" "6$
\"\"!*&\"\"#\"\"\"%#PiGF&" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT 
-1 48 "Since the sine function is periodic with period " }{XPPEDIT 18 
0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 15 ", the numbers: " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~
.~G" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "arcsin(3/5)-4*Pi;" "6#,&-%'arcs
inG6#*&\"\"$\"\"\"\"\"&!\"\"F)*&\"\"%F)%#PiGF)F+" }{TEXT -1 3 ",  " }
{XPPEDIT 18 0 "arcsin(3/5)-2*Pi;" "6#,&-%'arcsinG6#*&\"\"$\"\"\"\"\"&!
\"\"F)*&\"\"#F)%#PiGF)F+" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "arcsin(3/5
);" "6#-%'arcsinG6#*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT -1 3 ",  " }
{XPPEDIT 18 0 "arcsin(3/5)+2*Pi;" "6#,&-%'arcsinG6#*&\"\"$\"\"\"\"\"&!
\"\"F)*&\"\"#F)%#PiGF)F)" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "arcsin(3/5
)+4*Pi;" "6#,&-%'arcsinG6#*&\"\"$\"\"\"\"\"&!\"\"F)*&\"\"%F)%#PiGF)F)
" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "arcsin(3/5)+6*Pi;" "6#,&-%'arcsinG
6#*&\"\"$\"\"\"\"\"&!\"\"F)*&\"\"'F)%#PiGF)F)" }{TEXT -1 3 ",  " }
{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT -1 8 ", etc., " }}
{PARA 0 "" 0 "" {TEXT -1 23 "are solutions, as are: " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT 
-1 3 ",  " }{XPPEDIT 18 0 "Pi-arcsin(3/5)-4*Pi;" "6#,(%#PiG\"\"\"-%'ar
csinG6#*&\"\"$F%\"\"&!\"\"F,*&\"\"%F%F$F%F," }{TEXT -1 3 ",  " }
{XPPEDIT 18 0 "Pi-arcsin(3/5)-2*Pi;" "6#,(%#PiG\"\"\"-%'arcsinG6#*&\"
\"$F%\"\"&!\"\"F,*&\"\"#F%F$F%F," }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "Pi
-arcsin(3/5);" "6#,&%#PiG\"\"\"-%'arcsinG6#*&\"\"$F%\"\"&!\"\"F," }
{TEXT -1 3 ",  " }{XPPEDIT 18 0 "Pi-arcsin(3/5)+2*Pi;" "6#,(%#PiG\"\"
\"-%'arcsinG6#*&\"\"$F%\"\"&!\"\"F,*&\"\"#F%F$F%F%" }{TEXT -1 3 ",  " 
}{XPPEDIT 18 0 "Pi-arcsin(3/5)+4*Pi;" "6#,(%#PiG\"\"\"-%'arcsinG6#*&\"
\"$F%\"\"&!\"\"F,*&\"\"%F%F$F%F%" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "Pi
-arcsin(3/5)+6*Pi;" "6#,(%#PiG\"\"\"-%'arcsinG6#*&\"\"$F%\"\"&!\"\"F,*
&\"\"'F%F$F%F%" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.
~.~G" }{TEXT -1 8 ", etc., " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` . . . `" "6#%(~.~.
~.~G" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "-arcsin(3/5)-5*Pi;" "6#,&-%'ar
csinG6#*&\"\"$\"\"\"\"\"&!\"\"F+*&F*F)%#PiGF)F+" }{TEXT -1 3 ",  " }
{XPPEDIT 18 0 "-arcsin(3/5)-3*Pi;" "6#,&-%'arcsinG6#*&\"\"$\"\"\"\"\"&
!\"\"F+*&F(F)%#PiGF)F+" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "-arcsin(3/5)
+Pi;" "6#,&-%'arcsinG6#*&\"\"$\"\"\"\"\"&!\"\"F+%#PiGF)" }{TEXT -1 3 "
,  " }{XPPEDIT 18 0 "-arcsin(3/5)+3*Pi;" "6#,&-%'arcsinG6#*&\"\"$\"\"
\"\"\"&!\"\"F+*&F(F)%#PiGF)F)" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "-arcs
in(3/5)+5*Pi;" "6#,&-%'arcsinG6#*&\"\"$\"\"\"\"\"&!\"\"F+*&F*F)%#PiGF)
F)" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "-arcsin(3/5)+7*Pi;" "6#,&-%'arcs
inG6#*&\"\"$\"\"\"\"\"&!\"\"F+*&\"\"(F)%#PiGF)F)" }{TEXT -1 3 ",  " }
{XPPEDIT 18 0 "` . . . `" "6#%(~.~.~.~G" }{TEXT -1 8 ", etc., " }}
{PARA 257 "" 0 "" {TEXT -1 36 "In general, any number of the form: " }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta = arcsin(3/5)+
2*k*Pi;" "6#/%&thetaG,&-%'arcsinG6#*&\"\"$\"\"\"\"\"&!\"\"F+*(\"\"#F+%
\"kGF+%#PiGF+F+" }{TEXT -1 6 "  or  " }{XPPEDIT 18 0 "theta = (2*k+1)*
Pi-arcsin(3/5);" "6#/%&thetaG,&*&,&*&\"\"#\"\"\"%\"kGF*F*F*F*F*%#PiGF*
F*-%'arcsinG6#*&\"\"$F*\"\"&!\"\"F3" }{TEXT -1 2 ", " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{TEXT 356 29 "_____________________________" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 292 1 "k" 
}{TEXT -1 25 " is an integer, that is, " }{XPPEDIT 18 0 "k=` . . . `,-
4,-3,-2,-1,0,1,2,3,4,` . . . `" "6-/%\"kG%(~.~.~.~G,$\"\"%!\"\",$\"\"$
F(,$\"\"#F(,$\"\"\"F(\"\"!F.F,F*F'F%" }{TEXT -1 32 ", is a solution of
 the equation " }{XPPEDIT 18 0 "sin*theta = 3/5;" "6#/*&%$sinG\"\"\"%&
thetaGF&*&\"\"$F&\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 75 "The following picture shows the points of intersection P and Q \+
of the line " }{XPPEDIT 18 0 "y = 3/5;" "6#/%\"yG*&\"\"$\"\"\"\"\"&!\"
\"" }{TEXT -1 22 " with the unit circle " }{XPPEDIT 18 0 "x^2+y^2=1" "
6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 129 ". All the solutio
ns are associated with either P or Q, more precisely, the \"wrapping f
unction\" W, which associates a real number " }{XPPEDIT 18 0 "theta" "
6#%&thetaG" }{TEXT -1 16 " with the point " }{XPPEDIT 18 0 "W(theta)=`
`(cos*theta,sin*theta)" "6#/-%\"WG6#%&thetaG-%!G6$*&%$cosG\"\"\"F'F-*&
%$sinGF-F'F-" }{TEXT -1 39 ", maps each solution to either P or Q. " }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 257 247 247 {PLOTDATA 2 "
62-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4hRPij!**!#=$\"3Ikwb#=y_O
\"F/7$$\"3E8J#))4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\"3-N5')yke[#*F/$\"3gLj&[K
5J!QF/7$$\"3?goz=42`')F/$\"3j9NXR4U7]F/7$$\"3Sb](G._U!zF/$\"3t_H-qceDh
F/7$$\"3_\\$R'oTd#3(F/$\"3!GO#*3qU&fqF/7$$\"3?H$>jubk6'F/$\"3!\\@@&\\!
>8\"zF/7$$\"3VGuIc:x5]F/$\"3-$>p(Qh-a')F/7$$\"37C3nW'R,#QF/$\"3aK+16bc
T#*F/7$$\"3$oY@iF'QDDF/$\"3s&Q\"[M\"oen*F/7$$\"3OB^hAo8X8F/$\"33)fVPO<
\"4**F/7$$!3+qB/u(p5(f!#@$\"2%HhJ<#)******!#<7$$!33)\\T#fB[i8F/$\"3ap%
>wGZn!**F/7$$!3)e:d.:#=YEF/$\"3PA>=\\D`V'*F/7$$!3%*yV1J*3Jx$F/$\"3IzF#
e`m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'yI2&oN')F/7$$!3[igNw%*[RgF/$\"3!G;)RQ
+BqzF/7$$!3/XUwYjJ*3(F/$\"3AfvOg?x_qF/7$$!3&=e_b?/U!zF/$\"3u3<J%QZc7'F
/7$$!3w!G5)RnIf')F/$\"3D%H$\\4/k,]F/7$$!3/`k8ti$4B*F/$\"3In'[*>HvXQF/7
$$!3!p!RS4Nij'*F/$\"3o$p9m#Q%=d#F/7$$!3#p(pT>+.2**F/$\"31V?00]Ug8F/7$$
!3/gKG4>&*****F/$\"3vCA\\O%485$!#?7$$!3__Is=!Q%3**F/$!3q#4*>c=8]8F/7$$
!3)>b`sYlSn*F/$!3UNbFGIGKDF/7$$!37_J.8AEj#*F/$!3q4&=j`Bsw$F/7$$!3%\\F)
4Kh=u')F/$!3kENX')3zv\\F/7$$!3EB*z7xng%zF/$!37W(H!pUCrgF/7$$!3XD84Pfc5
rF/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[gF/$!3)4j2yeGL'zF/7$$!3g@UD=%)o!*\\F/
$!3I_Mh6Mil')F/7$$!3o+1s\"[\"ysPF/$!3#\\IN,%***4E*F/7$$!30yCH\"el'3EF/
$!3=V>(fp[Pl*F/7$$!3g67Cvnc\"H\"F/$!3;$RoI*>C;**F/7$$!3EzyOHMQdIFas$!3
W1O#>E`*****F/7$$\"3GIAj`Ks(G\"F/$!3lYZ3X=u;**F/7$$\"3EKRqX8/bDF/$!3U$
[o%H'z!o'*F/7$$\"3@%)yb&Q)zNQF/$!36.2<#>x]B*F/7$$\"3O7w4w0I.]F/$!3wHgb
5wMe')F/7$$\"3@\\#)[*R]$4hF/$!3/a*[=H2o\"zF/7$$\"33/wY'Q+$*4(F/$!3)4d)
eg?sUqF/7$$\"3[f=s$Ry,!zF/$!3D1$H<bQ38'F/7$$\"3%*R,@;vLu')F/$!3e>)zD(p
_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2ndo*fQF/7$$\"31mGAp))oa'*F/$!3%)f]nRQ=
0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4O\"F/7$F($\"36YKhSr8/#)!#F-%'COLOURG
6&%$RGBG$\"*++++\"!\")F+F+-%*THICKNESSG6#\"\"#-F$6&7$7$F+F+7$$\"3U++++
+++!)F/$\"3w**************fF/-Fjz6&F\\[l$\")#)eqkF_[l$\"))eqk\"F_[lFa
\\lF`[l-%*LINESTYLEG6#F)-F$6&7$Fg[l7$$!3U+++++++!)F/F[\\lF]\\lF`[lFc\\
l-F$6&7$Fi\\lFh[l-Fjz6&F\\[lF*F*F*-Fa[lFe\\l-Fd\\lFb[l-F$6&7$Fh[lFi\\l
-Fjz6&F\\[lF+F][lF][l-%'SYMBOLG6$%&CROSSG\"#5-%&STYLEG6#%&POINTG-F$6&F
e]lFf]l-Fi]l6$%(DIAMONDGF\\^lF]^l-F$6&Fe]lFf]l-Fi]l6$%'CIRCLEGF\\^lF]^
l-F$6&Fe]lF_]l-Fi]l6$Fj^l\"#7F]^l-%%TEXTG6%7$$\"##*!\"#$\"#lFf_lQ\"P6
\"-%&COLORG6&F\\[l$F)Ff_lF^`lF^`l-Fa_l6%7$$!##*Ff_lFg_lQ\"QFj_lF[`l-Fa
_l6%7$$\"$D\"Ff_l$!\"'Ff_lQ\"xFj_lF[`l-Fa_l6%7$Fj`lFh`lQ\"yFj_lF[`l-%(
SCALINGG6#%,CONSTRAINEDG-%*AXESTICKSG6$F*F*-%+AXESLABELSG6%Q!Fj_lF[bl-
%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!#6!\"\"$\"#8Ffbl;FdblFh`l" 1 2 0 1 10 
0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv
e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curv
e 10" "Curve 11" "Curve 12" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 4 "The " }{TEXT 259 12 "solution set" }{TEXT -1 17 " of the equatio
n " }{XPPEDIT 18 0 "sin*theta = 3/5;" "6#/*&%$sinG\"\"\"%&thetaGF&*&\"
\"$F&\"\"&!\"\"" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+
\{" }{XPPEDIT 18 0 "arcsin(3/5)+2*k*Pi;" "6#,&-%'arcsinG6#*&\"\"$\"\"
\"\"\"&!\"\"F)*(\"\"#F)%\"kGF)%#PiGF)F)" }{TEXT -1 2 " |" }{TEXT 357 
2 " k" }{TEXT -1 15 " is an integer\}" }{XPPEDIT 18 0 "``union``" "6#-
%&unionG6$%!GF&" }{TEXT -1 1 "\{" }{XPPEDIT 18 0 "(2*k+1)*Pi-arcsin(3/
5);" "6#,&*&,&*&\"\"#\"\"\"%\"kGF(F(F(F(F(%#PiGF(F(-%'arcsinG6#*&\"\"$
F(\"\"&!\"\"F1" }{TEXT -1 2 " |" }{TEXT 358 2 " k" }{TEXT -1 17 " is a
n integer\}. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 300 43 "_____
______________________________________" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 5 ":
 If " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 47 " is an angle i
n degrees, then the solutions of " }{XPPEDIT 18 0 "sin*theta = 3/5;" "
6#/*&%$sinG\"\"\"%&thetaGF&*&\"\"$F&\"\"&!\"\"" }{TEXT -1 9 " between \+
" }{XPPEDIT 18 0 "0^o;" "6#)\"\"!%\"oG" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "180^o;" "6#)\"$!=%\"oG" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "thet
a" "6#%&thetaG" }{TEXT -1 1 " " }{TEXT 305 1 "~" }{TEXT -1 3 " 0." }
{XPPEDIT 18 0 "6435011088*`.`*``(180^o/Pi);" "6#*(\"+)36]V'\"\"\"%\".G
F%-%!G6#*&)\"$!=%\"oGF%%#PiG!\"\"F%" }{TEXT -1 1 " " }{TEXT 359 1 "~" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "36.86989765^o" "6#)-%&FloatG6$\"+l(*)p
o$!\")%\"oG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "2.498091545*`.`*``(18
0^o/Pi)" "6#*(-%&FloatG6$\"+X:4)\\#!\"*\"\"\"%\".GF)-%!G6#*&)\"$!=%\"o
GF)%#PiG!\"\"F)" }{TEXT -1 1 " " }{TEXT 360 1 "~" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "143.13010235^o" "6#)-%&FloatG6$\",N-,8V\"!\")%\"oG" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 59 "The general solution ca
n be given in the approximate form: " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 1 " " }{TEXT 361 1 "~
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "36.86989765^o+k*360^o" "6#,&)-%&Floa
tG6$\"+l(*)po$!\")%\"oG\"\"\"*&%\"kGF+)\"$g$F*F+F+" }{TEXT -1 6 "  or \+
 " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 1 " " }{TEXT 362 1 "~
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "143.13010235^o+k*360^o" "6#,&)-%&Flo
atG6$\",N-,8V\"!\")%\"oG\"\"\"*&%\"kGF+)\"$g$F*F+F+" }{TEXT -1 3 ",  \+
" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 306 1 "k" }{TEXT -1 16 
" is an integer. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 72 "The previous two examples illust
rate the fact that, for any real number " }{TEXT 277 1 "a" }{TEXT -1 
6 " with " }{XPPEDIT 18 0 "-1<a" "6#2,$\"\"\"!\"\"%\"aG" }{XPPEDIT 18 
0 "``<1" "6#2%!G\"\"\"" }{TEXT -1 15 ", the equation " }{XPPEDIT 18 0 
"sin*theta = a;" "6#/*&%$sinG\"\"\"%&thetaGF&%\"aG" }{TEXT -1 9 " has \+
the " }{TEXT 259 16 "general solution" }{TEXT -1 2 ": " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta=arcsin*a+2*k*Pi" "6#/%&the
taG,&*&%'arcsinG\"\"\"%\"aGF(F(*(\"\"#F(%\"kGF(%#PiGF(F(" }{TEXT -1 6 
"  or  " }{XPPEDIT 18 0 "theta = (2*k+1)*Pi-arcsin*a;" "6#/%&thetaG,&*
&,&*&\"\"#\"\"\"%\"kGF*F*F*F*F*%#PiGF*F**&%'arcsinGF*%\"aGF*!\"\"" }
{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 388 27 "____
_______________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 
"where " }{TEXT 387 1 "k" }{TEXT -1 50 " is an integer, that is, it ha
s the solution set: " }}{PARA 256 "" 0 "" {TEXT -1 2 " \{" }{XPPEDIT 
18 0 "arcsin*a+2*k*Pi" "6#,&*&%'arcsinG\"\"\"%\"aGF&F&*(\"\"#F&%\"kGF&
%#PiGF&F&" }{TEXT -1 3 " | " }{TEXT 301 1 "k" }{TEXT -1 15 " is an int
eger\}" }{XPPEDIT 18 0 "``union``" "6#-%&unionG6$%!GF&" }{TEXT -1 1 "
\{" }{XPPEDIT 18 0 "(2*k+1)*Pi-arcsin*a;" "6#,&*&,&*&\"\"#\"\"\"%\"kGF
(F(F(F(F(%#PiGF(F(*&%'arcsinGF(%\"aGF(!\"\"" }{TEXT -1 3 " | " }{TEXT 
352 1 "k" }{TEXT -1 17 " is an integer\}. " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{TEXT 302 41 "_________________________________________" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 259 4 "Note" }{TEXT -1 45 ": It may be useful to note that the e
quation " }{XPPEDIT 18 0 "sin*theta = -a;" "6#/*&%$sinG\"\"\"%&thetaGF
&,$%\"aG!\"\"" }{TEXT -1 27 " has the general solution: " }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta = 2*k*Pi-arcsin*a;" "6#/%
&thetaG,&*(\"\"#\"\"\"%\"kGF(%#PiGF(F(*&%'arcsinGF(%\"aGF(!\"\"" }
{TEXT -1 6 "  or  " }{XPPEDIT 18 0 "theta = arcsin*a+(2*k+1)*Pi;" "6#/
%&thetaG,&*&%'arcsinG\"\"\"%\"aGF(F(*&,&*&\"\"#F(%\"kGF(F(F(F(F(%#PiGF
(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 
386 1 "k" }{TEXT -1 16 " is an integer. " }}{PARA 0 "" 0 "" {TEXT -1 
22 "We have the following " }{TEXT 259 13 "special cases" }{TEXT -1 2 
". " }}{PARA 0 "" 0 "" {TEXT -1 37 "The general solution of the equati
on " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*theta = 1;
" "6#/*&%$sinG\"\"\"%&thetaGF&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 2 "is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "the
ta = Pi/2+2*k*Pi;" "6#/%&thetaG,&*&%#PiG\"\"\"\"\"#!\"\"F(*(F)F(%\"kGF
(F'F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{TEXT 303 1 "k" }{TEXT -1 58 " is an integer, while the general soluti
on of the equation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"sin*theta = -1;" "6#/*&%$sinG\"\"\"%&thetaGF&,$F&!\"\"" }{TEXT -1 1 "
 " }}{PARA 0 "" 0 "" {TEXT -1 3 "is " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "theta = 3*Pi/2+2*k*Pi;" "6#/%&thetaG,&*(\"\"$\"\"\"
%#PiGF(\"\"#!\"\"F(*(F*F(%\"kGF(F)F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 
0 "" {TEXT -1 6 "where " }{TEXT 304 1 "k" }{TEXT -1 15 " is an integer
." }}{PARA 0 "" 0 "" {TEXT -1 43 "Also, the general solution of the eq
uation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*theta=
0" "6#/*&%$sinG\"\"\"%&thetaGF&\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 3 "is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
theta=k*Pi" "6#/%&thetaG*&%\"kG\"\"\"%#PiGF'" }{TEXT -1 2 ", " }}
{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 353 1 "k" }{TEXT -1 15 " is
 an integer." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 62 "Find the general solution of each of the following equati
ons. " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "tan*theta=1
" "6#/*&%$tanG\"\"\"%&thetaGF&F&" }{TEXT -1 9 "     (b) " }{XPPEDIT 
18 0 "tan*theta=-1" "6#/*&%$tanG\"\"\"%&thetaGF&,$F&!\"\"" }{TEXT -1 
9 "     (c) " }{XPPEDIT 18 0 "tan*theta=1/sqrt(3)" "6#/*&%$tanG\"\"\"%
&thetaGF&*&F&F&-%%sqrtG6#\"\"$!\"\"" }{TEXT -1 9 "     (d) " }
{XPPEDIT 18 0 "tan*theta = -sqrt(3);" "6#/*&%$tanG\"\"\"%&thetaGF&,$-%
%sqrtG6#\"\"$!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 
33 "In each of the following answers " }{TEXT 364 1 "k" }{TEXT -1 25 "
 is an arbitrary integer." }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }
{XPPEDIT 18 0 "theta = Pi/4+k*Pi" "6#/%&thetaG,&*&%#PiG\"\"\"\"\"%!\"
\"F(*&%\"kGF(F'F(F(" }{TEXT -1 9 "     (b) " }{XPPEDIT 18 0 "theta = -
Pi/4+k*Pi" "6#/%&thetaG,&*&%#PiG\"\"\"\"\"%!\"\"F**&%\"kGF(F'F(F(" }
{TEXT -1 9 "     (c) " }{XPPEDIT 18 0 "theta = Pi/6+k*Pi" "6#/%&thetaG
,&*&%#PiG\"\"\"\"\"'!\"\"F(*&%\"kGF(F'F(F(" }{TEXT -1 9 "     (d) " }
{XPPEDIT 18 0 "theta = -Pi/3+k*Pi;" "6#/%&thetaG,&*&%#PiG\"\"\"\"\"$!
\"\"F**&%\"kGF(F'F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 40 "____________________________________
____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 40 "________________________________________" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 120 "Find the general solution of each of the following equat
ions in terms of an appropriate inverse trigonometric function. " }}
{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "tan*theta = 3/2;" "6
#/*&%$tanG\"\"\"%&thetaGF&*&\"\"$F&\"\"#!\"\"" }{TEXT -1 9 "     (b) \+
" }{XPPEDIT 18 0 "tan*theta = -3/2;" "6#/*&%$tanG\"\"\"%&thetaGF&,$*&
\"\"$F&\"\"#!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 
33 "In each of the following answers " }{TEXT 363 1 "k" }{TEXT -1 25 "
 is an arbitrary integer." }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }
{XPPEDIT 18 0 "theta = arctan(3/2)+k*Pi;" "6#/%&thetaG,&-%'arctanG6#*&
\"\"$\"\"\"\"\"#!\"\"F+*&%\"kGF+%#PiGF+F+" }{TEXT -1 9 "     (b) " }
{XPPEDIT 18 0 "theta = -arctan(3/2)+k*Pi;" "6#/%&thetaG,&-%'arctanG6#*
&\"\"$\"\"\"\"\"#!\"\"F-*&%\"kGF+%#PiGF+F+" }{TEXT -1 4 "    " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "________
________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "____________________
____________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Find the general solution of each \+
of the following equations. " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }
{XPPEDIT 18 0 "cos*theta = 1/2;" "6#/*&%$cosG\"\"\"%&thetaGF&*&F&F&\"
\"#!\"\"" }{TEXT -1 9 "     (b) " }{XPPEDIT 18 0 "cos*theta = -1/2;" "
6#/*&%$cosG\"\"\"%&thetaGF&,$*&F&F&\"\"#!\"\"F+" }{TEXT -1 9 "     (c)
 " }{XPPEDIT 18 0 "cos*theta = 1/sqrt(2);" "6#/*&%$cosG\"\"\"%&thetaGF
&*&F&F&-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 9 "     (d) " }{XPPEDIT 18 0 "c
os*theta = -1/sqrt(2);" "6#/*&%$cosG\"\"\"%&thetaGF&,$*&F&F&-%%sqrtG6#
\"\"#!\"\"F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 
1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 33 "In \+
each of the following answers " }{TEXT 365 1 "k" }{TEXT -1 25 " is an \+
arbitrary integer." }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 
0 "theta=2*k*Pi" "6#/%&thetaG*(\"\"#\"\"\"%\"kGF'%#PiGF'" }{TEXT -1 1 
" " }{TEXT 366 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi/3;" "6#*&%#PiG
\"\"\"\"\"$!\"\"" }{TEXT -1 9 "     (b) " }{XPPEDIT 18 0 "theta=2*k*Pi
" "6#/%&thetaG*(\"\"#\"\"\"%\"kGF'%#PiGF'" }{TEXT -1 1 " " }{TEXT 367 
1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi/3;" "6#*(\"\"#\"\"\"%#PiGF%
\"\"$!\"\"" }{TEXT -1 9 "     (c) " }{XPPEDIT 18 0 "theta=2*k*Pi" "6#/
%&thetaG*(\"\"#\"\"\"%\"kGF'%#PiGF'" }{TEXT -1 1 " " }{TEXT 368 1 "+" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"\"" }
{TEXT -1 9 "     (d) " }{XPPEDIT 18 0 "theta=2*k*Pi" "6#/%&thetaG*(\"
\"#\"\"\"%\"kGF'%#PiGF'" }{TEXT -1 1 " " }{TEXT 369 1 "+" }{TEXT -1 1 
" " }{XPPEDIT 18 0 "3*Pi/4;" "6#*(\"\"$\"\"\"%#PiGF%\"\"%!\"\"" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 40 "________________________________________" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 40 "________________________________________" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "Find t
he general solution of each of the following equations in terms of an \+
appropriate inverse trigonometric function. " }}{PARA 0 "" 0 "" {TEXT 
-1 4 "(a) " }{XPPEDIT 18 0 "cos*theta = 3/5;" "6#/*&%$cosG\"\"\"%&thet
aGF&*&\"\"$F&\"\"&!\"\"" }{TEXT -1 9 "     (b) " }{XPPEDIT 18 0 "cos*t
heta = -3/5;" "6#/*&%$cosG\"\"\"%&thetaGF&,$*&\"\"$F&\"\"&!\"\"F," }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 
"" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 33 "In each of the fol
lowing answers " }{TEXT 370 1 "k" }{TEXT -1 25 " is an arbitrary integ
er." }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "theta=2*k*Pi
" "6#/%&thetaG*(\"\"#\"\"\"%\"kGF'%#PiGF'" }{TEXT -1 1 " " }{TEXT 371 
1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arccos(3/5);" "6#-%'arccosG6#*&
\"\"$\"\"\"\"\"&!\"\"" }{TEXT -1 9 "     (b) " }{XPPEDIT 18 0 "theta=2
*k*Pi" "6#/%&thetaG*(\"\"#\"\"\"%\"kGF'%#PiGF'" }{TEXT -1 1 " " }
{TEXT 372 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arccos(-3/5);" "6#-%'a
rccosG6#,$*&\"\"$\"\"\"\"\"&!\"\"F+" }{TEXT -1 20 " or, alternatively,
 " }{XPPEDIT 18 0 "theta = (2*k-1)*Pi;" "6#/%&thetaG*&,&*&\"\"#\"\"\"%
\"kGF)F)F)!\"\"F)%#PiGF)" }{TEXT -1 1 " " }{TEXT 373 1 "+" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "arccos(3/5);" "6#-%'arccosG6#*&\"\"$\"\"\"\"\"&!
\"\"" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 40 "________________________________________" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 40 "________________________________________" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Find th
e general solution of each of the following equations. " }}{PARA 0 "" 
0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "sin*theta = sqrt(3)/2;" "6#/*&%
$sinG\"\"\"%&thetaGF&*&-%%sqrtG6#\"\"$F&\"\"#!\"\"" }{TEXT -1 9 "     \+
(b) " }{XPPEDIT 18 0 "sin*theta = -sqrt(3)/2;" "6#/*&%$sinG\"\"\"%&the
taGF&,$*&-%%sqrtG6#\"\"$F&\"\"#!\"\"F/" }{TEXT -1 9 "     (c) " }
{XPPEDIT 18 0 "sin*theta = 1/sqrt(2);" "6#/*&%$sinG\"\"\"%&thetaGF&*&F
&F&-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 9 "     (d) " }{XPPEDIT 18 0 "sin*t
heta = -1/sqrt(2);" "6#/*&%$sinG\"\"\"%&thetaGF&,$*&F&F&-%%sqrtG6#\"\"
#!\"\"F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 
{PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 33 "In ea
ch of the following answers " }{TEXT 374 1 "k" }{TEXT -1 25 " is an ar
bitrary integer." }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "
theta = Pi/3+2*k*Pi;" "6#/%&thetaG,&*&%#PiG\"\"\"\"\"$!\"\"F(*(\"\"#F(
%\"kGF(F'F(F(" }{TEXT -1 6 "  or  " }{XPPEDIT 18 0 "theta = 2*Pi/3+2*k
*Pi;" "6#/%&thetaG,&*(\"\"#\"\"\"%#PiGF(\"\"$!\"\"F(*(F'F(%\"kGF(F)F(F
(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 
"theta = 4*Pi/3+2*k*Pi;" "6#/%&thetaG,&*(\"\"%\"\"\"%#PiGF(\"\"$!\"\"F
(*(\"\"#F(%\"kGF(F)F(F(" }{TEXT -1 6 "  or  " }{XPPEDIT 18 0 "theta = \+
5*Pi/3+2*k*Pi;" "6#/%&thetaG,&*(\"\"&\"\"\"%#PiGF(\"\"$!\"\"F(*(\"\"#F
(%\"kGF(F)F(F(" }{TEXT -1 21 ", or, alternatively, " }{XPPEDIT 18 0 "t
heta = -Pi/3+2*k*Pi;" "6#/%&thetaG,&*&%#PiG\"\"\"\"\"$!\"\"F**(\"\"#F(
%\"kGF(F'F(F(" }{TEXT -1 17 "  together with  " }{XPPEDIT 18 0 "theta \+
= -2*Pi/3+2*k*Pi;" "6#/%&thetaG,&*(\"\"#\"\"\"%#PiGF(\"\"$!\"\"F+*(F'F
(%\"kGF(F)F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }
{XPPEDIT 18 0 "theta = Pi/4+2*k*Pi;" "6#/%&thetaG,&*&%#PiG\"\"\"\"\"%!
\"\"F(*(\"\"#F(%\"kGF(F'F(F(" }{TEXT -1 6 "  or  " }{XPPEDIT 18 0 "the
ta = 3*Pi/4+2*k*Pi;" "6#/%&thetaG,&*(\"\"$\"\"\"%#PiGF(\"\"%!\"\"F(*(
\"\"#F(%\"kGF(F)F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(d
) " }{XPPEDIT 18 0 "theta = 5*Pi/4+2*k*Pi;" "6#/%&thetaG,&*(\"\"&\"\"
\"%#PiGF(\"\"%!\"\"F(*(\"\"#F(%\"kGF(F)F(F(" }{TEXT -1 6 "  or  " }
{XPPEDIT 18 0 "theta = 7*Pi/4+2*k*Pi;" "6#/%&thetaG,&*(\"\"(\"\"\"%#Pi
GF(\"\"%!\"\"F(*(\"\"#F(%\"kGF(F)F(F(" }{TEXT -1 21 ", or, alternative
ly, " }{XPPEDIT 18 0 "theta = -Pi/4+2*k*Pi;" "6#/%&thetaG,&*&%#PiG\"\"
\"\"\"%!\"\"F**(\"\"#F(%\"kGF(F'F(F(" }{TEXT -1 17 "  together with  \+
" }{XPPEDIT 18 0 "theta = -3*Pi/4+2*k*Pi;" "6#/%&thetaG,&*(\"\"$\"\"\"
%#PiGF(\"\"%!\"\"F+*(\"\"#F(%\"kGF(F)F(F(" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "_________________
_______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "_______________________________
_________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 120 "Find the general solution of each of the
 following equations in terms of an appropriate inverse trigonometric \+
function. " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "sin*th
eta = 3/4;" "6#/*&%$sinG\"\"\"%&thetaGF&*&\"\"$F&\"\"%!\"\"" }{TEXT 
-1 9 "     (b) " }{XPPEDIT 18 0 "sin*theta = -3/4;" "6#/*&%$sinG\"\"\"
%&thetaGF&,$*&\"\"$F&\"\"%!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "
" 0 "" {TEXT -1 33 "In each of the following answers " }{TEXT 375 1 "k
" }{TEXT -1 25 " is an arbitrary integer." }}{PARA 0 "" 0 "" {TEXT -1 
4 "(a) " }{XPPEDIT 18 0 "theta = arcsin(3/4)+2*k*Pi;" "6#/%&thetaG,&-%
'arcsinG6#*&\"\"$\"\"\"\"\"%!\"\"F+*(\"\"#F+%\"kGF+%#PiGF+F+" }{TEXT 
-1 6 "  or  " }{XPPEDIT 18 0 "theta = -arcsin(3/4)+(2*k+1)*Pi;" "6#/%&
thetaG,&-%'arcsinG6#*&\"\"$\"\"\"\"\"%!\"\"F-*&,&*&\"\"#F+%\"kGF+F+F+F
+F+%#PiGF+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }
{XPPEDIT 18 0 "theta = -arcsin(3/4)+2*k*Pi;" "6#/%&thetaG,&-%'arcsinG6
#*&\"\"$\"\"\"\"\"%!\"\"F-*(\"\"#F+%\"kGF+%#PiGF+F+" }{TEXT -1 6 "  or
  " }{XPPEDIT 18 0 "theta = arcsin(3/4)+(2*k+1)*Pi;" "6#/%&thetaG,&-%'
arcsinG6#*&\"\"$\"\"\"\"\"%!\"\"F+*&,&*&\"\"#F+%\"kGF+F+F+F+F+%#PiGF+F
+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 40 "________________________________________" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 40 "________________________________________" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 
";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for dr
awing pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 0 "" 0 "" {TEXT -1 15 "Arcsin graphs  " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 619 "p1 := plo
t(sin(Pi*x),x=-1..-1/2,color=red):\np2 := plot(sin(Pi*x),x=1/2..3/2,co
lor=red):\np3 := plot(sin(Pi*x),x=-1/2..1/2,color=blue,thickness=3):\n
p4 := plot([[[1/2,1],[-1/2,-1]]$3],style=point,\n                 symb
ol=[circle,diamond,cross],color=blue):\nt1 := plots[textplot]([[1.7,-.
1,`x`],[-.05,1.3,`y`]],\n                    color=COLOR(RGB,.01,.01,.
01),font=[HELVETICA,9]):\nplots[display]([p1,p2,p3,p4,t1],ytickmarks=3
,\nxtickmarks=[-1=`-p`,-.5=`-p/2`,0=`0`,.5=`p/2`,1=`p`,1.5=`3p/2`],\n \+
  font=[SYMBOL,9],labels=[``,``],label_font=[HELVETICA,9],title=\"y = \+
sin x\",\n   title_font=[HELVETICA,9],view=[-1..1.7,-1..1.3]);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
698 "p1 := plot(sin(Pi*x)/Pi,x=-1/2..1/2,color=blue):\np2 := plot(arcs
in(Pi*x)/Pi,x=-1/Pi..1/Pi,color=red,numpoints=100):\np3 := plot(x,x=-1
/2..1/2,color=green):\nt1 := plots[textplot]([.5,.27,`y = sin x`],\n  \+
                        color=blue,font=[HELVETICA,9]):\nt2 := plots[t
extplot]([.2,.55,`y = arcsin x`],\n                          color=red
,font=[HELVETICA,9]):\nt3 := plots[textplot]([[.6,-.03,`x`],[-.03,.6,`
y`]],\n                          color=black,font=[HELVETICA,9]):\nplo
ts[display]([p1,p2,p3,t1,t2,t3],labels=[``,``],\nxtickmarks=[-.5=`-p/2
`,-.31831=`-1`,0=`0`,.31831=`1`,.5=`p/2`],\nytickmarks=[-.5=`-p/2`,-.3
1831=`-1`,0=`0`,.31831=`1`,.5=`p/2`],\n   font=[SYMBOL,9],view=[-.5..0
.6,-0.5..0.6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 458 "p1 := plot(arcsin(Pi*x)/Pi,x=-1/Pi..1/Pi,colo
r=red,numpoints=100):\n\nt1 := plots[textplot]([.2,.48,`y = arcsin x`]
,\n                          color=red,font=[HELVETICA,9]):\nt2 := plo
ts[textplot]([[.37,-.03,`x`],[-.02,.6,`y`]],\n                        \+
  color=black,font=[HELVETICA,9]):\nplots[display]([p1,t1,t2],labels=[
``,``],\nxtickmarks=[-.31831=`-1`,0=`0`,.31831=`1`],\nytickmarks=[-.5=
`-p/2`,0=`0`,.5=`p/2`],font=[SYMBOL,9],\n  view=[-.33..0.37,-0.5..0.6]
);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 497 "p1 := plot([arcsin(x),1/sqrt(1-x^2)],x=-1..1,y=-Pi/2
..3,numpoints=100,\n      color=[red,blue]):\np2 := plot([[[-1,-Pi/2],
[-1,3]],[[1,-Pi/2],[1,3]]],\n          color=black,linestyle=3):\nt1 :
= plots[textplot]([-.6,-1.4,`y = arcsin x`],color=red):\nt2 := plots[t
extplot]([[1.2,-.1,`x`],[-.1,3,`y`]],\n                          color
=black):\nplots[display]([p1,p2,t1,t2],labels=[``,``],\n    xtickmarks
=[-1=`-1`,-.5=`-.5`,.5=`.5`,0=`0`,1=`1`],\n  ytickmarks=[-1=`-1`,1=`1`
,2=`2`],view=[-1.2..1.2,-1.6..3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 559 "f := x -> 1/sqrt(1-x^2):\n
p1 := plot(f(x),x=-1..1,y=0..3,color=red):\np2 := plot([[[-1,-Pi/2],[-
1,3]],[[1,-Pi/2],[1,3]]],\n          color=black,linestyle=3):\na := -
1/2: b := 1/2:\np3 := plot(f(x),x=a..b,color=COLOR(RGB,.87,.87,.93),fi
lled=true):\np4 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=blac
k):\nt1 := plots[textplot]([[1.1,-.1,`x`],[-.1,3,`y`]],\n             \+
             color=black):\nplots[display]([p1,p2,p3,p4,t1],labels=[``
,``],\nxtickmarks=[-1=`-1`,-.5=`-.5`,.5=`.5`,0=`0`,1=`1`],\n   ytickma
rks=[-1=`-1`,1=`1`,2=`2`],\n  view=[-1.1..1.1,-.1..3]);" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 15 "Arccos graphs  " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 604 "p1 := pl
ot(cos(Pi*x),x=-1/2..0,color=red):\np2 := plot(cos(Pi*x),x=1..2,color=
red):\np3 := plot(cos(Pi*x),x=0..1,color=blue,thickness=3):\np4 := plo
t([[[0,1],[1,-1]]$3],style=point,\n                 symbol=[circle,dia
mond,cross],color=blue):\nt1 := plots[textplot]([[2.2,-.1,`x`],[-.05,1
.3,`y`]],\n                    color=COLOR(RGB,.01,.01,.01),font=[HELV
ETICA,9]):\nplots[display]([p1,p2,p3,p4,t1],ytickmarks=3,\nxtickmarks=
[-.5=`-p/2`,0=`0`,.5=`p/2`,1=`p`,1.5=`3p/2`,2=`2p`],\n   font=[SYMBOL,
9],labels=[``,``],label_font=[HELVETICA,9],title=\"y = cos x\",\n   ti
tle_font=[HELVETICA,9],view=[-.5..2.2,-1..1.3]);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 712 "p1 := plot(
cos(Pi*x)/Pi,x=-0..1,color=blue):\np2 := plot(arccos(Pi*x)/Pi,x=-1/Pi.
.1/Pi,color=red,numpoints=100):\np3 := plot(x,x=-.2..0.7,color=green):
\nt1 := plots[textplot]([.5,.27,`y = cos x`],\n                       \+
   color=blue,font=[HELVETICA,9]):\nt2 := plots[textplot]([.2,.55,`y =
 arccos x`],\n                          color=red,font=[HELVETICA,9]):
\nt3 := plots[textplot]([[1.13,-.03,`x`],[-.03,1.13,`y`]],\n          \+
                color=black,font=[HELVETICA,9]):\nplots[display]([p1,p
2,p3,t1,t2,t3],labels=[``,``],\nxtickmarks=[-.31831=`-1`,0=`0`,.31831=
`1`,.5=`p/2`,.63662=`2`,1=`p`],\nytickmarks=[-.31831=`-1`,0=`0`,.31831
=`1`,.5=`p/2`,.63662=`2`,1=`p`],\n   font=[SYMBOL,9],view=[-.33..1.13,
-.33..1.13]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 457 "p1 := plot(arccos(Pi*x)/Pi,x=-1/Pi..1/Pi,color=
red,numpoints=100):\nt1 := plots[textplot]([-.2,.96,`y = arccos x`],\n
                          color=red,font=[HELVETICA,9]):\nt2 := plots[
textplot]([[.37,-.03,`x`],[-.02,1.13,`y`]],\n                         \+
 color=black,font=[HELVETICA,9]):\nplots[display]([p1,t1,t2],labels=[`
`,``],\nxtickmarks=[-.31831=`-1`,0=`0`,.31831=`1`],\nytickmarks=[0=`0`
,.5=`p/2`,1=`p`],\n   font=[SYMBOL,9],view=[-.33..0.37,-.03..1.13]);" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 15 "Arctan graphs  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
682 "p1 := plot(tan(Pi*x),x=-3/2..-1/2,y=-6..6,color=red,discont=true)
:\np2 := plot(tan(Pi*x),x=1/2..3/2,color=red,discont=true):\np3 := plo
t(tan(Pi*x),x=-1/2..1/2,color=blue,thickness=2):\np4 := plot([[[-1.5,-
6],[-1.5,6]],[[-.5,-6],[-.5,6]],\n        [[.5,-6],[.5,6]],[[1.5,-6],[
1.5,6]]],color=black,linestyle=3):\nt1 := plots[textplot]([[1.7,-.2,`x
`],[-.06,6,`y`]],\n                    color=COLOR(RGB,.01,.01,.01),fo
nt=[HELVETICA,9]):\nplots[display]([p1,p2,p3,p4,t1],ytickmarks=3,\nxti
ckmarks=[-1.5=`-3p/2`,-1=`-p`,-.5=`-p/2`,0=`0`,.5=`p/2`,1=`p`,1.5=`3p/
2`],\n   font=[SYMBOL,9],labels=[``,``],label_font=[HELVETICA,9],title
=\"y = tan x\",\n   title_font=[HELVETICA,9],view=[-1.6..1.7,-6..6]);
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1182 "p1 := plot(tan(Pi*x)/Pi,x=-1/2..1/2,y=-2.1..2.1,col
or=blue,discont=true):\np2 := plot(arctan(Pi*x)/Pi,x=-2.1..2.1,color=r
ed):\np3 := plot(x,x=-2..2,color=green):\np4 := plot([[[-.5,-2.1],[-.5
,2.1]],[[.5,-2.1],[.5,2.1]]],\n          color=COLOR(RGB,.3,0,.7),line
style=3):\np5 := plot([[[-2.1,-.5],[2.1,-.5]],[[-2.1,.5],[2.1,.5]]],\n
          color=brown,linestyle=3):\nt1 := plots[textplot]([.8,1.8,`y \+
= tan x`],\n                          color=blue,font=[HELVETICA,9]):
\nt2 := plots[textplot]([1.8,.35,`y = arctan x`],\n                   \+
       color=red,font=[HELVETICA,9]):\nt3 := plots[textplot]([[2.1,-.0
7,`x`],[-.07,2.1,`y`]],\n                          color=black,font=[H
ELVETICA,9]):\nt4 := plots[textplot]([[-.4,-.07,`-p/2`],[.4,-.07,`p/2`
],\n             [-.1,-.4,`-p/2`],[-.1,.4,`p/2`]],\n                  \+
        color=black,font=[SYMBOL,9]):\nplots[display]([p1,p2,p3,p4,p5,
t1,t2,t3,t4],labels=[``,``],\nxtickmarks=[-1.9099=`-6`,-1.2732=`-4`,-.
63662=`-2`,0=`0`,\n                       .63662=`2`,1.2732=`4`,1.9099
=`6`],\nytickmarks=[-1.9099=`-6`,-1.2732=`-4`,-.63662=`-2`,0=`0`,\n   \+
                    .63662=`2`,1.2732=`4`,1.9099=`6`],\n   font=[SYMBO
L,9],view=[-2.1..2.1,-2.1..2.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 702 "p1 := plot(arctan(Pi*x)/Pi,
x=-2.1..2.1,color=red):\np2 := plot([[[-2.1,-.5],[2.1,-.5]],[[-2.1,.5]
,[2.1,.5]]],\n          color=black,linestyle=3):\nt1 := plots[textplo
t]([1.8,.4,`y = arctan x`],\n                          color=red,font=
[HELVETICA,9]):\nt2 := plots[textplot]([[2.1,-.07,`x`],[-.07,.6,`y`]],
\n                          color=black,font=[HELVETICA,9]):\nt3 := pl
ots[textplot]([[-.13,-.44,`-p/2`],[-.1,.44,`p/2`]],\n                 \+
         color=black,font=[SYMBOL,9]):\nplots[display]([p1,p2,t1,t2,t3
],labels=[``,``],\nxtickmarks=[-1.9099=`-6`,-1.2732=`-4`,-.63662=`-2`,
0=`0`,\n                       .63662=`2`,1.2732=`4`,1.9099=`6`],\nyti
ckmarks=0,\n   font=[SYMBOL,9],view=[-2.1..2.1,-.6..0.6]);" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 534 "p
1 := plot([arctan(x),1/(1+x^2)],x=-6.5..6.5,color=[red,blue]):\np2 := \+
plot([-Pi/2,Pi/2],-6.5..6.5,color=black,linestyle=3):\nt1 := plots[tex
tplot]([4.5,1.02,`y = arctan x`],color=black):\nt2 := plots[textplot](
[[6.5,-.2,`x`],[-.2,1.9,`y`]],color=black):\nt3 := plots[textplot]([[-
.35,-1.4,`-p/2`],[-.3,1.4,`p/2`]],\n                          color=bl
ack,font=[SYMBOL,9]):\nplots[display]([p1,p2,t1,t2,t3],labels=[``,``],
xtickmarks=[-6=`-6`,-4=`-4`,-2=`-2`,0=`0`,2=`2`,4=`4`,6=`6`],\n  ytick
marks=[-1=`-1`,1=`1`],view=[-6.5..6.5,-1.7..1.9]);" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "read \"D:
\\\\Maple7/procdrs/colours.m\";\nread \"D:\\\\Maple7/procdrs/calculus.
m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "areaplot(1/(1+x^2),
x=0..6,color=[purple,cyan,red],areafunction=true,\n   tickmarks=[6,2])
;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 20 "tan equati
on graph  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 534 "cs := evalf(cos(Pi/3)):\nsn := evalf(sin(Pi/3)):\n
p1 := plot([cos(t),sin(t),t=0..2*Pi],color=red,thickness=2):\np2 := pl
ot([[-cs,-sn],[cs,sn]],color=brown,thickness=2):\np3 := plot([[[cs,sn]
,[-cs,-sn]]$4],style=point,\n   symbol=[cross,diamond,circle$2],symbol
size=[10$3,12],color=[cyan$3,black]):\nt1 := plots[textplot]([[.58,1.,
`P`],[-.58,-1.,`Q`],\n    [1.25,-.06,`x`],[-.06,1.25,`y`]],color=COLOR
(RGB,.01,.01,.01)):\nplots[display]([p1,p2,p3,t1],view=[-1.1..1.3,-1.1
..1.25],\n                          tickmarks=[0,0],scaling=constraine
d);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1523 "p1 := plot([tan(Pi*x),sqrt(3)],x=-3.55..3.55,y=-5..
5,color=[red,blue],\n                thickness=2,discont=true):\np2 :=
 plot([seq([[(2*j-1)/2,-5],[(2*j-1)/2,5]],j=-3..4)],color=black,linest
yle=3):\np3 := plot([seq([[1/3+j,0],[1/3+j,sqrt(3)]],j=-3..4)],color=n
avy,linestyle=2,thickness=1):\np4 := plot([seq([[1/3+j,sqrt(3)]],j=-3.
.4)],style=point,symbol=circle,color=green):\np5 := plot([seq([[1/3+j,
sqrt(3)]],j=-3..4)],style=point,symbol=diamond,color=green):\np6 := pl
ot([seq([[1/3+j,sqrt(3)]],j=-3..4)],style=point,symbol=cross,color=gre
en):\np7 := plot([seq([[1/3+j,sqrt(3)]],j=-3..4)],style=point,symbol=c
ircle,\n                      symbolsize=12,color=black):\np8 := plot(
[seq([[1/3+j,-.88],[1/3+j,0]],j=-3..4)],color=black):\np9 := plots[pol
ygonplot]([seq([[1/3+j-.04,-.1],[1/3+j,0],[1/3+j+.04,-.1]],j=-3..4)],
\n    color=coral):\np10 := seq(plottools[circle]([(3*j-2)/3,-1.1],.2,
color=black),j=-2..4):\nt1 := plots[textplot]([-.08,5,`y`],color=COLOR
(RGB,.01,.01,.01),font=[HELVETICA,10]):\nt2 := plots[textplot]([3.8,-.
2,`q`],color=COLOR(RGB,.01,.01,.01),font=[SYMBOL,11]):\nt3 := plots[te
xtplot]([seq([(3*j-2)/3,-1.1,cat(convert((3*j-2),string),\"p/3\")],\n \+
j=[-2,-1,0,2,3,4]),[1/3,-1.1,\"p/3\"]],color=COLOR(RGB,.01,.01,.01),fo
nt=[SYMBOL,10]):\nplots[display]([p||(1..10),t1,t2,t3],ytickmarks=3,xt
ickmarks=[-3.5=`-7p/2`,-3=`-3p`,\n -2.5=`-5p/2`,-2=`-2p`,-1.5=`-3p/2`,
-1=`-p`,-.5=`-p/2`,0=`0`,.5=`p/2`,1=`p`,\n  1.5=`3p/2`,2=`2p`,2.5=`5p/
2`,3=`3p`,3.5=`7p/2`],ytickmarks=5,font=[SYMBOL,9],\n    labels=[``,``
],view=[-3.55..3.8,-5..5]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 534 "cs := evalf(1/sqrt(5)):\nsn := evalf(2/sqrt(5)):\np1 := plot(
[cos(t),sin(t),t=0..2*Pi],color=red,thickness=2):\np2 := plot([[-cs,-s
n],[cs,sn]],color=brown,thickness=2):\np3 := plot([[[cs,sn],[-cs,-sn]]
$4],style=point,\n   symbol=[cross,diamond,circle$2],symbolsize=[10$3,
12],color=[cyan$3,black]):\nt1 := plots[textplot]([[.55,1.,`P`],[-.55,
-1.,`Q`],\n    [1.25,-.06,`x`],[-.06,1.25,`y`]],color=COLOR(RGB,.01,.0
1,.01)):\nplots[display]([p1,p2,p3,t1],view=[-1.1..1.3,-1.1..1.25],\n \+
                         tickmarks=[0,0],scaling=constrained);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 20 "cos equation graph  " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 604 "cs := evalf(cos(Pi/6)):\nsn := evalf(sin(Pi/6)):\np1 := plot([c
os(t),sin(t),t=0..2*Pi],color=red,thickness=2):\np2 := plot([[[0,0],[c
s,sn]],[[0,0],[cs,-sn]],[[cs,-sn],[cs,sn]]],\n   color=[brown$2,black]
,linestyle=[1$2,2],thickness=[2$2,1]):\np3 := plot([[[cs,sn],[cs,-sn]]
$4],style=point,\n   symbol=[cross,diamond,circle$2],symbolsize=[10$3,
12],color=[cyan$3,black]):\nt1 := plots[textplot]([[1.,.53,`P`],[1.,-.
53,`Q`],\n    [1.25,-.06,`x`],[-.06,1.25,`y`]],color=COLOR(RGB,.01,.01
,.01)):\nplots[display]([p1,p2,p3,t1],view=[-1.1..1.3,-1.1..1.25],\n  \+
                        tickmarks=[0,0],scaling=constrained);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1943 "p1 := plot([cos(Pi*x),sqrt(3)/2],x=-2.55..4.55,color=[red,blue],
\n                thickness=2):\np2 := plot([seq([[1/6+2*j,0],[1/6+2*j
,sqrt(3)/2]],j=-1..2),\n   seq([[-1/6+2*j,0],[-1/6+2*j,sqrt(3)/2]],j=-
1..2)],color=navy,linestyle=2):\np3 := plot([seq([[1/6+2*j,sqrt(3)/2]]
,j=-1..2),\n   seq([[-1/6+2*j,sqrt(3)/2]],j=-1..2)],style=point,symbol
=circle,color=green):\np4 := plot([seq([[1/6+2*j,sqrt(3)/2]],j=-1..2),
\n   seq([[1/6+2*j,sqrt(3)/2]],j=-1..2)],style=point,symbol=diamond,co
lor=green):\np5 := plot([seq([[1/6+2*j,sqrt(3)/2]],j=-1..2),\n   seq([
[-1/6+2*j,sqrt(3)/2]],j=-1..2)],style=point,symbol=cross,color=green):
\np6 := plot([seq([[1/6+2*j,sqrt(3)/2]],j=-1..2),\n   seq([[-1/6+2*j,s
qrt(3)/2]],j=-1..2)],style=point,symbol=circle,\n                     \+
 symbolsize=12,color=black):\np7 := plot([seq([[1/6+2*j,-.51],[1/6+2*j
,0]],j=-1..2),\n       seq([[-1/6+2*j,-.51],[-1/6+2*j,0]],j=-1..2)],co
lor=black):\np8 := plots[polygonplot]([seq([[1/6+2*j-.04,-.1],[1/6+2*j
,0],[1/6+2*j+.04,-.1]],j=-1..2),\n    seq([[-1/6+2*j-.04,-.1],[-1/6+2*
j,0],[-1/6+2*j+.04,-.1]],j=-1..2)],\n    color=coral):\np9 := seq(plot
tools[circle]([1/6+2*j+.07,-.7],.2,color=black),j=-1..2),\n      seq(p
lottools[circle]([-1/6+2*j-.07,-.7],.2,color=black),j=-1..2):\nt1 := p
lots[textplot]([-.08,1.29,`y`],color=COLOR(RGB,.01,.01,.01),font=[HELV
ETICA,10]):\nt2 := plots[textplot]([4.7,-.13,`q`],color=COLOR(RGB,.01,
.01,.01),font=[SYMBOL,11]):\nt3 := plots[textplot]([seq([1/6+2*j+.07,-
.7,cat(convert((1+12*j),string),\"p/6\")],\n   j=[-1,1,2]),seq([-1/6+2
*j-.07,-.7,cat(convert((-1+12*j),string),\"p/6\")],\n   j=[-1,1,2]),[1
/6+.07,-.7,\"p/6\"],[-1/6-.07,-.7,\"-p/6\"]],\n    color=COLOR(RGB,.01
,.01,.01),font=[SYMBOL,10]):\nplots[display]([p||(1..9),t1,t2,t3],ytic
kmarks=3,xtickmarks=[-2.5=`-5p/2`,-2=`-2p`,\n -1.5=`-3p/2`,-1=`-p`,-.5
=`-p/2`,0=`0`,.5=`p/2`,1=`p`,1.5=`3p/2`,2=`2p`,\n  2.5=`5p/2`,3=`3p`,3
.5=`7p/2`,4=`4p`,4.5=`9p/2`],\n   font=[SYMBOL,9],labels=[``,``],view=
[-2.55..4.7,-1.32..1.29]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 600 "cs := evalf(2/3):\nsn := evalf(sqr
t(5)/3):\np1 := plot([cos(t),sin(t),t=0..2*Pi],color=red,thickness=2):
\np2 := plot([[[0,0],[cs,sn]],[[0,0],[cs,-sn]],[[cs,-sn],[cs,sn]]],\n \+
  color=[brown$2,black],linestyle=[1$2,2],thickness=[2$2,1]):\np3 := p
lot([[[cs,sn],[cs,-sn]]$4],style=point,\n   symbol=[cross,diamond,circ
le$2],symbolsize=[10$3,12],color=[cyan$3,black]):\nt1 := plots[textplo
t]([[.81,.79,`P`],[.81,-.79,`Q`],\n    [1.25,-.06,`x`],[-.06,1.25,`y`]
],color=COLOR(RGB,.01,.01,.01)):\nplots[display]([p1,p2,p3,t1],view=[-
1.1..1.3,-1.1..1.25],\n                          tickmarks=[0,0],scali
ng=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 20 "si
n equation graph  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 604 "cs := evalf(cos(Pi/6)):\nsn := evalf(sin(Pi
/6)):\np1 := plot([cos(t),sin(t),t=0..2*Pi],color=red,thickness=2):\np
2 := plot([[[0,0],[cs,sn]],[[0,0],[-cs,sn]],[[-cs,sn],[cs,sn]]],\n   c
olor=[brown$2,black],linestyle=[1$2,2],thickness=[2$2,1]):\np3 := plot
([[[cs,sn],[-cs,sn]]$4],style=point,\n   symbol=[cross,diamond,circle$
2],symbolsize=[10$3,12],color=[cyan$3,black]):\nt1 := plots[textplot](
[[1.,.53,`P`],[-1.,.53,`Q`],\n    [1.25,-.06,`x`],[-.06,1.25,`y`]],col
or=COLOR(RGB,.01,.01,.01)):\nplots[display]([p1,p2,p3,t1],view=[-1.1..
1.3,-1.1..1.25],\n                          tickmarks=[0,0],scaling=co
nstrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1878 "p1 := plot([sin(Pi*x),1/2],x=-2.05..5.05,color=[
red,blue],\n                thickness=2):\np2 := plot([seq([[1/6+2*j,0
],[1/6+2*j,1/2]],j=-1..2),\n   seq([[5/6+2*j,0],[5/6+2*j,1/2]],j=-1..2
)],color=navy,linestyle=2):\np3 := plot([seq([[1/6+2*j,1/2]],j=-1..2),
\n   seq([[5/6+2*j,1/2]],j=-1..2)],style=point,symbol=circle,symbolsiz
e=10,color=green):\np4 := plot([seq([[1/6+2*j,1/2]],j=-1..2),\n   seq(
[[5/6+2*j,1/2]],j=-1..2)],style=point,symbol=diamond,symbolsize=10,col
or=green):\np5 := plot([seq([[1/6+2*j,1/2]],j=-1..2),\n   seq([[5/6+2*
j,1/2]],j=-1..2)],style=point,symbol=cross,symbolsize=10,color=green):
\np6 := plot([seq([[1/6+2*j,1/2]],j=-1..2),\n   seq([[5/6+2*j,1/2]],j=
-1..2)],style=point,symbol=circle,symbolsize=12,color=black):\np7 := p
lot([seq([[1/6+2*j,-.51],[1/6+2*j,0]],j=-1..2),\n       seq([[5/6+2*j,
-.51],[5/6+2*j,0]],j=-1..2)],color=black):\np8 := plots[polygonplot]([
seq([[1/6+2*j-.04,-.1],[1/6+2*j,0],[1/6+2*j+.04,-.1]],j=-1..2),\n    s
eq([[5/6+2*j-.04,-.1],[5/6+2*j,0],[5/6+2*j+.04,-.1]],j=-1..2)],\n    c
olor=coral):\np9 := seq(plottools[circle]([1/6+2*j+.07,-.7],.2,color=b
lack),j=-1..2),\n      seq(plottools[circle]([5/6+2*j-.07,-.7],.2,colo
r=black),j=-1..2):\nt1 := plots[textplot]([-.08,1.29,`y`],color=COLOR(
RGB,.01,.01,.01),font=[HELVETICA,10]):\nt2 := plots[textplot]([5.2,-.1
3,`q`],color=COLOR(RGB,.01,.01,.01),font=[SYMBOL,11]):\nt3 := plots[te
xtplot]([seq([1/6+2*j+.07,-.7,cat(convert((1+12*j),string),\"p/6\")],
\n   j=[-1,1,2]),seq([5/6+2*j-.07,-.7,cat(convert((5+12*j),string),\"p
/6\")],\n   j=[-1,1,2]),[1/6+.07,-.7,\"p/6\"],[5/6-.07,-.7,\"5p/6\"]],
\n    color=COLOR(RGB,.01,.01,.01),font=[SYMBOL,10]):\nplots[display](
[p||(1..9),t1,t2,t3],ytickmarks=3,xtickmarks=[-2=`-2p`,-1.5=`-3p/2`,\n
  -1=`-p`,-.5=`-p/2`,0=`0`,.5=`p/2`,1=`p`,1.5=`3p/2`,2=`2p`,2.5=`5p/2`
,3=`3p`,3.5=`7p/2`,\n   4=`4p`,4.5=`9p/2`,5=`5p`],\n   font=[SYMBOL,9]
,labels=[``,``],view=[-2.05..5.2,-1.32..1.29]);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 594 "cs := evalf
(4/5):\nsn := evalf(3/5):\np1 := plot([cos(t),sin(t),t=0..2*Pi],color=
red,thickness=2):\np2 := plot([[[0,0],[cs,sn]],[[0,0],[-cs,sn]],[[-cs,
sn],[cs,sn]]],\n   color=[brown$2,black],linestyle=[1$2,2],thickness=[
2$2,1]):\np3 := plot([[[cs,sn],[-cs,sn]]$4],style=point,\n   symbol=[c
ross,diamond,circle$2],symbolsize=[10$3,12],color=[cyan$3,black]):\nt1
 := plots[textplot]([[.92,.65,`P`],[-.92,.65,`Q`],\n    [1.25,-.06,`x`
],[-.06,1.25,`y`]],color=COLOR(RGB,.01,.01,.01)):\nplots[display]([p1,
p2,p3,t1],view=[-1.1..1.3,-1.1..1.25],\n                          tick
marks=[0,0],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 \+
0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }