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numbers" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Sto ne, 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 44 "Trigonometric functions of \+ angles less than " }{XPPEDIT 18 0 "90^o" "6#)\"#!*%\"oG" }{TEXT -1 18 " .. special values" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 89 "The trigonometric functions arise in conn ection with right-angled triangle trigonometry. 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" }}{PARA 0 "" 0 "" {TEXT -1 5 " The " }{TEXT 259 6 "cosine" }{TEXT -1 10 " of angle \+ " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "adjacent/hypotenuse;" "6#*&%)adjacentG\"\"\"%+hypotenuseG!\"\"" }{TEXT -1 11 " ----- " }{XPPEDIT 18 0 "cos*theta = b/c;" "6#/*&%$c osG\"\"\"%&thetaGF&*&%\"bGF&%\"cG!\"\"" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {TEXT -1 5 " The " }{TEXT 259 7 "tangent" }{TEXT -1 10 " of ang le " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 5 " is " } {XPPEDIT 18 0 "opposite/adjacent;" "6#*&%)oppositeG\"\"\"%)adjacentG! \"\"" }{TEXT -1 11 " ----- " }{XPPEDIT 18 0 "tan*theta = a/b;" "6# /*&%$tanG\"\"\"%&thetaGF&*&%\"aGF&%\"bG!\"\"" }{TEXT -1 4 ". " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follo ws that:" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "PIECEWI SE([a = c*`.`*sin*theta, `opposite = hypotenuse times sine of angle`], [b = c*`.`*cos*theta, `adjacent = hypotenuse times cosine of angle`],[ a = b*`.`*tan*theta, `opposite = adjacent times tangent of angle`]);" "6#-%*PIECEWISEG6%7$/%\"aG**%\"cG\"\"\"%\".GF+%$sinGF+%&thetaGF+%Joppo site~=~hypotenuse~times~sine~of~angleG7$/%\"bG**F*F+F,F+%$cosGF+F.F+%L adjacent~=~hypotenuse~times~cosine~of~angleG7$/F(**F2F+F,F+%$tanGF+F.F +%Kopposite~=~adjacent~times~tangent~of~angleG" }{TEXT -1 2 " " }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 27 "\nPyt hagoras' theorem gives " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2 + b^2 = c^2" "6#/,&*$%\"aG\"\"#\"\"\"*$%\"bGF'F(*$%\"cGF'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b = 1" "6#/%\"bG\"\"\"" }{TEXT -1 12 ", the ang le " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "45^o" "6#)\"#X%\"oG" }{TEXT -1 35 ", and the length of the hypot enuse " }{TEXT 269 1 "c" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 2 ". 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" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {TEXT 268 22 "______________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "The graph the sine of an acute an gle given in degrees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 167 "The following picture shows the graph of the sine func tion as a function of an angle given in degrees. The indicated points \+ are obtained by using the \"special values\":" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*0^o=0" "6#/*&%$sinG\"\"\")\"\"!%\"o GF&F(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin*30^o=1/2" "6#/*&%$sinG\"\" \")\"#I%\"oGF&*&F&F&\"\"#!\"\"" }{XPPEDIT 18 0 "`` = 0;" "6#/%!G\"\"! " }{TEXT -1 4 ".5, " }{XPPEDIT 18 0 "sin*45^o=sqrt(2)/2" "6#/*&%$sinG \"\"\")\"#X%\"oGF&*&-%%sqrtG6#\"\"#F&F.!\"\"" }{TEXT -1 1 " " }{TEXT 292 1 "~" }{TEXT -1 10 " 0.70711, " }{XPPEDIT 18 0 "sin*60^o=sqrt(3)/2 " "6#/*&%$sinG\"\"\")\"#g%\"oGF&*&-%%sqrtG6#\"\"$F&\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 293 1 "~" }{TEXT -1 14 " 0.866025 and " }{XPPEDIT 18 0 "sin*90^o=1" "6#/*&%$sinG\"\"\")\"#!*%\"oGF&F&" }{TEXT -1 2 ". 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The vertex P o f the triangle lying on the circle of unit radius with its centre at t he origin has coordinates" }{XPPEDIT 18 0 "``(cos*theta,sin*theta)" "6 #-%!G6$*&%$cosG\"\"\"%&thetaGF(*&%$sinGF(F)F(" }{TEXT -1 15 " as indic ated. 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"6#%&thetaG" }{TEXT -1 81 " to vary with the poi nt P remaining on the unit circle, and even allow the angle " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 20 " to increase beyond \+ " }{XPPEDIT 18 0 "90^o" "6#)\"#!*%\"oG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "The sine and cosine of an angle " }{XPPEDIT 18 0 "th eta" "6#%&thetaG" }{TEXT -1 23 " which is greater than " }{XPPEDIT 18 0 "90^o" "6#)\"#!*%\"oG" }{TEXT -1 27 " can be defined as follows." }} {PARA 0 "" 0 "" {TEXT -1 44 "Starting form the positive direction of t he " }{TEXT 270 1 "x" }{TEXT -1 42 " axis, rotate a line OP through th e angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 30 ". Then the cosine and sine of " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 18 " are given by the " }{TEXT 271 1 "x" }{TEXT -1 5 " and " }{TEXT 272 1 "y" }{TEXT -1 42 " coordinates respectively of the point P. 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" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 245 243 243 {PLOTDATA 2 "6<-%'CU RVESG6%7$7$$\"\"!F)F(7$$\"\"%F)F+-%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))e qk\"F3F4-%*THICKNESSG6#\"\"#-F$6%7$F*7$F+F(-%&COLORG6&F0F($\"\"(!\"\"F (F6-F$6%7$F=F'-F.6&F0$\"*++++\"F3F(F(F6-F$6$7S7$$\"\"\"F)F(7$$\"3up#G) eY`)***!#=$\"3kR6'\\'3'=r\"!#>7$$\"3)GQeAkv[***FT$\"3&ze,`N^4?$FW7$$\" 3iT\"[OY6\"))**FT$\"35f`P-Xsu[FW7$$\"3ADk*zwr%y**FT$\"31JaML*4#elFW7$$ \"37&ouVygg'**FT$\"3,\\A-1L%=B)FW7$$\"3YoPS&fY?&**FT$\"3IE_U7ZX\"y*FW7 $$\"3wI&[c$o*\\$**FT$\"34yEd\\([$Q6FT7$$\"3(*3b6,:l9**FT$\"3+f@&H\"*>P I\"FT7$$\"3%R$='HFT7$$\"3_i#>**f$4t(*FT$\"3Ti@qr$p\"=@FT7$$\"3U[mcG:1P(*FT$\"3 acMmal2yAFT7$$\"3ADXH%3u?q*FT$\"3W$3\")3WeFU#FT7$$\"3EfT.HMnd'*FT$\"3M (G*\\/z4%f#FT7$$\"3ZkJ$G[#p<'*FT$\"333B;+7hQFFT7$$\"3;IRr6_Yo&*FT$\"3d Fa.Fw$f!HFT7$$\"3!*\\r\\j:YA&*FT$\"3[*)\\'***QJ`IFT7$$\"31*3aXSx$p%*FT $\"3qU)\\EVqT@$FT7$$\"3R!)QzNII;%*FT$\"3mH\"p*et[mLFT7$$\"3WL+\"R29$e$ *FT$\"3\"oIsjxzW_$FT7$$\"3i!Q\\p'ps-$*FT$\"3Ontk4zooOFT7$$\"3qo!4z:y-C *FT$\"3g2O0&[_K#QFT7$$\"3OyQf)p#ps\"*FT$\"3!)f[Jngn#)RFT7$$\"3;-MR*[>; 6*FT$\"32yp))GU[?TFT7$$\"3%psc'yPLV!*FT$\"39#GlnGo#oUFT7$$\"3a<\"H!36F q*)FT$\"3tI$[\\Ab(>WFT7$$\"3i>f?&*zL'*))FT$\"3+c+[OhumXFT7$$\"3X:CC*>6 D#))FT$\"3_o;?/5z2ZFT7$$\"3W)*f'>K?zt)FT$\"3q+^z')y*H'[FT7$$\"3M#*4Zi@ ef')FT$\"3xy%e$)zN2H>iFT7$$\"3Rc)*o V5$ps(FT$\"3q$4N#zxzZjFT7$$\"3$*fktYboHwFT$\"3YWC:(*fNkkFT7$$\"3jPaW,I (FT$\"3[VJ'oU(FT$\"3!*4WXbfP\\pFT7$$\"3yD.96y1rqF T$\"3mD1f7y1rqFT-F.6&F0$\")!\\DP\"F3F_]l$\")viobF3-F$6$7%7$$\"3#)***** ********z$!#Fd^lFb_lF i`lFGFb`l-F_^l6&F`_lFi`lFGFb`l-F_^l6&7$$\"$y\"Fd^l$!#PFd^lQ\"vFh^lFG-F j^l6$F\\_l\"\")-F_^l6%7$$\"#WFC$\"#@FCFd_lF>-F_^l6%7$F_bl$\"# -F_^l6&7$$\"$F%Fd^l$\"$9#Fd^lFa`lF>Fb`l-F_^l6&7$F_bl$\"$[#Fd^lFi`lF>Fb `l-F_^l6&7$$\"$M%Fd^l$\"$6#Fd^lFi`lF>Fb`l-F_^l6&7$F_blFiclFi`lF>Fb`l-F _^l6&7$$\"#UFC$\"$.#Fd^lFhalF>Fial-F_^l6%7$$\"\"'FC$\"#CFd^lQ#45Fh^lF] ]l-F_^l6%7$$\"#xFd^l$F,FCQ\"oFh^lF]]l-%+AXESLABELSG6%Q!Fh^lFgel-Fj^l6# %(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$F jelFjel" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curv e 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" "Cur ve 21" "Curve 22" }}{TEXT -1 2 " " }{GLPLOT2D 275 275 275 {PLOTDATA 2 "68-%'CURVESG6%7$7$$\"\"!F)F(7$$\"\"\"F)$\"3?x)ov!30K NaiN'*RFT$\"3^!Hj`AZpq\"!#>7$$\"3%ee-GBZ:*RFT$\"3Us=>D:0*f#Fjn7$$\"3sD 1m3`p%)RFT$\"3]0^-=)ed\\$Fjn7$$\"33f%*3qg(e(RFT$\"3_g9dZ/X'Q%Fjn7$$\"3 3:)zN,@f'RFT$\"38S]]OKG5_Fjn7$$\"3iM$oa^9Q&RFT$\"39$=5K$p%41'Fjn7$$\"3 .0OKn\\PRRFT$\"3,2SQW!ex$pFjn7$$\"3H)))p?'o/BRFT$\"3\\PP0L,N3yFjn7$$\" 37FI>gVC/RFT$\"3-!fp$H>')*p)Fjn7$$\"3;P6Qv[+')QFT$\"3kg#*3GBQ\"[*Fjn7$ $\"3S5bmd-gjQFT$\"3NX1N#ogc.\"FT7$$\"3;#[d3b?\"RQFT$\"3)33<>'3,B6FT7$$ \"3A#QF+MhO\"QFT$\"3P(3c&RHk17FT7$$\"3u!*>I3M'*)y$FT$\"3yrot$f!4#G\"FT 7$$\"3)H)3.YNldPFT$\"3O^5yVa9r8FT7$$\"3wNHe$*=\\HPFT$\"3))p\"RU3\"*fW \"FT7$$\"3kaqhK&e[p$FT$\"3AP#H\"p^KK:FT7$$\"3A*y'>%HMDm$FT$\"3duCS;b03 ;FT7$$\"3C?E@fe.p\"FT7$$\"3cvh12[4)e$FT$\"3c;KVl(Hzw\" FT7$$\"3#QLBmNAva$FT$\"3w3EF%*[*z%=FT7$$\"3=FKL&H!p3NFT$\"3cLW$3-,2#>F T7$$\"3qb-9C\\8lMFT$\"3=NOi(=4#)*>FT7$$\"3jVB$e#33=MFT$\"3QS=\\3Enx?FT 7$$\"3!)QW**oejvLFT$\"3i:#*4\"fXf9#FT7$$\"3*[Lz)QJEGLFT$\"3=#4(zRyq=AF T7$$\"3+tRJ!*enxKFT$\"3M_CJP,y#H#FT7$$\"3/'G:())4fEKFT$\"3)4,xh)*HTO#F T7$$\"3]*3pcL&ovJFT$\"3]I%ow&>3KCFT7$$\"3++=m!)z[)))GFT$\"3cp%f_IHHV#FT$\"3a#R#y.j.vJFT7$$\"3kW5`XHFg^lQ\"oF[_lFa]l-%+AXESLABELSG6%Q!F[_lF]dl-F`al6#%(DE FAULTG-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$F`dlF `dl" 1 2 0 1 10 0 2 9 1 1 1 1.000000 47.000000 43.000000 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" "Cu rve 15" "Curve 16" "Curve 17" "Curve 18" }}{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 16 "For example, if " }{XPPEDIT 18 0 "theta = 120^o;" "6#/%&thetaG)\"$?\"%\"oG" }{TEXT -1 37 ", the right-angled triangle OP Q is a " }{XPPEDIT 18 0 "30,60,90;" "6%\"#I\"#g\"#!*" }{TEXT -1 17 " d egree triangle." }}{PARA 0 "" 0 "" {TEXT -1 11 "Hence QO = " } {XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 10 " and QP = \+ " }{XPPEDIT 18 0 "sqrt(3)/2" "6#*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" } {TEXT -1 18 ". This means that " }{XPPEDIT 18 0 "cos*120^o = -1/2;" "6 #/*&%$cosG\"\"\")\"$?\"%\"oGF&,$*&F&F&\"\"#!\"\"F-" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "sin*120^o = sqrt(3)/2;" "6#/*&%$sinG\"\"\")\"$?\"% \"oGF&*&-%%sqrtG6#\"\"$F&\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "For another example, if " }{XPPEDIT 18 0 "theta = 135^o;" "6#/%&thetaG)\"$N\"%\"oG" }{TEXT -1 37 ", the right-angled triangle OPQ is a " }{XPPEDIT 18 0 "45,45,90 ;" "6%\"#XF#\"#!*" }{TEXT -1 17 " degree triangle." }}{PARA 0 "" 0 "" {TEXT -1 11 "Hence QO = " }{XPPEDIT 18 0 "sqrt(2)/2;" "6#*&-%%sqrtG6# \"\"#\"\"\"F'!\"\"" }{TEXT -1 10 " and QP = " }{XPPEDIT 18 0 "sqrt(2)/ 2;" "6#*&-%%sqrtG6#\"\"#\"\"\"F'!\"\"" }{TEXT -1 18 ". This means that " }{XPPEDIT 18 0 "cos*135^o = -sqrt(2)/2;" "6#/*&%$cosG\"\"\")\"$N\"% \"oGF&,$*&-%%sqrtG6#\"\"#F&F/!\"\"F0" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "sin*135^o = sqrt(2)/2;" "6#/*&%$sinG\"\"\")\"$N\"%\"oGF&*&-%%sqr tG6#\"\"#F&F.!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 14 "radian measure" } {TEXT -1 13 " of an angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 8 " is the " }{TEXT 259 10 "arc length" }{TEXT -1 41 " cut of f on the unit circle, as shown in " }{TEXT 256 4 "blue" }{TEXT -1 33 " in the last two circle pictures." }}{PARA 0 "" 0 "" {TEXT -1 110 "Thi s makes sense because the arc length cut off on the unit circle is pro portional to the corresponding angle." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi radians = 360^ o" "6#/*(\"\"#\"\"\"%#PiGF&%(radiansGF&)\"$g$%\"oG" }{TEXT -1 9 ", and so " }{XPPEDIT 18 0 "Pi radians = 180^o" "6#/*&%#PiG\"\"\"%(radiansGF &)\"$!=%\"oG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "To convert from " }{TEXT 259 18 "degrees to rad ians" }{TEXT -1 14 ", multiply by " }{XPPEDIT 18 0 "Pi/180" "6#*&%#PiG \"\"\"\"$!=!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "To co nvert from " }{TEXT 259 18 "radians to degrees" }{TEXT -1 14 ", multip ly by " }{XPPEDIT 18 0 "180/Pi" "6#*&\"$!=\"\"\"%#PiG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "For example, 1 radian is " } {XPPEDIT 18 0 "180/Pi;" "6#*&\"$!=\"\"\"%#PiG!\"\"" }{TEXT -1 25 " deg rees, which is about " }{XPPEDIT 18 0 "57.296^o" "6#)-%&FloatG6$\"&'Hd !\"$%\"oG" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 5 "r ange" }{TEXT -1 81 " of both the sine and the cosine function is the s et of all real numbers between " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\" " }{TEXT -1 28 " and 1 inclusive, that is, \{" }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 3 " | " }{XPPEDIT 18 0 "theta in R" "6#-%#inG6$ %&thetaG%\"RG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-1<=theta" "6#1,$\"\" \"!\"\"%&thetaG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 2 " \+ \}" }{XPPEDIT 18 0 "``=[-1,1]" "6#/%!G7$,$\"\"\"!\"\"F'" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "A \+ procedure which animates the drawing of these graphs in relation to a \+ point rotating around the unit circle is given in the next section." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 22 ": The Maple procedure " }{TEXT 0 15 "convert/radians" } {TEXT -1 48 " can be used to convert from degrees to radians." }} {PARA 0 "" 0 "" {TEXT -1 72 "The expression to be converted must be of degree 1 in the indeterminate " }{TEXT 274 7 "degrees" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "convert(120*degrees,radians);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"\"\"$!\"\"%#PiGF&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 15 "co nvert/degrees" }{TEXT -1 48 " can be used to convert from radians to d egrees." }}{PARA 0 "" 0 "" {TEXT -1 61 "The converted expression is of degree 1 in the indeterminate " }{TEXT 274 7 "degrees" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "convert(2*Pi/3,degrees);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *&\"$?\"\"\"\"%(degreesGF&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 53 "The graph the sine of an acute angle given in radians" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 170 "The fo llowing picture shows the graph of the sine function as a function of \+ an acute angle given in radians, or equivalently, as a function of rea l numbers between 0 and " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#! \"\"" }{TEXT -1 67 ". Points corresponding to the following \"special \+ values\" are shown:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*0 = 0;" "6#/*&%$sinG\"\"\"\"\"!F&F'" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "sin(Pi/6) = 1/2;" "6#/-%$sinG6#*&%#PiG\"\"\"\"\"'!\"\"* &F)F)\"\"#F+" }{XPPEDIT 18 0 "`` = 0;" "6#/%!G\"\"!" }{TEXT -1 4 ".5, \+ " }{XPPEDIT 18 0 "sin(Pi/4) = sqrt(2)/2;" "6#/-%$sinG6#*&%#PiG\"\"\"\" \"%!\"\"*&-%%sqrtG6#\"\"#F)F0F+" }{TEXT -1 1 " " }{TEXT 294 1 "~" } {TEXT -1 10 " 0.70711, " }{XPPEDIT 18 0 "sin(Pi/3) = sqrt(3)/2;" "6#/- %$sinG6#*&%#PiG\"\"\"\"\"$!\"\"*&-%%sqrtG6#F*F)\"\"#F+" }{TEXT -1 1 " \+ " }{TEXT 295 1 "~" }{TEXT -1 11 " 0.866025, " }{XPPEDIT 18 0 "sin(Pi/2 ) = 1;" "6#/-%$sinG6#*&%#PiG\"\"\"\"\"#!\"\"F)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Points co rresponding to the following extra special values are also shown:" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi/12)=(sqrt(6)-s qrt(2))/4" "6#/-%$sinG6#*&%#PiG\"\"\"\"#7!\"\"*&,&-%%sqrtG6#\"\"'F)-F/ 6#\"\"#F+F)\"\"%F+" }{TEXT -1 1 " " }{TEXT 297 1 "~" }{TEXT -1 10 " 0. 25882, " }{XPPEDIT 18 0 "sin(5*Pi/12)=(sqrt(6)+sqrt(2))/4" "6#/-%$sinG 6#*(\"\"&\"\"\"%#PiGF)\"#7!\"\"*&,&-%%sqrtG6#\"\"'F)-F06#\"\"#F)F)\"\" %F," }{TEXT -1 1 " " }{TEXT 296 1 "~" }{TEXT -1 10 " 0.96593. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "The last two special values can be obtained by using the addition formulas whi ch will be derived later." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 779 "p1:=plot(sin(x),x=0..Pi/2,thicknes s=2):\np2:=plot([[[Pi/12,0],[Pi/12,(sqrt(6)-sqrt(2))/4],[0,(sqrt(6)-sq rt(2))/4]],\n [[Pi/6,0],[Pi/6,1/2],[0,1/2]],[[Pi/4,0],[Pi/4,sqrt(2)/2 ],[0,sqrt(2)/2]],\n [[Pi/3,0],[Pi/3,sqrt(3)/2],[0,sqrt(3)/2]],\n [ [5*Pi/12,0],[5*Pi/12,(sqrt(6)+sqrt(2))/4],[0,(sqrt(6)+sqrt(2))/4]],\n \+ [[Pi/2,0],[Pi/2,1],[0,1]]],color=black,linestyle=2):\np3:=plot([[[0, 0],[Pi/12,(sqrt(6)-sqrt(2))/4],[Pi/4,sqrt(2)/2],[Pi/6,1/2],[Pi/3,sqrt( 3)/2],\n [5*Pi/12,(sqrt(6)+sqrt(2))/4],[Pi/2,1]]$3],color=black,style= point,\n symbol=[circle,diamond,cross]):\npi := evalf(Pi):\nplots[di splay]([p1,p2,p3],xtickmarks=[pi/12=`p/12`,pi/6=`p/6`,pi/4=`p/4`,pi/3= `p/3`,\n 5*pi/12=`5p/12`,pi/2=`p/2`],font=[SYMBOL,11],labels=[`angle i n radians`,`sine of angle`],\n labelfont=[HELVETICA,10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 606 425 425 {PLOTDATA 2 "60-%'CURVESG6%7S7$$\"\"!F)F (7$$\"3NGK5j*))QU$!#>$\"3QsLeI+ABMF-7$$\"3DXXYUk*HS'F-$\"3]DS+L@i)R'F- 7$$\"3=N68yVJ`(*F-$\"3_b&ehJeyt*F-7$$\"3%3h%eRSe78!#=$\"3<&Q%f]#=)38F= 7$$\"3k+,hQPB[;F=$\"3V-yj65yS;F=7$$\"3'H#*ed(RUf>F=$\"3q)*)ys&)4p%>F=7 $$\"3kHX[TMk\"G#F=$\"3G3RBS#)*=E#F=7$$\"3Saq%HF=$\"3=7dd?+e/HF=7$$\"3)pq'4OKt)G$F=$\"3c0jx$Rp(HKF=7 $$\"3#f/N#RTo*e$F=$\"3ML>uye38NF=7$$\"3Ut79wN[GRF=$\"3V3zj5M@GQF=7$$\" 3/VrOc6q%F=7$$\"31\\,oR/A[_F=$\"3SMGBr+f5]F=7$$\"3uvW/ X(*RG,:eF=7$$\"3.n3p46^WlF=$\"33g0zi&Qs3'F=7$$\"3oHyF.C6no F=$\"3h+)=j,t*RjF=7$$\"3(*zx.5Ir.sF=$\"3Qz3MFxj'f'F=7$$\"3'pz!QUu\"G^( F=$\"3$[8q$y.wDoF=7$$\"3A\"3@7LGi%yF=$\"3wm47uKelqF=7$$\"3)3\"HIT&[D>) F=$\"3M(RvZssjI(F=7$$\"3gv)\\AI@S\\)F=$\"3C>#p\"po&)3vF=7$$\"3vn()=V,i >))F=$\"3%4zeG%\\()>xF=7$$\"3'G:[dg&*f:*F=$\"31()\\)y]!GHzF=7$$\"3sSt6 rL2&[*F=$\"3g&H%*=Uja7)F=7$$\"3mka(*HIZ.)*F=$\"3%3R7+w2pI)F=7$$\"3Em\" [l:+d,\"!#<$\"3Q?Zs+w\\)\\)F=7$$\"3c!3XyEmu/\"F_u$\"3Ih!)R3tfh')F=7$$ \"3sJ_*>P$Q\"3\"F_u$\"3VZPF_)*3E))F=7$$\"3.M\"G%4t676F_u$\"3kW#3Hj\"Qm *)F=7$$\"3co*Q4iq'*z.@\"F_u$\"3Y(Q,up+vN*F=7$$\"34Q$[)>&*oU7F_u$\"3OZ!) zP7am%*F=7$$\"3I^lOFY^w7F_u$\"3-w%)pSt5q&*F=7$$\"3`!)f6EA448F_u$\"3-KP DS[]f'*F=7$$\"3=;T7EvSU8F_u$\"3B\"='y#[C.u*F=7$$\"3a_wiepWv8F_u$\"3$oY $>Q\"*z4)*F=7$$\"3U$obdw1eS\"F_u$\"3#4Wi\"*p+U')*F=7$$\"3s$)o#3`-1W\"F _u$\"3aZ!3__n`\"**F=7$$\"3-#*)4zFC)e#F=7$F(Fi[l-Fhz6&F jzF)F)F)-%*LINESTYLEGF`[l-F$6%7%7$$\"3;))H)fv()fB&F=F(7$Fd\\l$\"3+++++ +++]F=7$F(Fg\\lF\\\\lF^\\l-F$6%7%7$$\"3!G[uRj\")R&yF=F(7$F^]l$\"3sva'= \"y1rqF=7$F(Fa]lF\\\\lF^\\l-F$6%7%7$$\"3k(f'>^v>Z5F_uF(7$Fh]l$\"3'fQWy .a-m)F=7$F(F[^lF\\\\lF^\\l-F$6%7%7$$\"3@Zd**Qp**38F_uF(7$Fb^l$\"3-#o!* GEe#f'*F=7$F(Fe^lF\\\\lF^\\l-F$6%7%7$$\"3c'*[zEjzq:F_uF(7$F\\_lFez7$F( FezF\\\\lF^\\l-F$6&7)F'Fh[lF`]lFf\\lFj]lFd^lF^_l-%'SYMBOLG6#%'CIRCLEGF \\\\l-%&STYLEG6#%&POINTG-F$6&Fb_l-Fd_l6#%(DIAMONDGF\\\\lFg_l-F$6&Fb_l- Fd_l6#%&CROSSGF\\\\lFg_l-%*AXESTICKSG6$7(/$\"+yQ*zh#!#5%%p/12G/$\"+dx) fB&F\\al%$p/6G/$\"+N;)R&yF\\al%$p/4G/$\"+^v>Z5!\"*%$p/3G/$\"+Rp**38Fia l%&5p/12G/$\"+Fjzq:Fial%$p/2G%(DEFAULTG-%+AXESLABELSG6%%1angle~in~radi ansG%.sine~of~angleG-%%FONTG6$%*HELVETICAGF\\[l-Fjbl6$Fd_l\"#6-%%VIEWG 6$;F(F`blFcbl" 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 373 "In general it is not straightforward to obtain a numeric al value for the sine of an angle by means of the arithmetic operation s of addition, subtraction, multiplication, division and taking square roots. However, scientific calculators are programmed to perform the \+ necessary computation needed to obtain an accurate value for the sine \+ of a real number (angle in radians). " }}{PARA 0 "" 0 "" {TEXT -1 81 " Maple can quickly compute many digits in the sine of a number (angle i n radians)." }}{PARA 0 "" 0 "" {TEXT -1 76 "The first 100 digits in th e value for the sine of 1 radian are given below. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sin(1);\neva lf[100](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qfp_IaV(R'[*o'R7R//\">**4 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 69 "A procedure for the animated construction of sine and c osine curves: " }{TEXT 0 13 "triganimation" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "triganimation: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 18 "Calling S equence:\n" }}{PARA 0 "" 0 "" {TEXT 261 4 " " }{TEXT -1 17 "trigani mation( )\n" }}{PARA 0 "" 0 "" {TEXT -1 38 " triganimation( . . opt ions . . ) \n" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 7 " None" }} {PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "De scription:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 13 "triganimation" }{TEXT -1 152 " pro vides animations which shows the graph of the the sine and cosine func tions being drawn alongside a point rotating around a circle of unit r adius. \n" }}{PARA 0 "" 0 "" {TEXT 262 8 "Options:" }{TEXT -1 1 "\n" } }{PARA 0 "" 0 "" {TEXT -1 16 "function=sin/cos" }}{PARA 0 "" 0 "" {TEXT -1 68 "Which graph to draw. The default graph is that of the sin e function." }}{PARA 0 "" 0 "" {TEXT -1 74 "This options can also be e ntered as \"function=sine\" or \"function=cosine\". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "quadrants=n\nThe numbe r of quadrants of the unit circle to show, where n = 1,2 or 4. The def ault value of \"quadrants\" is 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 85 "frames=n\nThe number of frames for the an imation. The default value of \"frames\" is 40." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To ma ke the procedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the s ubsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "triganimation: impl ementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 4532 "triganimation := proc()\n local i,j,h,cosdata,s indata,circledata,leftcircledata,\n rightcircledata,cutsindata ,cutcosdata,theta,costheta,\n sintheta,piby2,origin,left,botto m1,bottom2,frms,pt1,xpt1,\n pt2,xpt2,x1,delta,arc,toplot1,topl ot2,frame,seqn,quad,\n fcn,Options,saveDigits;\n \n # Get \+ the options we need to process.\n frms := 40;\n fcn := 'sin';\n \+ quad := 4;\n if nargs>0 then\n Options:=[args[1..nargs]];\n \+ if not type(Options,list(equation)) then\n error \"each opti onal argument must be an equation\"\n end if;\n if hasoption (Options,'function','fcn','Options') then\n if not member(fcn, \{'sin','cos','sine','cosine'\}) then\n error \"\\\"functio n\\\" must be 'sin' <-> 'sine' or 'cos' <-> 'cosine'\"\n end i f;\n if fcn='sine' then fcn := 'sin' end if;\n if fcn= 'cosine' then fcn := 'cos' end if;\n end if;\n if hasoption( Options,'frames','frms','Options') then\n if not type(frms,pos int) or frms<2 then\n error \"\\\"frames\\\" must be an int eger greater than 1\" \n end if;\n end if;\n if has option(Options,'quadrants','quad','Options') then\n if not (qu ad=1 or quad=2 or quad=4) then\n error \"\\\"quadrants\\\" \+ must be 1,2 or 4\"\n end if;\n end if;\n if nops(Opt ions)>0 then\n error \"%1 is not a valid option for %2\",op(1, Options),procname;\n end if;\n end if;\n \n saveDigits := D igits;\n Digits := 14;\n piby2 := evalf(Pi/2);\n origin := [-pib y2,0];\n arc := piby2;\n delta := evalf(piby2/frms);\n\n if quad =1 then\n left := -piby2;\n bottom1 := -0.1;\n bottom2 \+ := bottom1;\n elif quad=2 then\n delta := delta*2;\n arc : = arc*2;\n left := -1.0-piby2;\n bottom1 := -0.1;\n if \+ fcn=sin then bottom2 := bottom1 else bottom2 := -1.1 end if;\n elif \+ quad=4 then\n delta := delta*4;\n arc := arc*4;\n left \+ := -1.0-piby2;\n bottom1 := -1.1;\n bottom2 := bottom1;\n \+ end if;\n\n # compute all the trig function values needed.\n h := \+ arc/50.0;\n cosdata := NULL;\n sindata := NULL;\n circledata := \+ NULL;\n for i from 0 to 50 do\n theta := i*h;\n costheta : = cos(theta);\n sintheta := sin(theta);\n cosdata := cosdata ,[theta,costheta];\n sindata := sindata,[theta,sintheta];\n \+ circledata := circledata,[costheta,sintheta];\n end do;\n\n # Cons truct the animation frames.\n h := evalf(arc/frms);\n seqn := NULL ;\n for i from 0 to frms do\n theta := i*h;\n costheta := \+ cos(theta);\n sintheta := sin(theta);\n x1 := costheta-piby2 ;\n pt1 := [x1,sintheta];\n xpt1 := [x1,0];\n xpt2 := [ theta,0];\n \n # Partition data and shift circle to left.\n \+ leftcircledata := NULL;\n rightcircledata := NULL;\n cu tcosdata := NULL;\n cutsindata := NULL;\n for j from 1 to 51 do\n if sindata[j][1]<=theta then\n cutcosdata := \+ cutcosdata,cosdata[j];\n cutsindata := cutsindata,sindata[j ];\n leftcircledata := \n leftcircledata,[circle data[j][1]-piby2,circledata[j][2]];\n else\n rightc ircledata :=\n rightcircledata,[circledata[j][1]-piby2,circ ledata[j][2]];\n end if;\n end do;\n \n leftcircl edata := leftcircledata,pt1;\n rightcircledata := pt1,rightcircle data;\n toplot1 := CURVES([leftcircledata],COLOR(RGB,0,0,1)),\n \+ CURVES([rightcircledata],COLOR(RGB,1,0,0)),\n CURVES([[-piby2 ,bottom1],[-piby2,1.1]]);\n\n if fcn='sin' then\n pt2:=[t heta,sintheta];\n cutsindata := cutsindata,pt2;\n topl ot2 := \n POINTS(pt1,pt2,SYMBOL(CIRCLE)),CURVES([origin,pt1]), \n CURVES([[0,0],xpt2],COLOR(RGB,0,0,1),THICKNESS(2)),\n \+ CURVES([xpt1,pt1],[xpt2,pt2],COLOR(RGB,0,0.8,0)),\n CURVES( [cutsindata],COLOR(RGB,1,0,0));\n else\n pt2:=[theta,cost heta];\n cutcosdata := cutcosdata,pt2;\n toplot2 := \n POINTS(pt1,pt2,SYMBOL(CIRCLE)),\n CURVES([origin,pt1] ,[xpt1,pt1]),\n CURVES([[0,0],xpt2],COLOR(RGB,0,0,1),THICKNESS (2)),\n CURVES([origin,xpt1],COLOR(RGB,0,0.8,0),THICKNESS(2)), \n CURVES([xpt2,pt2],COLOR(RGB,0.0,0.8,0.0)),\n CURVES ([cutcosdata],COLOR(RGB,1,0,0));\n end if;\n frame := PLOT(t oplot1,toplot2);\n seqn := seqn,frame;\n end do:\n Digits := \+ saveDigits;\n plots[display]([seqn],insequence=true,tickmarks=[0,0], \n scaling=constrained,view=[left..arc,bottom2..1.1]);\nend \+ proc: # of triganimation" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next sectio n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "triganimati on" }{TEXT -1 10 ": examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 115 "To run the animation, click on the graphic and then use \+ the controls in the context bar, or use the animation menu." }}{PARA 15 "" 0 "" {TEXT -1 116 "You should probably delete the animation pict ures from one example before drawing another, so as to conserve memory ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "triganimation();" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "triganimation(function='cos' );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "triganimation(quadrants=1,function='sin',frames=40); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "triganimation(quadrants=1,function='cosine',frames=40 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "The wrappi ng function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "One can visualise the sine and cosine functions as functions o f real numbers with the aid of the following " }{TEXT 259 17 "wrapping function" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 91 "Consider th e real line situated in the coordinate plane as the vertical line with equation " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 43 ". This real line is tangential at the point" }{XPPEDIT 18 0 " ``(1,0)" "6#-% !G6$\"\"\"\"\"!" }{TEXT -1 50 " to the unit circle with its centre at \+ the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 452 388 388 {PLOTDATA 2 "6/-%'CURVESG6%7$7$$\"\"!F)$!3++++++ ++N!#<7$F($\"3++++++++NF,-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%*THICKN ESSG6#\"\"#-F$6$7$7$$!\"\"F)F*7$F?F.-F16&F3F)F)F)-F$6$7S7$F(F(7$$!3%R# 3*QgiPO*!#?$\"3Ikwb#=y_O\"!#=7$$!3Sn)o<,z>E$!#>$\"3%[#\\ff*)GLDFN7$$!3 !)\\'*Q6_89vFR$\"3gLj&[K5J!QFN7$$!3#)RJ?\"3HpM\"FN$\"3j9NXR4U7]FN7$$!3 eW\\7nzu&4#FN$\"3t_H-qceDhFN7$$!3[]1OJeU8\"zFN7$$!3crDpV%G#*)\\FN$\"3-$>p(Qh-a')FN7$$!3 Uw\"H`Ng)zhFN$\"3aK+16bcT#*FN7$$!3sL&yPs8YZ(FN$\"3s&Q\"[M\"oen*FN7$$!3 ?x[QxJ'[l)FN$\"33)fVPO<\"4**FN7$$!3B/u(p5(f+5F,$\"2%HhJ<#)******F,7$$! 3q\\T#fB[i8\"F,$\"3ap%>wGZn!**FN7$$!3[:d.:#=YE\"F,$\"3PA>=\\D`V'*FN7$$ !3%zV1J*3Jx8F,$\"3IzF#e`m3E*FN7$$!3u\\AOF:B/:F,$\"3vU'yI2&oN')FN7$$!3O 1cjZ*[Rg\"F,$\"3!G;)RQ+BqzFN7$$!3]CknM;$*3XOJF O4B>F,$\"3In'[*>HvXQFN7$$!3q!RS4Nij'>F,$\"3o$p9m#Q%=d#FN7$$!3!ypT>+.2* >F,$\"31V?00]Ug8FN7$$!3+E$G4>&****>F,$\"3vCA\\O%485$FK7$$!3O0B(=!Q%3*> F,$!3q#4*>c=8]8FN7$$!3?b`sYlSn>F,$!3UNbFGIGKDFN7$$!3@:LI@iKE>F,$!3q4&= j`Bsw$FN7$$!3gF)4Kh=u'=F,$!3kENX')3zv\\FN7$$!3K#*z7xng%z\"F,$!37W(H!pU CrgFN7$$!3mK\"4Pfc5r\"F,$!3Y?#o?\"yMJqFN7$$!3Q%y6Ike[g\"F,$!3)4j2yeGL' zFN7$$!35Aa#=%)o!*\\\"F,$!3I_Mh6Mil')FN7$$!3'*f?<[\"ysP\"F,$!3#\\IN,%* **4E*FN7$$!3wZ#H\"el'3E\"F,$!3=V>(fp[Pl*FN7$$!33@T_xm:H6F,$!3;$RoI*>C; **FN7$$!3!zOHMQdI+\"F,$!3W1O#>E`*****FN7$$!3spxOYnF7()FN$!3lYZ3X=u;**F N7$$!3ungHa'e\\W(FN$!3U$[o%H'z!o'*FN7$$!3C:@W9;?khFN$!36.2<#>x]B*FN7$$ !3k(Q-RU*p'*\\FN$!3wHgb5wMe')FN7$$!3!3v60g\\1*QFN$!3/a*[=H2o\"zFN7$$!3 #fRKNh*p+HFN$!3)4d)eg?sUqFN7$$!3]S\"yig@)*4#FN$!3D1$H)zD(p_v\\FN7$$!3-s;%yY!**\\xFR$!3i&>2ndo*fQFN7$$!3OR8x2 86`MFR$!3%)f]nRQ=0EFN7$$!3Hz?W.kW/$*FK$!3/up91t'4O\"FN7$F($\"36YKhSr8/ #)!#F-F16&F3F4F(F(-F$6&7#FGFB-%'SYMBOLG6#%'CIRCLEG-%&STYLEG6#%&POINTG- %%TEXTG6%7$$\"\"%F@$F@F@Q\"x6\"-%%FONTG6$%*HELVETICAG\"#5-Fb]l6%7$$!#6 F@$\"#NF@Q\"yFi]lFj]l-Fb]l6%7$$\"\"$F@Fj^lQ&(1,0)Fi]lFj]l-Fb]l6%7$$!\" #F@$\"#MF@Q\"qFi]l-F[^l6$Fj\\l\"#6-%*AXESTICKSG6$F)7&/$!+aEfTJ!\"*%#-p G/$!+Fjzq:F_`l%%-p/2G/$\"+Fjzq:F_`l%$p/2G/$\"+aEfTJF_`l%\"pG-%(SCALING G6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!Fi]lFdal-F[^l6#%(DEFAULTGFe_l-%%VIE WG6$;$!#@F@Fe]l;$!#NF@Fd^l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 202 "We now imagine wrapping the upper part of this real line (which corresponds to the positive real numbers) around the unit circle in a counter-clockwise direction as indicated by the following animation." }}{PARA 0 "" 0 "" {TEXT -1 119 "To play the animation cli ck on the graph and use the controls in the context bar or the command s in the animation menu." }}{PARA 0 "" 0 "" {TEXT -1 69 "The animation just shows the wrapping for real numbers between 0 and " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". 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3y:oS\"G2o\"zFH7$$\"3AR8!))R+$*4(FH$!3w^d:[?sUqFH7$$\"3#ei&p/%y,!zFH$! 3pZ&)eP&Q38'FH7$$\"3!>TO`_PVn)FH$!33p!pm&p_v\\FH7$$\"3kj)[/'4+D#*FH$!3 Ee3Uf&o*fQFH7$$\"3EvY@u))oa'*FH$!3&*[c<@Q=0EFH7$$\"3`s$G'Qb&p!**FH$!3# pdjnGn4O\"FH7$F,$\"3rm?azQT?GFac\\rF4F;-F(6$7SF]\\l7$F,$\"3eeYNa<3S')F `\\l7$F,$\"31\"*f07hR>!*F`\\l7$F,$\"3Co&pMWqfW*F`\\l7$F,$\"3y!zdA=p`() *F`\\l7$F,$\"3,!\\Y#*Rt-.\"Fac\\r7$F,$\"3g7$\\&Q]*)p5Fac\\r7$F,$\"3y+@ 6w<#46\"Fac\\r7$F,$\"3Y&f*RZ4N`6Fac\\r7$F,$\"3-w))QFUk&>\"Fac\\r7$F,$ \"3'y?kX4[\"R7Fac\\r7$F,$\"3OI`7wnYx7Fac\\r7$F,$\"3/ELhDQg?8Fac\\r7$F, $\"3wh_y5&=RO\"Fac\\r7$F,$\"3$))[(zV$fcS\"Fac\\r7$F,$\"3y]MRA]cV9Fac\\ r7$F,$\"3u#RO3IQ')[\"Fac\\r7$F,$\"37,$*4\"4@o_\"Fac\\r7$F,$\"3GXKLHiBr :Fac\\r7$F,$\"3Kr73Z9b5;Fac\\r7$F,$\"3gFC@?no`;Fac\\r7$F,$\"3gz)4%>9w% p\"Fac\\r7$F,$\"3upcp#p=wt\"Fac\\r7$F,$\"3q@%H[lvpx\"Fac\\r7$F,$\"3()3 udIqU>=Fac\\r7$F,$\"3IT=SO:_j=Fac\\r7$F,$\"3g@g;Vf!>!>Fac\\r7$F,$\"3lW R3fDOV>Fac\\r7$F,$\"3Q/.m59>')>Fac\\r7$F,$\"3!yl4wA\"4G?Fac\\r7$F,$\"3 iI+Ab.jo?Fac\\r7$F,$\"3'HVIvBVO6#Fac\\r7$F,$\"3EPnS1\"*3a@Fac\\r7$F,$ \"3u'RHv6us>#Fac\\r7$F,$\"3([aWB[0kB#Fac\\r7$F,$\"3]gYYgs=zAFac\\r7$F, $\"3))*)Qi%RS%>BFac\\r7$F,$\"3I$=ds%y^hBFac\\r7$F,$\"3,MC72cl-CFac\\r7 $F,$\"3Qt_^!\\AdW#Fac\\r7$F,$\"3k3@2*>-s[#Fac\\r7$F,$\"3@Z,/C)>'HDFac \\r7$F,$\"3%R#3r1vorDFac\\r7$F,$\"3z7I*y!GM5EFac\\r7$F,$\"3uugkNgkaEFa c\\r7$F,$\"37wDmF7F%p#Fac\\r7$F,$\"3or3t)4?lt#Fac\\r7$F,$\"3q)R,S(z&px #Fac\\rFjfb`lFa\\l-F(6$7$Fi\\l7$F,$\"3>+++BNT?GFac\\rFj\\l-F(6&7#F``c` lFj\\lF^]lFb]lFf]lF_^l-%*AXESTICKSG6$F.F.-%(SCALINGG6#%,CONSTRAINEDGF_ ^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" }}{TEXT -1 1 " " }}{PARA 257 " " 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 207 "In a similar wa y the lower part of the real line (which corresponds to the negative r eal numbers) can be wrapped in a clockwise direction around the unit c ircle starting around the lower part of the circle. " }}{PARA 0 "" 0 " " {TEXT -1 66 "If we denote the wrapping function by W then for each r eal number " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 62 " corres ponding to a point on the vertical number line, we have" }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "W(theta)=``(cos*theta,sin*theta) " "6#/-%\"WG6#%&thetaG-%!G6$*&%$cosG\"\"\"F'F-*&%$sinGF-F'F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Examples: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "W(0)=``(1,0)" "6#/-%\"WG6#\"\"!-%!G6 $\"\"\"F'" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "W(Pi/4) = ``(sqrt(2)/2,sqrt(2)/2);" "6#/-%\"WG6#*&%#PiG \"\"\"\"\"%!\"\"-%!G6$*&-%%sqrtG6#\"\"#F)F3F+*&-F16#F3F)F3F+" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "W(Pi/2)=`` (0,1)" "6#/-%\"WG6#*&%#PiG\"\"\"\"\"#!\"\"-%!G6$\"\"!F)" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "W(3*Pi/4) = ``(- sqrt(2)/2,sqrt(2)/2);" "6#/-%\"WG6#*(\"\"$\"\"\"%#PiGF)\"\"%!\"\"-%!G6 $,$*&-%%sqrtG6#\"\"#F)F5F,F,*&-F36#F5F)F5F," }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "W(Pi)=``(-1,0)" "6#/-%\"WG6 #%#PiG-%!G6$,$\"\"\"!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "W(5*Pi/4) = ``(-sqrt(2)/2, -sqrt(2)/2); " "6#/-%\"WG6#*(\"\"&\"\"\"%#PiGF)\"\"%!\"\"-%!G6$,$*&-%%sqrtG6#\"\"#F )F5F,F,,$*&-F36#F5F)F5F,F," }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "W(3*Pi/2)=``(0,-1)" "6#/-%\"WG6#*(\"\"$\"\"\" %#PiGF)\"\"#!\"\"-%!G6$\"\"!,$F)F," }{TEXT -1 1 " " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "W(7*Pi/4) = ``(sqrt(2)/2,-sqrt(2)/2); " "6#/-%\"WG6#*(\"\"(\"\"\"%#PiGF)\"\"%!\"\"-%!G6$*&-%%sqrtG6#\"\"#F)F 4F,,$*&-F26#F4F)F4F,F," }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "W(2*Pi)=``(1,0)" "6#/-%\"WG6#*&\"\"#\"\"\"%#PiGF )-%!G6$F)\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 " The tangent function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 7 "tangent" }{TEXT -1 24 " function is defined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "tan*theta = sin*theta/(cos*theta);" "6#/*&%$tanG\"\"\"% &thetaGF&*(%$sinGF&F'F&*&%$cosGF&F'F&!\"\"" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT 278 8 "________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "for any real number " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 11 " (or angle " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 12 ") such that " } {XPPEDIT 18 0 "cos*theta <> 0;" "6#0*&%$cosG\"\"\"%&thetaGF&\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 65 "This is consistent with the definition for right angle triangles." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan*theta" "6 #*&%$tanG\"\"\"%&thetaGF%" }{TEXT -1 8 " is the " }{TEXT 259 8 "gradie nt" }{TEXT -1 48 " of the line OP joining the origin to the point " } {XPPEDIT 18 0 "P(cos*theta,sin*theta);" "6#-%\"PG6$*&%$cosG\"\"\"%&the taGF(*&%$sinGF(F)F(" }{TEXT -1 2 ". 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" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[tan*0 = 0, ` |`, tan(2*Pi/3) = -sqrt(3)], [tan(Pi/6) = sqrt(3)/3, `|`, tan(3*Pi/4) \+ = -1], [tan(Pi/4) = 1, `|`, tan(5*Pi/6) = -sqrt(3)/3], [tan(Pi/3) = sq rt(3), `|`, tan*Pi = 0]]);" "6#-%'matrixG6#7&7%/*&%$tanG\"\"\"\"\"!F+F ,%\"|grG/-F*6#*(\"\"#F+%#PiGF+\"\"$!\"\",$-%%sqrtG6#F4F57%/-F*6#*&F3F+ \"\"'F5*&-F86#F4F+F4F5F-/-F*6#*(F4F+F3F+\"\"%F5,$F+F57%/-F*6#*&F3F+FGF 5F+F-/-F*6#*(\"\"&F+F3F+F?F5,$*&-F86#F4F+F4F5F57%/-F*6#*&F3F+F4F5-F86# F4F-/*&F*F+F3F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "The tangent function is periodic with per iod " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 1 ":" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(theta+Pi) = tan*theta;" "6#/-%$t anG6#,&%&thetaG\"\"\"%#PiGF)*&F%F)F(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "The following picture shows the graph of " } {XPPEDIT 18 0 "y=tan*theta" "6#/%\"yG*&%$tanG\"\"\"%&thetaGF'" }{TEXT -1 2 ". 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These are the values of " }{XPPEDIT 18 0 "theta" " 6#%&thetaG" }{TEXT -1 11 " for which " }{XPPEDIT 18 0 "cos*theta=0" "6 #/*&%$cosG\"\"\"%&thetaGF&\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 12 " ap proaches " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 26 " through values less than " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\" \"\"\"\"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tan*theta" "6#*&%$tan G\"\"\"%&thetaGF%" }{TEXT -1 34 " gets larger and larger, that is, " } {XPPEDIT 18 0 "tan*theta" "6#*&%$tanG\"\"\"%&thetaGF%" }{TEXT -1 12 " \+ approaches " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 13 "We write: as " }{XPPEDIT 18 0 "theta- >Pi/2" "6#f*6#%&thetaG7\"6$%)operatorG%&arrowG6\"*&%#PiG\"\"\"\"\"#!\" \"F*F*F*" }{XPPEDIT 18 0 "``^`-`" "6#)%!G%\"-G" }{TEXT -1 6 ", tan " } {XPPEDIT 18 0 "theta->infinity" "6#f*6#%&thetaG7\"6$%)operatorG%&arrow G6\"%)infinityGF*F*F*" }{TEXT -1 19 ", or equivalently, " }{XPPEDIT 18 0 "Limit(tan*theta,theta=Pi/2,left)=infinity" "6#/-%&LimitG6%*&%$ta nG\"\"\"%&thetaGF)/F**&%#PiGF)\"\"#!\"\"%%leftG%)infinityG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Similarly, as " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 12 " approaches " }{XPPEDIT 18 0 "-Pi/2 " "6#,$*&%#PiG\"\"\"\"\"#!\"\"F(" }{TEXT -1 29 " through values greate r than " }{XPPEDIT 18 0 "-Pi/2" "6#,$*&%#PiG\"\"\"\"\"#!\"\"F(" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "tan*theta" "6#*&%$tanG\"\"\"%&thetaGF% " }{TEXT -1 60 " gets larger and larger in the negative direction, tha t is, " }{XPPEDIT 18 0 "tan*theta" "6#*&%$tanG\"\"\"%&thetaGF%" } {TEXT -1 12 " approaches " }{XPPEDIT 18 0 "-infinity;" "6#,$%)infinity G!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "We write: as \+ " }{XPPEDIT 18 0 "theta->Pi/2" "6#f*6#%&thetaG7\"6$%)operatorG%&arrowG 6\"*&%#PiG\"\"\"\"\"#!\"\"F*F*F*" }{XPPEDIT 18 0 "``^`+`;" "6#)%!G%\"+ G" }{TEXT -1 6 ", tan " }{XPPEDIT 18 0 "theta->infinity" "6#f*6#%&thet aG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 19 ", or equ ivalently, " }{XPPEDIT 18 0 "Limit(tan*theta,theta = -Pi/2,right) = -i nfinity;" "6#/-%&LimitG6%*&%$tanG\"\"\"%&thetaGF)/F*,$*&%#PiGF)\"\"#! \"\"F0%&rightG,$%)infinityGF0" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 6 "domain " }{TEXT -1 37 " of the tangent function is the set \{" }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 3 " | " }{XPPEDIT 18 0 "theta in R" " 6#-%#inG6$%&thetaG%\"RG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta <> k*P i/2" "6#0%&thetaG*(%\"kG\"\"\"%#PiGF'\"\"#!\"\"" }{TEXT -1 8 ", where \+ " }{TEXT 281 1 "k" }{TEXT -1 22 " is an odd integer \}. " }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 259 5 "range" }{TEXT -1 56 " of the ta ngent function is the set of all real numbers " }{TEXT 280 1 "R" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 88 "Calcu lating values of the sine, cosine and tangent functions in terms of sp ecial values " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Given an angle (or real number) " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 26 " the following associated " }{TEXT 259 18 " reference triangle" }{TEXT -1 26 " can be used to determine " } {XPPEDIT 18 0 "cos*theta" "6#*&%$cosG\"\"\"%&thetaGF%" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "sin*theta" "6#*&%$sinG\"\"\"%&thetaGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "tan*theta" "6#*&%$tanG\"\"\"%&thetaGF%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 56 "To draw the reference t riangle rotate through the angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG " }{TEXT -1 36 " from the positive direction of the " }{TEXT 282 1 "x " }{TEXT -1 42 " axis in a counter-clockwise direction if " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 45 " is positive and in a clockwis e direction if " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 48 " is negative (in both cases through an angle of " }{XPPEDIT 18 0 "abs(the ta)" "6#-%$absG6#%&thetaG" }{TEXT -1 11 " radians). " }}{PARA 0 "" 0 " " {TEXT -1 71 "The terminal side of the angle intersects the unit circ le in the point " }{XPPEDIT 18 0 "W(theta);" "6#-%\"WG6#%&thetaG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "Then try to determine t he coordinates of " }{XPPEDIT 18 0 "W(theta)" "6#-%\"WG6#%&thetaG" } {TEXT -1 41 " by making use of the special triangles. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 285 8 "Quest ion" }{TEXT -1 27 ": Find the exact values of " }{XPPEDIT 18 0 "cos(7* Pi/4)" "6#-%$cosG6#*(\"\"(\"\"\"%#PiGF(\"\"%!\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "sin(7*Pi/4)" "6#-%$sinG6#*(\"\"(\"\"\"%#PiGF(\"\"%!\"\" " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "tan(7*Pi/4)" "6#-%$tanG6#*(\"\"( \"\"\"%#PiGF(\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "tt := 7*Pi/4:\np1 := plot([[cos(t),sin(t),t=0..tt],[cos(t),sin(t) ,t=tt..2*Pi]],\n color=[blue,red],thickness=2):\np2 := plot([[0,0],[co s(tt),sin(tt)],[cos(tt),0]],color=black):\np3 := plot([[[cos(tt),sin(t t)]]$3],color=black,style=point,symbol=[circle,cross,diamond]):\nplots [display]([p||(1..3)],scaling=constrained,tickmarks=[0,0]);" }}{PARA 13 "" 1 "" {GLPLOT2D 228 186 186 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"\" \"\"\"!$F*F*7$$\"3c8V\"oT#GG**!#=$\"313_*>)\\\\&>\"F/7$$\"3uom)QCM*\\( *F/$\"3!3b:@6PBA#F/7$$\"3D\"QnTT#)HU*F/$\"3?e1#=(euZLF/7$$\"3ADh5vu;j* )F/$\"3$>lak_VTV%F/7$$\"3]!f$oNUn\"Q)F/$\"3OK_Vq-8aaF/7$$\"3n%GdnyO\"R xF/$\"3/d-*[+6HL'F/7$$\"3y^ixj3GxpF/$\"3=f#4n*pijrF/7$$\"3/85M%*zC'4'F /$\"3pQa^e5!p#zF/7$$\"3uF&z-\"))eN^F/$\"3+;#pvaV0e)F/7$$\"3q+s=ZvAvSF/ $\"3mphgF0&>8*F/7$$\"3s3F/ $\"3*RvX3vq)3)*F/7$$\"3sX1Fm5]ow!#>$\"37CRW\"p`0(**F/7$$!3kqlx%*HL(z$F fo$\"3srs2Ivy#***F/7$$!3(Q0M#zR,<9F/$\"3].xq;A&HEF /$\"3-:+8w&)3['*F/7$$!3m.9%p7fei$F/$\"3c_nj0N]>$*F/7$$!3`:)=<6'*Qt%F/$ \"3$G.oO_J&3))F/7$$!3G99jTcRccF/$\"3i%Qz[$e_Y#)F/7$$!35\"zXY)=@#f'F/$ \"3Z'pV$Q3\\>vF/7$$!3!4/_YYhuR(F/$\"3$y\"fh=0')GnF/7$$!3'*H-[K73P\")F/ $\"3MVND#zOF\"eF/7$$!3vtsUa?9<()F/$\"3yo/Ylj9+\\F/7$$!3o3rxImPG#*F/$\" 3iJ,\"H1\"*=&QF/7$$!3x=`J*[>ki*F/$\"3[-[sOax2FF/7$$!3AYV'Hdj!e)*F/$\"3 EY'f?[j)y;F/7$$!3#\\cO>GH])**F/$\"3kmC<1&G)paFfo7$$!3'3j\"p,z:!)**F/$! 3_Gju>PT'H'Ffo7$$!3!Q$)oSft;%)*F/$!3?KIVumTs*)fj^$e*F/$!3KQ 8WV'3f&GF/7$$!3O#[C%R;xd\"*F/$!3dwG0kn'o,%F/7$$!3s#Gj%Hwab')F/$!37Q!RF *G93]F/7$$!33\\i4i9^,!)F/$!35iP?)=%)z*fF/7$$!3!HIJP'3J6tF/$!3\"[&))*f2 qB#oF/7$$!3W[[5O7MgkF/$!3o3\")\\%H&3LwF/7$$!33*4nL-Ezd&F/$!3YF[6*o.)*H )F/7$$!3cLZ@!H+Ge%F/$!3Wl%*)ohy!))))F/7$$!3qe_ZGdd]NF/$!3Qp=PiVW[$*F/7 $$!3!4I\"F/$!3uHP\"eQ$)[\"**F/ 7$$!31%y+\"ov@'R\"Ffo$!3?$[V2CD!****F/7$$\"3y7Y^/H+:5F/$!3C)p#Q&4b$[** F/7$$\"3'e-*R!R&Rk?F/$!3-lrZ(e$f%y*F/7$$\"3D)Rto(F/$!3Ym\\$\\\"[)*[pF/7$$\"3#\\?Xjy&y$H(F/$!3=^71,H5ToF/7$$\"32I% pSUctS(F/$!34iV9qi'zr'F/7$$\"3NOvF/$!3GJ]Wz&4@f'F/7$$\"3\"ymJkQ[ \"HwF/$!3ca4&z$**)\\Y'F/7$$\"3+8_xJH\")GxF/$!3[NhUOj]XjF/7$$\"3XC*QbvP +$yF/$!3Hj856&p,A'F/7$$\"3&*)*)*e-leKzF/$!3'eFNf6[))3'F/7$$\"3Qi:WxZhK !)F/$!3Ky*>\\fli&fF/7$$\"3sFy)p9%>L\")F/$!3RFBAnV<=eF/7$$\"3pU@F3$>)># )F/$!3['*)R5pN^p&F/7$$\"33+FPq26:$)F/$!3'e`\\kr\"3bbF/7$$\"3>R2-'o#R3% )F/$!3JQrEG)[GT&F/7$$\"3s0bY]Ck[F/7$$\"3LmIWS[t?))F/$!3C&>\"RJ%=6r%F/7$$\"3MeOz5hT#*))F /$!3K?oox\"yVd%F/7$$\"3[/f2(oA'o*)F/$!3+9mtj#*4BWF/7$$\"3&zD!\\sxzQ!*F /$!3w$\\mgolyF%F/7$$\"3s4w+%*3^4\"*F/$!31WT.)yV^7%F/7$$\"3IfikAabRel#HNF/7$$\"3Fm-g!4CFT*F/$!3 cSJB&=\"[wLF/7$$\"3G!f`>Uy\"o%*F/$!3z48+M1q&*F/$! 3c!*o7eP[hIF/7$$\"3_9/5T%otc*F/$!3F/7$$\"3%[2?X,k!Q)*F/$!3 ym<^TQM#z\"F/7$$\"3'[t\")Q%Rtl)*F/$!3.zgRa^=L;F/7$$\"3y0]`AI%>*)*F/$!3 !eVFWy/hY\"F/7$$\"3ce)fCz5X\"**F/$!3*fPf4a*y/8F/7$$\"3#o&\\#3-p[$**F/$ !3**oGx1PYR6F/7$$\"3S-9p%>OB&**F/$!3/$\\.$pZ%>v*Ffo7$$\"3]7qb]B*f'**F/ $!3(=2r>?C,C)Ffo7$$\"3k/v&>X>)y**F/$!3#*zo3oH40lFfo7$$\"31Dh;@Mt()**F/ $!3O4Jf%fw:&\\Ffo7$$\"3q_'fMmtX***F/$!3'*=NK;E*QH$Ffo7$$\"3!**H(fbRa)* **F/$!3-6fu@AU1 " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "The reference triangle is a 45 ,45,90 degree triangle. Since the hypotenuse has length 1, the shorter sides are both of length " }{XPPEDIT 18 0 "1/sqrt(2)=sqrt(2)/2" "6#/* &\"\"\"F%-%%sqrtG6#\"\"#!\"\"*&-F'6#F)F%F)F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "W(7*Pi/4)" "6#-%\"WG6#*(\"\"(\"\"\"%#PiGF(\"\"%!\"\"" } {TEXT -1 32 " is in quadrant IV, we see that " }{XPPEDIT 18 0 "W(7*Pi/ 4)=``(sqrt(2)/2,-sqrt(2)/2)" "6#/-%\"WG6#*(\"\"(\"\"\"%#PiGF)\"\"%!\" \"-%!G6$*&-%%sqrtG6#\"\"#F)F4F,,$*&-F26#F4F)F4F,F," }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Hence " }{XPPEDIT 18 0 "cos(7*Pi/4)=sqrt(2)/2" "6#/-%$cosG6#*(\"\"(\"\"\"%#PiG F)\"\"%!\"\"*&-%%sqrtG6#\"\"#F)F1F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 " sin(7*Pi/4)=-sqrt(2)/2" "6#/-%$sinG6#*(\"\"(\"\"\"%#PiGF)\"\"%!\"\",$* &-%%sqrtG6#\"\"#F)F2F,F," }{TEXT -1 6 " and " }{XPPEDIT 18 0 "tan(7*P i/4)=-1" "6#/-%$tanG6#*(\"\"(\"\"\"%#PiGF)\"\"%!\"\",$F)F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "tt := 7*Pi/4:\n'cos'(tt)=cos(tt),'sin'(tt)=sin(tt),'t an'(tt)=tan(tt);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/-%$cosG6#,$*(\"\" (\"\"\"\"\"%!\"\"%#PiGF*F*,$*&\"\"#F,F0#F*F0F*/-%$sinGF&,$*&F0F,F0F1F, /-%$tanGF&F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Examp le 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 287 8 "Question" }{TEXT -1 27 ": Find the exact values of " }{XPPEDIT 18 0 "cos(2*Pi/3);" "6#- %$cosG6#*(\"\"#\"\"\"%#PiGF(\"\"$!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(2*Pi/3);" "6#-%$sinG6#*(\"\"#\"\"\"%#PiGF(\"\"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "tan(2*Pi/3);" "6#-%$tanG6#*(\"\"#\"\"\"%# PiGF(\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 288 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "tt := \+ 2*Pi/3:\np1 := plot([[cos(t),sin(t),t=0..tt],[cos(t),sin(t),t=tt..2*Pi ]],\n color=[blue,red],thickness=2):\np2 := plot([[0,0],[cos(tt),sin(t t)],[cos(tt),0]],color=black):\np3 := plot([[[cos(tt),sin(tt)]]$3],col or=black,style=point,symbol=[circle,cross,diamond]):\nplots[display]([ p||(1..3)],scaling=constrained,tickmarks=[0,0]);" }}{PARA 13 "" 1 "" {GLPLOT2D 228 186 186 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F* 7$$\"3GZJD^8e*)**!#=$\"3O!p]!\\(*fjX!#>7$$\"3)z/)eR\"zN'**F/$\"3gCJ(*G :'p_)F27$$\"3\"R&p`R;c:**F/$\"3)>/FghznH\"F/7$$\"3S**>VyeCZ)*F/$\"3se? 7@<>T#z**z@F/7$$\"3'HB?r&=mg'*F/$\"3]F )**HZYHe#F/7$$\"3_?](z_63a*F/$\"3c`qj'\\#[&*HF/7$$\"3'R5,f\"RM)R*F/$\" 3)*>)Qxz.jT$F/7$$\"3l/rx^3(yB*F/$\"3f(fgwVl!HQF/7$$\"3%**)[BH]!R0*F/$ \"3!p\\?4O)zXUF/7$$\"3ig\\IjBHw))F/$\"32&*o1Yse0YF/7$$\"3yQ.a(oe#f')F/ $\"3vEc9,Os,]F/7$$\"3O(ybNDeNU)F/$\"3Apm\"3g<#*Q&F/7$$\"3)>4T)=g,!=)F/ $\"3g>ik9%f@v&F/7$$\"33JB#\\j/`%zF/$\"3eQ?sMHCsgF/7$$\"33xi%G@]*\\wF/$ \"35&)3CE>OSkF/7$$\"3=)3&*zA(Q'Q(F/$\"3Bwy=O],TnF/7$$\"3+v%4uYu\\1(F/$ \"3/j&G+\"f:xqF/7$$\"3g')4I>u4[>O(F/7$$\"3DraB&yf$GkF/$ \"3WSNVU&Q+m(F/7$$\"3Q]D:F/.$4'F/$\"3#*>X#f^u$HzF/7$$\"3lE)pYeV6t&F/$ \"3%Hjvb+aZ>)F/7$$\"3+`zn\"[M')Q&F/$\"3Sx_i#HJRU)F/7$$\"3=[tz[+&*3]F/$ \"3_Mxra03b')F/7$$\"3o@NFgc4/YF/$\"3Io]?Mh1x))F/7$$\"3b()e:&*okVUF/$\" 3mmf2SO\"\\0*F/7$$\"3!4$f%oRpm%QF/$\"3:zQ3,WbI#*F/7$$\"3rij&p/\\*GMF/$ \"3YDMKCxt$R*F/7$$\"3l(p:[v6O,$F/$\"3YcwvQ15N&*F/7$$\"3T=U&Ho@ig#F/$\" 3#4#ywh(3Wl*F/7$$\"3wE$[189%[@F/$\"3M7l'zZ*[m(*F/7$$\"3k2oCxJ&Ht\"F/$ \"3'paOc1*p[)*F/7$$\"3o\\]&oVZfG\"F/$\"3j$ei$zA(p\"**F/7$$\"35/8A'H\") fy)F2$\"3C()[0#\\G8'**F/7$$\"3/K:&o$=%fJ%F2$\"3'*esv\")>o!***F/7$$\"3# R7v7(*G\">5!#?$\"25Nzo![******!#<7$$!37RD[hU..VF2$\"3#GxB&ewt!***F/7$$ !3hx2@I6m,')F2$\"3?#=7(Hq$H'**F/7$$!3n!3hF/p%38F/$\"3\"fHu0'e-9**F/7$$ !3`H!4`*HtPV/HMF/$\"3')>!Rp%Hq$R*F/7$$!3O\"*Gy*fyd\"QF/$\"3)oFx(4qOV#*F/7$ $!3=YOt\"zl3A%F/$\"3Yw=v[_bl!*F/7$$!32E3Pb#f3g%F/$\"3mE0-\\Ruy))F/7$$! 3$>3b_++++&F/$\"3/k.\"[.a-m)F/-%'COLOURG6&%$RGBGF+F+$\"*++++\"!\")-%*T HICKNESSG6#\"\"#-F$6%7SFez7$$!3gkn!\\p)yodF/$\"3oU![\"oqHo\")F/7$$!3-3 #Hs%\\#))R'F/$\"3Ye\"=cmFZo(F/7$$!3?Y]VZ)e*eqF/$\"331I_vg:$3(F/7$$!31f &\\Ue*emwF/$\"36Am6raa?kF/7$$!3qV1d#*R\")4#)F/$\"3Sx-U(Q\\&4dF/7$$!3iW t3m.#[l)F/$\"3VnK\"yg*R4]F/7$$!3e,m^0Vz_!*F/$\"3y`cu0h;[UF/7$$!3G1NIWf 3%R*F/$\"3&f!>^M\\*zU$F/7$$!3ern%[E20m*F/$\"3+'pK_gCNe#F/7$$!3%3\\9so0 b&)*F/$\"3%*[WBEX\"Qp\"F/7$$!3)=DCRlt&f**F/$\"3_%eekD?F)*)F27$$!2/'Rp2 #*******F]w$!3+>s][&=2)R!#@7$$!31QW'R[=&e**F/$!319M&4\"Q&*)4*F27$$!3H< l!fjU5%)*F/$!3)4Hx\">m\"fx\"F/7$$!3UDR?6x?p'*F/$!3mtoD]*o2b#F/7$$!31%y ))[N>dQ*F/$!3cr41[k$3X$F/7$$!3sMy&3Db+3*F/$!3I#\\_s\">e*=%F/7$$!3k_/Y3 wi^')F/$!3J[SYO<\"\\,&F/7$$!34GHm0P%)4#)F/$!3!RE_Cm1&4dF/7$$!3=U=&e6c7 m(F/$!3[X-qW'3pU'F/7$$!3C*fuiU=23(F/$!3[-hD3SShqF/7$$!3.oi!*38B>kF/$!3 )pV@_'**onwF/7$$!3u2t1v17mdF/$!3m#*H-\"o!=q\")F/7$$!3ehnoVZ*y,&F/$!3%) R>fAy*)\\')F/7$$!3+tmJz0\"))>%F/$!3I\"[l+T\"zv!*F/7$$!37i!p[T2kX$F/$!3 ;(*))y1#pOQ*F/7$$!3H0B7!\\g'HEF/$!3;P:*)[^)F/7$$\"3n.6B.5M5kF/$!3'[ d@u)R7vwF/7$$\"3M+%4#=6kvqF/$!3+'[gda\"\\mqF/7$$\"3%[qI1bx?m(F/$!3'o=P '=$HfU'F/7$$\"3:q;5qU*>?)F/$!3FA)HX/x2s&F/7$$\"3YL@m&*)eNn)F/$!3E@y?uT )o(\\F/7$$\"3ezJ!QBNw/*F/$!3+SC4;F9fUF/7$$\"3.&**)p_xM.%*F/$!3UzZ=#e1D S$F/7$$\"3s,gJnx.`'*F/$!3DZMra]H6EF/7$$\"375'=)GO.Y)*F/$!3+\">c'*oL![< F/7$$\"3#4?&p*>6'e**F/$!3!p(38a1!))3*F27$F($\"36YKhSr8/#)!#F-F[[l6&F][ lF^[lF+F+Fa[l-F$6$7%7$F+F+7$$!3++++++++]F/$\"3'fQWy.a-m)F/7$F_[mF+-F[[ l6&F][lF*F*F*-F$6&7#F^[m-%'SYMBOLG6#%'CIRCLEGFd[m-%&STYLEG6#%&POINTG-F $6&Fh[m-Fj[m6#%&CROSSGFd[mF]\\m-F$6&Fh[m-Fj[m6#%(DIAMONDGFd[mF]\\m-%*A XESTICKSG6$F*F*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!6\"Fe]m-%% FONTG6#%(DEFAULTG-%%VIEWG6$Fj]mFj]m" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The refer ence triangle is a 30,60,90 degree triangle. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "W( 2*Pi/3);" "6#-%\"WG6#*(\"\"#\"\"\"%#PiGF(\"\"$!\"\"" }{TEXT -1 32 " is in quadrant II, we see that " }{XPPEDIT 18 0 "W(2*Pi/3) = ``(-1/2,sqr t(3)/2);" "6#/-%\"WG6#*(\"\"#\"\"\"%#PiGF)\"\"$!\"\"-%!G6$,$*&F)F)F(F, F,*&-%%sqrtG6#F+F)F(F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Hence " }{XPPEDIT 18 0 "cos(2*Pi/3) \+ = -1/2;" "6#/-%$cosG6#*(\"\"#\"\"\"%#PiGF)\"\"$!\"\",$*&F)F)F(F,F," } {TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(2*Pi/3) = sqrt(3)/2;" "6#/-%$sinG6 #*(\"\"#\"\"\"%#PiGF)\"\"$!\"\"*&-%%sqrtG6#F+F)F(F," }{TEXT -1 6 " and " }{XPPEDIT 18 0 "tan(2*Pi/3) = -sqrt(3);" "6#/-%$tanG6#*(\"\"#\"\" \"%#PiGF)\"\"$!\"\",$-%%sqrtG6#F+F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "tt := 2*Pi /3:\n'cos'(tt)=cos(tt),'sin'(tt)=sin(tt),'tan'(tt)=tan(tt);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/-%$cosG6#,$*(\"\"#\"\"\"\"\"$!\"\"%#PiGF*F* #F,F)/-%$sinGF&,$*&F)F,F+#F*F)F*/-%$tanGF&,$*$F+F4F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 289 8 "Question" }{TEXT -1 27 ": Find the exact v alues of " }{XPPEDIT 18 0 "cos(7*Pi/6);" "6#-%$cosG6#*(\"\"(\"\"\"%#Pi GF(\"\"'!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(7*Pi/6);" "6#-%$si nG6#*(\"\"(\"\"\"%#PiGF(\"\"'!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "tan(7*Pi/6);" "6#-%$tanG6#*(\"\"(\"\"\"%#PiGF(\"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "tt := 7*Pi/6:\np1 := plot([ [cos(t),sin(t),t=0..tt],[cos(t),sin(t),t=tt..2*Pi]],\n color=[blue,red ],thickness=2):\np2 := plot([[0,0],[cos(tt),sin(tt)],[cos(tt),0]],colo r=black):\np3 := plot([[[cos(tt),sin(tt)]]$3],color=black,style=point, symbol=[circle,cross,diamond]):\nplots[display]([p||(1..3)],scaling=co nstrained,tickmarks=[0,0]);" }}{PARA 13 "" 1 "" {GLPLOT2D 228 186 186 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"3]A0>;V5o**!#=$\" 3Cf)G,cy0)z!#>7$$\"37jiw*y+')))*F/$\"35!***[u0[)[\"F/7$$\"3l,,A+&e@u*F /$\"3KM%=M!*zhD#F/7$$\"3w&pUpG\\Y`*F/$\"35jj\"\\YR],$F/7$$\"3U!Rz!>8`p #*F/$\"3QO\"p[Yr)F/$\"3S>Yr(R<)HdF/7$$\"3*>v])pE UFxF/$\"3OXKH%H*>ZjF/7$$\"3mW;fLEQ(>(F/$\"30%zR&H^XUpF/7$$\"3)zE2UqKDp 'F/$\"3&e]j#*fV.V(F/7$$\"3]nb'*p0&[3'F/$\"31Y^L:LlNzF/7$$\"3;4f)GQPkV& F/$\"33Z6)=#3;$R)F/7$$\"3MW8([qG\"zZF/$\"35_!y3XsSy)F/7$$\"3uop`!*y!z: %F/$\"3eb-e.Hg%4*F/7$$\"3)*zANHfO$R$F/$\"31N?A$=]mS*F/7$$\"3'HevzH#QFF F/$\"3?\"*yO(z#)3i*F/7$$\"3A*=jn(o6O>F/$\"3!Ha-rS#y5)*F/7$$\"31yZS.p%[ A\"F/$\"3LL&)**GSqC**F/7$$\"3i_f%\\8yHP%F2$\"3CAalbRV!***F/7$$!3MJ\"QL 2oC:$F2$\"3M]xMP(H]***F/7$$!3UcHCO#z%)4\"F/$\"3=kRUfS[R**F/7$$!3O2$p9, \")=\"=F/$\"3,')z]#f%[M)*F/7$$!3'pm6om62d#F/$\"3??,aZZ#Rm*F/7$$!3iJ%*) HA'RULF/$\"3uB'z$)H\")[U*F/7$$!3Mb\"3K.kl*RF/$\"3spo$))F/7$$!3C\")4&y7Z#f`F/$\"3A$yv+xdEW)F/7$$!3u \"[%[Lr3\"*fF/$\"3DRji)*on1!)F/7$$!37m=T)*e&)olF/$\"34*=>Um-*RvF/7$$!3 mLGKJ2ynrF/$\"3O#Gj&F/7$$!3yOq>Cy eB*)F/$\"3\\\"o/u9nK^%F/7$$!3>-BPqb&>B*F/$\"3)4VlLR0L%QF/7$$!3UlP!Q%fe +&*F/$\"3aE67Rdr?JF/7$$!3u'f!Qt^k3(*F/$\"3o_+y`1H'R#F/7$$!3U/!fR*QNn)* F/$\"3*fKL\"o!pLi\"F/7$$!35.]W?09i**F/$\"3WU^EWUU$p)F27$$!3ych))pnc*** *F/$\"3yaj%Rs$G3$*!#?7$$!3'Hj'\\=d.x**F/$!3YC:lb'pJx'F27$$!30_!*)p)=2/ **F/$!3IU$)e`fz\"Q\"F/7$$!3gXW+X=Qf(*F/$!3Y=?]-QZ!=#F/7$$!3/wVX**)fad* F/$!3y9.HGb!G)GF/7$$!3[QQKvTzB$*F/$!3\\&Q+%*\\6[h$F/7$$!3V?j!e\\q0.*F/ $!3[/8eDi?&H%F/7$$!3=Y[yGSDg')F/$!3q%e\"p:+++]F/-%'COLOURG6&%$RGBGF+F+ $\"*++++\"!\")-%*THICKNESSG6#\"\"#-F$6%7SFdz7$$!3c%Q)=1!))4O)F/$!3SEL6 t))y&[&F/7$$!3=#pUtg;%y!)F/$!3GMFf$o&*R*eF/7$$!3?&[)yfT)ot(F/$!3\"G&G? IGmNjF/7$$!3i2pO>ItotF/$!3S#)3y(p3.w'F/7$$!33WtHb0AzpF/$!3#f'H<]sthrF/ 7$$!3MOzL!Re&)f'F/$!3`IVtAA#R^(F/7$$!3+Gfvhtr&='F/$!3[(RY8H$GdyF/7$$!3 NNIuM.,SdF/$!3]30GTda)=)F/7$$!35=vCfY6y_F/$!3#=+b8e:O\\)F/7$$!3'[-6XqQ hy%F/$!3'3WX4\"\\D!y)F/7$$!37(HRcn**)RVF/$!3c]ljKq<4!*F/7$$!3&Q)HHHc`C QF/$!33!3%\\SruR#*F/7$$!3+&pS,P\"z%H$F/$!3!zF+!e&G;W*F/7$$!3OPn;****F/7$$\"3%=c:_]`SN%F2$!3 u^ZI7m^!***F/7$$\"3*\\6m$*3jnr*F2$!3gk8$)*H!o_**F/7$$\"3_;%=Z@,#G:F/$! 3g(4!G7-a#))*F/7$$\"3_XMui82N?F/$!3syKiFYt!z*F/7$$\"3Gq%fEZ1dd#F/$!3?( >:7p%fi'*F/7$$\"3=mVj=F$)GJF/$!3yJa6ne\"z\\*F/7$$\"3$y2_g$F/$!3$Qm \"oScyG$*F/7$$\"3^'pW!>+g-TF/$!3!erA*=go>\"*F/7$$\"3&Hbyiwfrg%F/$!3ub/ giiZv))F/7$$\"3kg:qN]x'3&F/$!3cDBe-9d4')F/7$$\"3.c-TM2GObF/$!3$*yg#*f/ kF$)F/7$$\"3VkfnBe1G(F/7$$\"3z63xZu_&F/ 7$$\"3whXYa!=6#))F/$!3<1gxg3S5ZF/7$$\"3uSLkY*[P1*F/$!3ouYF\">VZA%F/7$$ \"3YrOhd3C%G*F/$!3cT'G;i@_r$F/7$$\"3qhvN.kku%*F/$!33eS\"yC0')>$F/7$$\" 31>t`M#*HC'*F/$!3ap)*yp=I:FF/7$$\"3%)3&H3t*\\l(*F/$!3ud#)[*\\3H:#F/7$$ \"3gspCbi)R')*F/$!3Tm+iYxqV;F/7$$\"3!HU8&*[i(R**F/$!33QA@dp&f4\"F/7$$ \"3A5K?(pDQ)**F/$!3!)eddvJG&o&F27$F($\"36YKhSr8/#)!#F-Fjz6&F\\[lF][lF+ F+F`[l-F$6$7%7$F+F+7$$!3'fQWy.a-m)F/$!3++++++++]F/7$F][mF+-Fjz6&F\\[lF *F*F*-F$6&7#F\\[m-%'SYMBOLG6#%'CIRCLEGFb[m-%&STYLEG6#%&POINTG-F$6&Ff[m -Fh[m6#%&CROSSGFb[mF[\\m-F$6&Ff[m-Fh[m6#%(DIAMONDGFb[mF[\\m-%*AXESTICK SG6$F*F*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!6\"Fc]m-%%FONTG6# %(DEFAULTG-%%VIEWG6$Fh]mFh]m" 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The refer ence triangle is a 30,60,90 degree triangle. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "W( 7*Pi/6);" "6#-%\"WG6#*(\"\"(\"\"\"%#PiGF(\"\"'!\"\"" }{TEXT -1 33 " is in quadrant III, we see that " }{XPPEDIT 18 0 "W(7*Pi/6) = ``(-sqrt(3 )/2,-1/2);" "6#/-%\"WG6#*(\"\"(\"\"\"%#PiGF)\"\"'!\"\"-%!G6$,$*&-%%sqr tG6#\"\"$F)\"\"#F,F,,$*&F)F)F6F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Hence " }{XPPEDIT 18 0 "cos(7*Pi/6) = -sqrt(3)/2;" "6#/-%$cosG6#*(\"\"(\"\"\"%#PiGF)\"\"'!\" \",$*&-%%sqrtG6#\"\"$F)\"\"#F,F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin (7*Pi/6) = -1/2;" "6#/-%$sinG6#*(\"\"(\"\"\"%#PiGF)\"\"'!\"\",$*&F)F) \"\"#F,F," }{TEXT -1 6 " and " }{XPPEDIT 18 0 "tan(7*Pi/6) = 1/sqrt(3 );" "6#/-%$tanG6#*(\"\"(\"\"\"%#PiGF)\"\"'!\"\"*&F)F)-%%sqrtG6#\"\"$F, " }{XPPEDIT 18 0 "`` = sqrt(3)/3;" "6#/%!G*&-%%sqrtG6#\"\"$\"\"\"F)!\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "tt := 7*Pi/6:\n'cos'(tt)=cos(tt),'sin'(tt)= sin(tt),'tan'(tt)=tan(tt);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/-%$cosG 6#,$*(\"\"(\"\"\"\"\"'!\"\"%#PiGF*F*,$*&\"\"#F,\"\"$#F*F0F,/-%$sinGF&# F,F0/-%$tanGF&,$*&F1F,F1F2F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 4 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 283 8 "Question" }{TEXT -1 27 ": Find the exact values of " }{XPPEDIT 18 0 "cos(-5*Pi/4);" "6#-%$cosG6#,$*(\"\"&\"\"\"%#PiGF)\"\"%!\"\"F," } {TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(-5*Pi/4);" "6#-%$sinG6#,$*(\"\"&\" \"\"%#PiGF)\"\"%!\"\"F," }{TEXT -1 5 " and " }{XPPEDIT 18 0 "tan(-5*Pi /4);" "6#-%$tanG6#,$*(\"\"&\"\"\"%#PiGF)\"\"%!\"\"F," }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 8 "Solut ion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 330 "tt := -5*Pi/4:\np1 := plot([[cos(t),sin( t),t=0..tt],[cos(t),sin(t),t=tt..2*Pi]],\n color=[blue,red],thickness= 2):\np2 := plot([[0,0],[cos(tt),sin(tt)],[cos(tt),0]],color=black):\np 3 := plot([[[cos(tt),sin(tt)]]$3],color=black,style=point,symbol=[circ le,cross,diamond]):\nplots[display]([p||(1..3)],scaling=constrained,ti ckmarks=[0,0]);" }}{PARA 13 "" 1 "" {GLPLOT2D 228 186 186 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$!3%3>/Z!y1rq!#=$\"3_fn->y1rqF*7$$!3KJV8+Tq\\wF*$\" 3'3Oe4Da1W'F*7$$!3BAn(pURx5)F*$\"3A[M<')Qf`eF*7$$!357J!f^G!p&)F*$\"3%G a0UOzZ:&F*7$$!3sc0IU,ps*)F*$\"3V@vR&>U[T%F*7$$!3TCk#)3X96$*F*$\"3]ZG** *Hrsk$F*7$$!3crUk9PWm&*F*$\"3Z)*[vhbe7HF*7$$!3%o`P0.\"zp(*F*$\"3&G5+PB ]L8#F*7$$!3*o.P$y0V8**F*$\"3`aaI.:(HJ\"F*7$$!3!y#GsFd<))**F*$\"3)HYe9= R:'[!#>7$$!3m?-)y-NK***F*$!3!zkf\"3^oxOFZ7$$!3U%f/(p*>t$**F*$!3j.\"GC@ ()y6\"F*7$$!3#HQd:1Cr!)*F*$!3-3Qd'GjX&>F*7$$!38Cd*yyYcg*F*$!3ENvf]kc!y #F*7$$!3%=eqA'\\wX$*F*$!3uB?8=CidNF*7$$!3Yl?GHUNb!*F*$!3]ySddmqUUF*7$$ !3AOT:+F#\\k)F*$!3ucpoJ6YE]F*7$$!3&H=)>![\\TC)F*$!3s:M*yfe)fcF*7$$!3<* R*G;([)>xF*$!3qA.AL!4kN'F*7$$!35psjz$fm?(F*$!3-h,Nj[#G$pF*7$$!3eY7GN#[ Vf'F*$!3s%H2')R,w\\)F*7$$!3k9zKj\\&**f%F*$!3AkC'R[7#z))F*7$$!3>f5E')[tWQF*$!3 DS+2O.OJ#*F*7$$!3TrsgB,3KIF*$!3Pr#QH%RCH&*F*7$$!3D:W/;/&fI#F*$!36xvSp! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Sinc e " }{XPPEDIT 18 0 "W(-5*Pi/4);" "6#-%\"WG6#,$*(\"\"&\"\"\"%#PiGF)\"\" %!\"\"F," }{TEXT -1 32 " is in quadrant II, we see that " }{XPPEDIT 18 0 "W(-5*Pi/4) = ``(-sqrt(2)/2,sqrt(2)/2);" "6#/-%\"WG6#,$*(\"\"&\" \"\"%#PiGF*\"\"%!\"\"F--%!G6$,$*&-%%sqrtG6#\"\"#F*F6F-F-*&-F46#F6F*F6F -" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Hence " }{XPPEDIT 18 0 "cos(-5*Pi/4) = -sqrt(2)/2;" "6#/- %$cosG6#,$*(\"\"&\"\"\"%#PiGF*\"\"%!\"\"F-,$*&-%%sqrtG6#\"\"#F*F3F-F- " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(-5*Pi/4) = sqrt(2)/2;" "6#/-%$s inG6#,$*(\"\"&\"\"\"%#PiGF*\"\"%!\"\"F-*&-%%sqrtG6#\"\"#F*F2F-" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "tan(-5*Pi/4) = -1;" "6#/-%$tanG6#, $*(\"\"&\"\"\"%#PiGF*\"\"%!\"\"F-,$F*F-" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "tt := -5*Pi/4:\n'cos'(tt)=cos(tt),'sin'(tt)=sin(tt),'tan'(tt)=tan(tt);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%/-%$cosG6#,$*(\"\"&\"\"\"\"\"%!\"\"%#P iGF*F,,$*&\"\"#F,F0#F*F0F,/-%$sinGF&,$*&F0F,F0F1F*/-%$tanGF&F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "Solving some simple trigonometric equatio ns " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{PARA 0 "" 0 "" {TEXT 302 8 "Questi on" }{TEXT -1 18 ": Find all angles " }{XPPEDIT 18 0 "theta" "6#%&thet aG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "0<=theta" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 11 " s uch that " }{XPPEDIT 18 0 "cos*theta=-1/2" "6#/*&%$cosG\"\"\"%&thetaGF &,$*&F&F&\"\"#!\"\"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 303 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta" "6#*&%$cosG\"\"\"%&t hetaGF%" }{TEXT -1 8 " is the " }{TEXT 304 1 "x" }{TEXT -1 25 " coordi nate of the point " }{XPPEDIT 18 0 "P(cos*theta,sin*theta)" "6#-%\"PG6 $*&%$cosG\"\"\"%&thetaGF(*&%$sinGF(F)F(" }{TEXT -1 51 " obtained by ro tating the line OP through an angle " }{XPPEDIT 18 0 "abs(theta)" "6#- %$absG6#%&thetaG" }{TEXT -1 36 " from the positive direction of the " }{TEXT 305 1 "x" }{TEXT -1 44 " axis in the counter-clockwise directio n if " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 48 " is positive, and in the clockwise direction if " }{XPPEDIT 18 0 "theta" "6#%&theta G" }{TEXT -1 15 " is negative. " }}{PARA 0 "" 0 "" {TEXT -1 32 "Hence we draw the vertical line " }{XPPEDIT 18 0 "x=-1/2" "6#/%\"xG,$*&\"\" \"F'\"\"#!\"\"F)" }{TEXT -1 56 " and observe where this line intersect s the unit circle " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"\"#\"\" \"*$%\"yGF'F(F(" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 327 295 295 {PLOTDATA 2 "6;-%'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/$\"3u3HvXQF/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)zD(p_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2ndo *fQF/7$$\"31mGAp))oa'*F/$!3%)f]nRQ=0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4O \"F/7$F($\"36YKhSr8/#)!#F-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%*THICKN ESSG6#\"\"#-F$6%7$7$F+F+7$$!3++++++++]F/$\"3a+++SSDg')F/-Fjz6&F\\[l$\" )#)eqkF_[l$\"))eqk\"F_[lFa\\lF`[l-F$6%7$Fg[l7$Fi[l$!3a+++SSDg')F/F]\\l F`[l-F$6%7$7$Fi[l$!3/++++++]5Fdo7$Fi[l$\"3%**************>\"Fdo-%&COLO RG6&F\\[lF+$\"\"(!\"\"F+F`[l-F$6$7S7$$\"3++++++++DF/F+7$$\"3#oG8yL&R( \\#F/$\"3fsEEP***39\"!#>7$$\"3)>,(*[y%*3\\#F/$\"3:\"GVAQS<8#Fc^l7$$\"3 [QU))4/*)yCF/$\"3,0w1S!\\>C$Fc^l7$$\"3%)**zgp9\"=Y#F/$\"3!o90GIzHN%Fc^ l7$$\"3&))o9*Q?()RCF/$\"3?!3*y/[**\\aFc^l7$$\"3Ce+Gka;:CF/$\"3uo&*\\#= mtX'Fc^l7$$\"39bP*>)G?&Q#F/$\"3%Rj#fTiq)[(Fc^l7$$\"3+w_(*yfe\\BF/$\"3' *\\qM%\\f2a)Fc^l7$$\"3;wU%Hrn%4BF/$\"3)R\\^TfjEd*Fc^l7$$\"3[A(3tDwME#F /$\"3AC,B!f\\91\"F/7$$\"3;Si#34t!>AF/$\"3yBn^6oR^6F/7$$\"3q%3&)=n9[;#F /$\"3q1kG+4V]7F/7$$\"3%o%*)Qj&*)e5#F/$\"3InT?+WIZ8F/7$$\"3*HF5Z+/]/#F/ $\"3!\\bhO&)R!Q9F/7$$\"3w#eI(ehK')>F/$\"3k40oL21=:F/7$$\"3Fp:@`v[7>F/$ \"3G@-c\"[!45;F/7$$\"3/s()*p!ofY=F/$\"31pp/fPD&o\"F/7$$\"3uoB&ohViw\"F /$\"3wSr]x*)Gpp\"F/$\"3b)zaV-([S=F/7$$\"3\"y')3j%**32 ;F/$\"36&Q3cj4]\">F/7$$\"3gP\")y1wDB:F/$\"3)*H6)*GOM#)>F/7$$\"3mcu;'*e yK9F/$\"3C3RR,&)o[?F/7$$\"3E)[>/ierM\"F/$\"3N>j:BG)f5#F/7$$\"3/P$*>7vB _7F/$\"3kL%z'Q,xj@F/7$$\"3U!Qo]\"R-^6F/$\"32n7bLlE>AF/7$$\"3)=(*)yB<\" 41\"F/$\"3m\"**=+TGPE#F/7$$\"3DF[6#\\tmh*Fc^l$\"3zp4F+'QwI#F/7$$\"3y14 RW!>-OT#F/7$$\"3!p\"3iE`.r`Fc^l$\"34G;\\pBiTCF/7$$\" 36>q6VHQKVFc^l$\"3uO\"4kwu@Y#F/7$$\"3DCw8#fo[@$Fc^l$\"3!fkS)pICzCF/7$$ \"3-E`0C`\\'>#Fc^l$\"3!=s8I7K.\\#F/7$$\"3-$)G@fa)*y5Fc^l$\"3u9$Ra\\qw \\#F/7$$\"3!)4y=GC#ya#Fao$\"3sP)>Y#F/7$$!3+ipxt=ILaFc^l$\"3 %\\zsIJW-W#F/7$$!3U:zV`Nn-lFc^l$\"3F)*R(o`\\RT#F/7$$!35y?-M.CuuFc^l$\" 3OgkGzll&Q#F/7$$!3)ep%f)z5Ed)Fc^l$\"3)\\vMntD%[BF/7$$!3RGsX*\\Y%R&*Fc^ l$\"3A>VW_<%3J#F/7$$!3a6M$zW;_0\"F/$\"37pz=7))QmAF/7$$!3_1F%Q\"[@]6F/$ \"3nJ^D()fo>AF/7$$!3[qPJ,++]7F/$\"3-\"f-(3N1l@F/-Fjz6&F\\[l$\")!\\DP\" F_[lFa]m$\")viobF_[l-F$6&7$Fh[lFf\\l-%'SYMBOLG6#%'CIRCLEG-Fjz6&F\\[lF* F*F*-%&STYLEG6#%&POINTG-F$6&Fg]m-Fi]m6#%(DIAMONDGF\\^mF^^m-F$6&Fg]m-Fi ]m6#%&CROSSGF\\^mF^^m-%%TEXTG6%7$$!#A!\"#$\"#aFb_mQ\"16\"F\\^m-F]_m6%7 $$Fg]lFg]l$!\"'Fb_mQ\"OFf_mF\\^m-F]_m6%7$$!#dFb_mF[`mQ\"QFf_mF\\^m-F]_ m6%7$Fa`m$\"##*Fb_mQ\"PFf_mF\\^m-F]_m6%7$Fa`m$!##*Fb_mQ\"RFf_mF\\^m-F] _m6%7$F`_m$!#aFb_mFe_mF\\^m-F]_m6%7$$\"$:\"Fb_m$!\"%Fb_mQ\"xFf_mF\\^m- F]_m6%7$$!\"&Fb_mFhamQ\"yFf_mF\\^m-F]_m6%7$$!#tFb_m$\"#DFb_mQ&x~=~-Ff_ mF\\^m-F]_m6%7$Fa`m$\"#LFb_mFe_mF\\^m-F]_m6%7$Fa`m$\"#JFb_mQ\"_Ff_mF\\ ^m-F]_m6%7$Fa`m$\"#=Fb_mQ\"2Ff_mF\\^m-F]_m6&7$$\"#&)!\"$$\"#8Fb_mQ\"qF f_mF\\^m-%%FONTG6$Fi]m\"#6-%+AXESLABELSG6%Q!Ff_mF\\em-Ffdm6#%(DEFAULTG -%(SCALINGG6#%,CONSTRAINEDG-%*AXESTICKSG6$F*F*-%%VIEWG6$;$!$0\"Fb_m$\" $=\"Fb_m;F[fm$\"#7Fg]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" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "The line " }{XPPEDIT 18 0 "x=-1/2" "6#/%\"xG,$*&\"\"\"F'\"\"#!\"\"F )" }{TEXT -1 87 " intersects the unit circle at the two points P and R in the previous picture. Since QO" }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G *&\"\"\"F&\"\"#!\"\"" }{TEXT -1 7 " and OP" }{XPPEDIT 18 0 "`` = 1;" " 6#/%!G\"\"\"" }{TEXT -1 4 ", QP" }{XPPEDIT 18 0 "``=sqrt(3)/2" "6#/%!G *&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 147 " by Pythagoras' theo rem and we recognise the triangle OPQ to be a (special) 30,60,90 degre e triangle with hypotenuse 1. The fact that angle QOP is " }{XPPEDIT 18 0 "60^o=Pi/3" "6#/)\"#g%\"oG*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 40 " radians means that the angle marked as " }{XPPEDIT 18 0 "theta" "6#%& thetaG" }{TEXT -1 19 " in the picture is " }{XPPEDIT 18 0 "120^o = 2*P i/3;" "6#/)\"$?\"%\"oG*(\"\"#\"\"\"%#PiGF)\"\"$!\"\"" }{TEXT -1 10 " r adians. " }{XPPEDIT 18 0 "theta=2*Pi/3" "6#/%&thetaG*(\"\"#\"\"\"%#PiG F'\"\"$!\"\"" }{TEXT -1 40 " provides one solution for the equation " }{XPPEDIT 18 0 "cos*theta=-1/2" "6#/*&%$cosG\"\"\"%&thetaGF&,$*&F&F&\" \"#!\"\"F+" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "0<=theta" "6#1\"\"! %&thetaG" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "Note that P is the poin t" }{XPPEDIT 18 0 "``(-1/2,sqrt(3)/2)" "6#-%!G6$,$*&\"\"\"F(\"\"#!\"\" F**&-%%sqrtG6#\"\"$F(F)F*" }{TEXT -1 46 " given in terms of the wrappi ng function W as " }{XPPEDIT 18 0 "W(2*Pi/3)" "6#-%\"WG6#*(\"\"#\"\"\" %#PiGF(\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 100 "The triangle OQR is also a (special) 30, 60,90 degree triangle with hypotenuse 1. Since angle ROQ is " } {XPPEDIT 18 0 "60^o=Pi/3" "6#/)\"#g%\"oG*&%#PiG\"\"\"\"\"$!\"\"" } {TEXT -1 40 " radians, it follows that R is the point" }{XPPEDIT 18 0 "``(-1/2,-sqrt(3)/2)" "6#-%!G6$,$*&\"\"\"F(\"\"#!\"\"F*,$*&-%%sqrtG6# \"\"$F(F)F*F*" }{TEXT -1 46 " given in terms of the wrapping function \+ W as " }{XPPEDIT 18 0 "W(4*Pi/3)" "6#-%\"WG6#*(\"\"%\"\"\"%#PiGF(\"\"$ !\"\"" }{TEXT -1 31 ". This gives a second solution " }{XPPEDIT 18 0 " theta=4*Pi/3" "6#/%&thetaG*(\"\"%\"\"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 18 " for the equation " }{XPPEDIT 18 0 "cos*theta=-1/2" "6#/*&%$cosG\" \"\"%&thetaGF&,$*&F&F&\"\"#!\"\"F+" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "0<=theta" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*& \"\"#\"\"\"%#PiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 39 "Indeed there are exactly two solutions \+ " }{XPPEDIT 18 0 "theta=2*Pi/3" "6#/%&thetaG*(\"\"#\"\"\"%#PiGF'\"\"$! \"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta=4*Pi/3" "6#/%&thetaG*( \"\"%\"\"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 18 " for the equation " } {XPPEDIT 18 0 "cos*theta = -1/2;" "6#/*&%$cosG\"\"\"%&thetaGF&,$*&F&F& \"\"#!\"\"F+" }{TEXT -1 12 " satisfying " }{XPPEDIT 18 0 "0<=theta" "6 #1\"\"!%&thetaG" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF' " }{TEXT -1 83 " given by angles obtained by rotating clockwise from t he positive direction of the " }{TEXT 306 1 "x" }{TEXT -1 95 " axis un til the terminal side of the angle coincides with either of the radius lines OP or OR. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 10 "Conclusion" }{TEXT -1 3 ": " }{XPPEDIT 18 0 "cos*theta \+ = -1/2;" "6#/*&%$cosG\"\"\"%&thetaGF&,$*&F&F&\"\"#!\"\"F+" }{TEXT -1 14 " exactly when " }{XPPEDIT 18 0 "theta = 2*Pi/3;" "6#/%&thetaG*(\" \"#\"\"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "theta = 4*Pi/3;" "6#/%&thetaG*(\"\"%\"\"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 7 " f or " }{XPPEDIT 18 0 "0<=theta" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``< 2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 6 " }} {PARA 0 "" 0 "" {TEXT 307 8 "Question" }{TEXT -1 18 ": Find all angles " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "0<=theta" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*& \"\"#\"\"\"%#PiGF'" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "sin*the ta = 1/2;" "6#/*&%$sinG\"\"\"%&thetaGF&*&F&F&\"\"#!\"\"" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 308 8 "Sol ution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*theta;" "6#*&%$sinG\"\"\"%&thetaGF%" }{TEXT -1 8 " is the " } {TEXT 309 1 "y" }{TEXT -1 25 " coordinate of the point " }{XPPEDIT 18 0 "P(cos*theta,sin*theta)" "6#-%\"PG6$*&%$cosG\"\"\"%&thetaGF(*&%$sinG F(F)F(" }{TEXT -1 51 " obtained by rotating the line OP through an ang le " }{XPPEDIT 18 0 "abs(theta)" "6#-%$absG6#%&thetaG" }{TEXT -1 36 " \+ from the positive direction of the " }{TEXT 310 1 "x" }{TEXT -1 44 " a xis in the counter-clockwise direction if " }{XPPEDIT 18 0 "theta" "6# %&thetaG" }{TEXT -1 48 " is positive, and in the clockwise direction i f " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 15 " is negative. \+ " }}{PARA 0 "" 0 "" {TEXT -1 34 "Hence we draw the horizontal line " } {XPPEDIT 18 0 "y = 1/2;" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 56 " and observe where this line intersects the unit circle " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 344 313 313 {PLOTDATA 2 "6>-%'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$$\"3?goz=42`')F/$\"3j9NXR4U 7]F/7$$\"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%>wGZn!**F/7$$!3)e:d.:#=YEF/$\"3PA> =\\D`V'*F/7$$!3%*yV1J*3Jx$F/$\"3IzF#e`m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'y I2&oN')F/7$$!3[igNw%*[RgF/$\"3!G;)RQ+BqzF/7$$!3/XUwYjJ*3(F/$\"3AfvOg?x _qF/7$$!3&=e_b?/U!zF/$\"3u3HvXQF/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*z 7xng%zF/$!37W(H!pUCrgF/7$$!3XD84Pfc5rF/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[g F/$!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/$!3 6.2<#>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< bQ38'F/7$$\"3%*R,@;vLu')F/$!3e>)zD(p_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2nd o*fQF/7$$\"31mGAp))oa'*F/$!3%)f]nRQ=0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4 O\"F/7$F($\"36YKhSr8/#)!#F-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%*THICK NESSG6#\"\"#-F$6%7$7$F+F+7$$\"3a+++SSDg')F/$\"3++++++++]F/-Fjz6&F\\[l$ \")#)eqkF_[l$\"))eqk\"F_[lFa\\lF`[l-F$6%7$Fg[l7$$!3a+++SSDg')F/F[\\lF] \\lF`[l-F$6%7$7$$!3/++++++]5FdoF[\\l7$$\"3%**************>\"FdoF[\\l-% &COLORG6&F\\[lF+$\"\"(!\"\"F+F`[l-F$6$7S7$$\"3-+++++++QF/F+7$$\"3W?'*z ;Dv*z$F/$\"3!*=+Kz=$oL%Fas7$$\"3'pzX=^M\"*z$F/$\"3T!**oSUY)4\")Fas7$$ \"3%HS?s$>*zz$F/$\"3(f@FC?-_B\"!#>7$$\"3%p]&o&Rjjz$F/$\"3o(4w\"[g2i;F] _l7$$\"3Q15&3CmUz$F/$\"35`$o4d7n3#F]_l7$$\"3]R/r-w*=z$F/$\"31$Q0vys,[# F]_l7$$\"3E;A&*p],*y$F/$\"3!*3T`2jH()GF]_l7$$\"3)z;_DCVHd*=xPF/$\"3o`r ]n,RdTF]_l7$$\"3MU95=!HGx$F/$\"3,ou!*H34OXF]_l7$$\"3G]+9)yluw$F/$\"3O( ei>'*p='\\F]_l7$$\"3KHjA!3(fhPF/$\"3'y@l8eg()Q&F]_l7$$\"3s$zy%>Q[bPF/$ \"3')))**3&y$\\*z&F]_l7$$\"3m>=YKQa\\PF/$\"3'yIALT6w)pF]_l7$$\"3 o-ffQV#os$F/$\"3y!=Wetd9U(F]_l7$$\"3bXB6d$))*=PF/$\"3eAa*Q*)\\Y!yF]_l7 $$\"3,z*H)4y$*4PF/$\"3i<>n#p>TA)F]_l7$$\"3\"\\/&Gx(z3q$F/$\"3fG\"))**3 !eA')F]_l7$$\"3%\\hK1[s4p$F/$\"3%f&Hb+\"os.*F]_l7$$\"3Ks:,&>l9o$F/$\"3 7cY/A.3<%*F]_l7$$\"3Srb+*>s2n$F/$\"33z@m&>Pc#)*F]_l7$$\"3eu(*3S^=fOF/$ \"3Nq.&*GF([-\"F/7$$\"3QJ)\\X^,([OF/$\"3EIt\"3&=fh5F/7$$\"3D%4Rg2lpj$F /$\"37b\\I#)*G65\"F/7$$\"3D&Hs3H!RCOF/$\"3vA+TQ%Q=9\"F/7$$\"3Z\\x3rtk6 OF/$\"3Y#\\=:#e_\"=\"F/7$$\"3g1cv\"G/*)f$F/$\"3^m0!R8!z>7F/7$$\"3OZPau .G%e$F/$\"3O*=o'p\\6i7F/7$$\"3i'))p'GbrqNF/$\"3e2=t*Q'***H\"F/7$$\"3u@ Vuq-zbNF/$\"3!zXq@*=GS8F/7$$\"3?6LB/K(=a$F/$\"35s.VM(QmP\"F/7$$\"3!G%H p\")GBENF/$\"3>KbM44A;9F/7$$\"3?#[%=8E66NF/$\"3q1s*4_-LX\"F/7$$\"3Mypb z2*[\\$F/$\"3')>!)y$*3*=\\\"F/7$$\"3k([()Q'4iyMF/$\"3kgN'y4V%H:F/7$$\" 31utZ$pb6Y$F/$\"3kK?pFmco:F/7$$\"3#fgk&y&=RW$F/$\"3a$RSgyegg\"F/7$$\"3 #yA]i9reU$F/$\"39nV4HS?W;F/7$$\"3![]0v*eb2MF/$\"3+0h6oJ$=o\"F/7$$\"3=q (\\nlh.R$F/$\"3?q^112B;k$=N$F/$\"3S`&[P<0.z\"F/7$$\"3!zxP6mH=L$F/$\"3$4Z'oc(os#=F/7$$\"3CO! )RK))H7LF/$\"3-#[KjIQC'=F/7$$\"3#[j(*R`'*3H$F/$\"3QxPm++++>F/-Fjz6&F\\ [l$\")!\\DP\"F_[lFa]m$\")viobF_[l-F$6&7$Ff\\lFh[l-%'SYMBOLG6#%'CIRCLEG -Fjz6&F\\[lF*F*F*-%&STYLEG6#%&POINTG-F$6&Fg]m-Fi]m6#%(DIAMONDGF\\^mF^^ m-F$6&Fg]m-Fi]m6#%&CROSSGF\\^mF^^m-F$6%7$7$Fi[lF+Fh[lF\\^m-%*LINESTYLE GFb[l-F$6%7$7$Fg\\lF+Ff\\lF\\^mF`_m-%%TEXTG6%7$$\"#V!\"#$\"#MF\\`mQ\"1 6\"F\\^m-Fg_m6%7$$Fg]lFg]l$!\"'F\\`mQ\"OF``mF\\^m-Fg_m6%7$$\"\"*Fg]l$ \"\"'Fg]lQ\"PF``mF\\^m-Fg_m6%7$$\"#()F\\`m$!\"(F\\`mQ\"SF``mF\\^m-Fg_m 6%7$$!#()F\\`mFeamQ\"TF``mF\\^m-Fg_m6%7$$!\"*Fg]lF]amQ\"RF``mF\\^m-Fg_ m6%7$$!#VF\\`mF]`mF_`mF\\^m-Fg_m6%7$$\"$:\"F\\`m$!\"%F\\`mQ\"xF``mF\\^ m-Fg_m6%7$$!\"&F\\`mF\\cmQ\"yF``mF\\^m-Fg_m6%7$$!#XF\\`mFe]lQ$y~=F``mF \\^m-Fg_m6%7$$!#JF\\`m$\"#yF\\`mF_`mF\\^m-Fg_m6%7$F`dm$\"#vF\\`mQ\"_F` `mF\\^m-Fg_m6%7$F`dm$\"#iF\\`mQ\"2F``mF\\^m-Fg_m6&7$$\"#GF\\`m$\"\")F \\`mQ\"qF``mF\\^m-%%FONTG6$Fi]m\"#6-%*AXESTICKSG6$F*F*-%+AXESLABELSG6% Q!F``mFbfm-Fiem6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$0 \"F\\`m$\"$=\"F\\`m;F^gm$\"#7Fg]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" "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" } }{TEXT -1 3 " " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 9 "The line " }{XPPEDIT 18 0 "y = 1/2;" "6#/%\"yG*&\"\"\"F& \"\"#!\"\"" }{TEXT -1 87 " intersects the unit circle at the two point s P and R in the previous picture. Since SP" }{XPPEDIT 18 0 "`` = 1/2; " "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 7 " and OP" }{XPPEDIT 18 0 "` ` = 1;" "6#/%!G\"\"\"" }{TEXT -1 4 ", OS" }{XPPEDIT 18 0 "``=sqrt(3)/2 " "6#/%!G*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 147 " by Pythago ras' theorem and we recognise the triangle OSP to be a (special) 30,60 ,90 degree triangle with hypotenuse 1. The fact that angle SOP is " } {XPPEDIT 18 0 "30^o = Pi/6;" "6#/)\"#I%\"oG*&%#PiG\"\"\"\"\"'!\"\"" } {TEXT -1 40 " radians means that the angle marked as " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 19 " in the picture is " }{XPPEDIT 18 0 "30^o = Pi/6;" "6#/)\"#I%\"oG*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 10 " radians. " }{XPPEDIT 18 0 "theta = Pi/6;" "6#/%&thetaG*&%#PiG\"\"\"\" \"'!\"\"" }{TEXT -1 40 " provides one solution for the equation " } {XPPEDIT 18 0 "sin*theta = 1/2;" "6#/*&%$sinG\"\"\"%&thetaGF&*&F&F&\" \"#!\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "0<=theta" "6#1\"\"!%& thetaG" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "Note that P is the point" } {XPPEDIT 18 0 "``(sqrt(3)/2,1/2);" "6#-%!G6$*&-%%sqrtG6#\"\"$\"\"\"\" \"#!\"\"*&F+F+F,F-" }{TEXT -1 46 " given in terms of the wrapping func tion W as " }{XPPEDIT 18 0 "W(Pi/6);" "6#-%\"WG6#*&%#PiG\"\"\"\"\"'!\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 100 "The triangle OTR is also a (special) 30,60,90 degree t riangle with hypotenuse 1. Since angle TOR is " }{XPPEDIT 18 0 "30^o = Pi/6;" "6#/)\"#I%\"oG*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 40 " radians, it follows that R is the point" }{XPPEDIT 18 0 "``(-sqrt(3)/2,1/2);" "6#-%!G6$,$*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F.*&F,F,F-F." }{TEXT -1 46 " given in terms of the wrapping function W as " }{XPPEDIT 18 0 "W( 5*Pi/6);" "6#-%\"WG6#*(\"\"&\"\"\"%#PiGF(\"\"'!\"\"" }{TEXT -1 31 ". T his gives a second solution " }{XPPEDIT 18 0 "theta = 5*Pi/6;" "6#/%&t hetaG*(\"\"&\"\"\"%#PiGF'\"\"'!\"\"" }{TEXT -1 18 " for the equation \+ " }{XPPEDIT 18 0 "sin*theta = 1/2;" "6#/*&%$sinG\"\"\"%&thetaGF&*&F&F& \"\"#!\"\"" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "0<=theta" "6#1\"\"!% &thetaG" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "Indeed there are exactly two solutions " }{XPPEDIT 18 0 "theta = P i/6;" "6#/%&thetaG*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "theta = 5*Pi/6;" "6#/%&thetaG*(\"\"&\"\"\"%#PiGF'\"\"'! \"\"" }{TEXT -1 18 " for the equation " }{XPPEDIT 18 0 "sin*theta = 1/ 2;" "6#/*&%$sinG\"\"\"%&thetaGF&*&F&F&\"\"#!\"\"" }{TEXT -1 12 " satis fying " }{XPPEDIT 18 0 "0<=theta" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "` `<2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 83 " given by angles ob tained by rotating clockwise from the positive direction of the " } {TEXT 311 1 "x" }{TEXT -1 95 " axis until the terminal side of the ang le coincides with either of the radius lines OP or OR. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 313 10 "Conclusion" }{TEXT -1 3 ": " }{XPPEDIT 18 0 "sin*theta = 1/2;" "6#/*&%$sinG\"\"\"%&theta GF&*&F&F&\"\"#!\"\"" }{TEXT -1 14 " exactly when " }{XPPEDIT 18 0 "the ta = Pi/6;" "6#/%&thetaG*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 4 " or " } {XPPEDIT 18 0 "theta = 5*Pi/6;" "6#/%&thetaG*(\"\"&\"\"\"%#PiGF'\"\"'! \"\"" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "0<=theta" "6#1\"\"!%&theta G" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 7 " }}{PARA 0 "" 0 "" {TEXT 298 8 "Question" }{TEXT -1 18 ": Find all angles " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "0<=theta" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 11 " such that " } {XPPEDIT 18 0 "tan*theta = -1;" "6#/*&%$tanG\"\"\"%&thetaGF&,$F&!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 299 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan*theta;" "6#*&%$tanG\"\"\"%&thetaGF%" }{TEXT -1 52 " is the gradient of the line OP where P is the point" }{XPPEDIT 18 0 "``(cos*theta,sin*theta)" "6#-%!G6$*&%$cosG\"\"\"%&thetaGF(*&%$si nGF(F)F(" }{TEXT -1 49 " and OP is obtained by rotating through an ang le " }{XPPEDIT 18 0 "abs(theta)" "6#-%$absG6#%&thetaG" }{TEXT -1 36 " \+ from the positive direction of the " }{TEXT 300 1 "x" }{TEXT -1 44 " a xis in the counter-clockwise direction if " }{XPPEDIT 18 0 "theta" "6# %&thetaG" }{TEXT -1 48 " is positive, and in the clockwise direction i f " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 15 " is negative. \+ " }}{PARA 0 "" 0 "" {TEXT -1 23 "Hence we draw the line " }{XPPEDIT 18 0 "y = -x;" "6#/%\"yG,$%\"xG!\"\"" }{TEXT -1 110 ", which passes th rough the origin and has gradient -1, and observe where this line inte rsects the unit circle " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"\"# \"\"\"*$%\"yGF'F(F(" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "______________________________ ___" }}{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 33 "_________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q2 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Find the \+ exact radian measure of the following angles given in degrees." }} {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "210^o" "6#)\"$5#%\"o G" }{TEXT -1 6 " (b) " }{XPPEDIT 18 0 "225^o" "6#)\"$D#%\"oG" }{TEXT -1 6 " (c) " }{XPPEDIT 18 0 "75^o" "6#)\"#v%\"oG" }{TEXT -1 6 " (d) \+ " }{XPPEDIT 18 0 "-40^o" "6#,$)\"#S%\"oG!\"\"" }{TEXT -1 6 " (e) " } {XPPEDIT 18 0 "-105^o" "6#,$)\"$0\"%\"oG!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) \+ " }{XPPEDIT 18 0 "7*Pi/6" "6#*(\"\"(\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 6 " (b) " }{XPPEDIT 18 0 "5*Pi/4" "6#*(\"\"&\"\"\"%#PiGF%\"\"%!\" \"" }{TEXT -1 6 " (c) " }{XPPEDIT 18 0 "5*Pi/12" "6#*(\"\"&\"\"\"%#Pi GF%\"#7!\"\"" }{TEXT -1 6 " (d) " }{XPPEDIT 18 0 "-2*Pi/9" "6#,$*(\" \"#\"\"\"%#PiGF&\"\"*!\"\"F)" }{TEXT -1 6 " (e) " }{XPPEDIT 18 0 "-7* Pi/12" "6#,$*(\"\"(\"\"\"%#PiGF&\"#7!\"\"F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "______________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Find the exact degree measure of the following angles giv en in radians." }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "3* Pi/4" "6#*(\"\"$\"\"\"%#PiGF%\"\"%!\"\"" }{TEXT -1 7 " (b) " } {XPPEDIT 18 0 "4*Pi/3;" "6#*(\"\"%\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT -1 6 " (c) " }{XPPEDIT 18 0 "Pi/12;" "6#*&%#PiG\"\"\"\"#7!\"\"" }{TEXT -1 6 " (d) " }{XPPEDIT 18 0 "-2*Pi/5;" "6#,$*(\"\"#\"\"\"%#PiGF&\"\"& !\"\"F)" }{TEXT -1 6 " (e) " }{XPPEDIT 18 0 "-7*Pi/6;" "6#,$*(\"\"(\" \"\"%#PiGF&\"\"'!\"\"F)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "135^ o;" "6#)\"$N\"%\"oG" }{TEXT -1 6 " (b) " }{XPPEDIT 18 0 "240^o" "6#) \"$S#%\"oG" }{TEXT -1 6 " (c) " }{XPPEDIT 18 0 "18^o" "6#)\"#=%\"oG" }{TEXT -1 6 " (d) " }{XPPEDIT 18 0 "-72^o" "6#,$)\"#s%\"oG!\"\"" } {TEXT -1 6 " (e) " }{XPPEDIT 18 0 "- 210^o" "6#,$)\"$5#%\"oG!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "____________ _____________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Find the exact values of " } {XPPEDIT 18 0 "cos*theta" "6#*&%$cosG\"\"\"%&thetaGF%" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "sin*theta" "6#*&%$sinG\"\"\"%&thetaGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "tan*theta" "6#*&%$tanG\"\"\"%&thetaGF%" } {TEXT -1 33 " in each of the following cases. " }}{PARA 0 "" 0 "" {TEXT -1 88 "In each case draw a picture that shows the appropriate re ference triangle for the angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "th eta=3*Pi/4" "6#/%&thetaG*(\"\"$\"\"\"%#PiGF'\"\"%!\"\"" }{TEXT -1 7 " \+ (b) " }{XPPEDIT 18 0 "theta=5*Pi/6" "6#/%&thetaG*(\"\"&\"\"\"%#PiGF' \"\"'!\"\"" }{TEXT -1 7 " (c) " }{XPPEDIT 18 0 "theta=4*Pi/3" "6#/%& thetaG*(\"\"%\"\"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 7 " (d) " }{XPPEDIT 18 0 "theta=-3*Pi/4" "6#/%&thetaG,$*(\"\"$\"\"\"%#PiGF(\"\"%!\"\"F+" } {TEXT -1 7 " (e) " }{XPPEDIT 18 0 "theta=-7*Pi/6" "6#/%&thetaG,$*(\" \"(\"\"\"%#PiGF(\"\"'!\"\"F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "cos(3*Pi/4)=-sqrt(2)/2" "6#/-%$cosG6 #*(\"\"$\"\"\"%#PiGF)\"\"%!\"\",$*&-%%sqrtG6#\"\"#F)F2F,F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(3*Pi/4)=sqrt(2)/2" "6#/-%$sinG6#*(\"\"$\"\" \"%#PiGF)\"\"%!\"\"*&-%%sqrtG6#\"\"#F)F1F," }{TEXT -1 2 ", " } {XPPEDIT 18 0 "tan(3*Pi/4)=-1" "6#/-%$tanG6#*(\"\"$\"\"\"%#PiGF)\"\"%! \"\",$F)F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " } {XPPEDIT 18 0 "cos(5*Pi/6)=-sqrt(3)/2" "6#/-%$cosG6#*(\"\"&\"\"\"%#PiG F)\"\"'!\"\",$*&-%%sqrtG6#\"\"$F)\"\"#F,F," }{TEXT -1 2 ", " } {XPPEDIT 18 0 "sin(5*Pi/6)=1/2" "6#/-%$sinG6#*(\"\"&\"\"\"%#PiGF)\"\"' !\"\"*&F)F)\"\"#F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tan(5*Pi/6)=-sqrt (3)/3" "6#/-%$tanG6#*(\"\"&\"\"\"%#PiGF)\"\"'!\"\",$*&-%%sqrtG6#\"\"$F )F2F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " } {XPPEDIT 18 0 "cos(4*Pi/3)=-1/2" "6#/-%$cosG6#*(\"\"%\"\"\"%#PiGF)\"\" $!\"\",$*&F)F)\"\"#F,F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(4*Pi/3)= -sqrt(3)/2" "6#/-%$sinG6#*(\"\"%\"\"\"%#PiGF)\"\"$!\"\",$*&-%%sqrtG6#F +F)\"\"#F,F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tan(4*Pi/3)=sqrt(3)" "6 #/-%$tanG6#*(\"\"%\"\"\"%#PiGF)\"\"$!\"\"-%%sqrtG6#F+" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }{XPPEDIT 18 0 "cos(-3*Pi/4)=-sq rt(2)/2" "6#/-%$cosG6#,$*(\"\"$\"\"\"%#PiGF*\"\"%!\"\"F-,$*&-%%sqrtG6# \"\"#F*F3F-F-" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(-3*Pi/4)=-sqrt(2)/ 2" "6#/-%$sinG6#,$*(\"\"$\"\"\"%#PiGF*\"\"%!\"\"F-,$*&-%%sqrtG6#\"\"#F *F3F-F-" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tan(-3*Pi/4)=1" "6#/-%$tanG6 #,$*(\"\"$\"\"\"%#PiGF*\"\"%!\"\"F-F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(e) " }{XPPEDIT 18 0 "cos(-7*Pi/6)=-sqrt(3)/2" "6#/-%$c osG6#,$*(\"\"(\"\"\"%#PiGF*\"\"'!\"\"F-,$*&-%%sqrtG6#\"\"$F*\"\"#F-F- " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(-7*Pi/6)=1/2" "6#/-%$sinG6#,$*( \"\"(\"\"\"%#PiGF*\"\"'!\"\"F-*&F*F*\"\"#F-" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "tan(-7*Pi/6)=-sqrt(3)/3" "6#/-%$tanG6#,$*(\"\"(\"\"\"%# PiGF*\"\"'!\"\"F-,$*&-%%sqrtG6#\"\"$F*F3F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 33 "_____________ ____________________" }}{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 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 33 "_________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }} {PARA 0 "" 0 "" {TEXT -1 48 "Solve the following trigonometric equatio ns for " }{XPPEDIT 18 0 "0<=theta" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 " ``<2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "cos*theta = sqrt(3)/2;" "6#/*&%$c osG\"\"\"%&thetaGF&*&-%%sqrtG6#\"\"$F&\"\"#!\"\"" }{TEXT -1 6 " (b) \+ " }{XPPEDIT 18 0 "sin*theta=sqrt(2)/2" "6#/*&%$sinG\"\"\"%&thetaGF&*&- %%sqrtG6#\"\"#F&F,!\"\"" }{TEXT -1 6 " (c) " }{XPPEDIT 18 0 "tan*thet a=sqrt(3)" "6#/*&%$tanG\"\"\"%&thetaGF&-%%sqrtG6#\"\"$" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }{XPPEDIT 18 0 "cos*theta=-sqrt( 2)/2" "6#/*&%$cosG\"\"\"%&thetaGF&,$*&-%%sqrtG6#\"\"#F&F-!\"\"F." } {TEXT -1 6 " (e) " }{XPPEDIT 18 0 "sin*theta=-sqrt(3)/2" "6#/*&%$sinG \"\"\"%&thetaGF&,$*&-%%sqrtG6#\"\"$F&\"\"#!\"\"F/" }{TEXT -1 6 " (f) \+ " }{XPPEDIT 18 0 "tan*theta=-sqrt(3)/3" "6#/*&%$tanG\"\"\"%&thetaGF&,$ *&-%%sqrtG6#\"\"$F&F-!\"\"F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "thet a=Pi/6" "6#/%&thetaG*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 4 " or " } {XPPEDIT 18 0 "theta=11*Pi/6" "6#/%&thetaG*(\"#6\"\"\"%#PiGF'\"\"'!\" \"" }{TEXT -1 6 " (b) " }{XPPEDIT 18 0 "theta=Pi/4" "6#/%&thetaG*&%#P iG\"\"\"\"\"%!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "theta=3*Pi/4" " 6#/%&thetaG*(\"\"$\"\"\"%#PiGF'\"\"%!\"\"" }{TEXT -1 6 " (c) " } {XPPEDIT 18 0 "theta=Pi/3" "6#/%&thetaG*&%#PiG\"\"\"\"\"$!\"\"" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "theta=4*Pi/3" "6#/%&thetaG*(\"\"%\" \"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "( d) " }{XPPEDIT 18 0 "theta = 3*Pi/4;" "6#/%&thetaG*(\"\"$\"\"\"%#PiGF' \"\"%!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "theta = 5*Pi/4;" "6#/%& thetaG*(\"\"&\"\"\"%#PiGF'\"\"%!\"\"" }{TEXT -1 6 " (e) " }{XPPEDIT 18 0 "theta = 4*Pi/3;" "6#/%&thetaG*(\"\"%\"\"\"%#PiGF'\"\"$!\"\"" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "theta = 5*Pi/3;" "6#/%&thetaG*(\"\"& \"\"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 6 " (f) " }{XPPEDIT 18 0 "theta = \+ 5*Pi/6;" "6#/%&thetaG*(\"\"&\"\"\"%#PiGF'\"\"'!\"\"" }{TEXT -1 4 " or \+ " }{XPPEDIT 18 0 "theta = 11*Pi/6;" "6#/%&thetaG*(\"#6\"\"\"%#PiGF'\" \"'!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 33 "_________________________________" }}{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 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 33 "_______________________________ __" }}{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 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 32 "Code for general rig ht triangle " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 564 "darkgreen := COLOR(RGB,0,.7,0):\np1 := plot([[[0, 0],[4,3]],[[4,3],[4,0]],[[4,0],[0,0]]],\n color=[brown,darkgreen,red ],thickness=2):\np2 := plot([[0.84*cos(t),0.84*sin(t),t=0..arctan(0.75 )],\n[[3.8,0],[3.8,.2],[4,.2]]],color=navy):\nt1 := plots[textplot]([[ 4.15,1.5,`a`],[4.6,2,`opposite`]],color=darkgreen):\nt2 := plots[textp lot]([[2,-.15,`b`],[2.7,-.35,`adjacent`]],color=red):\nt3 := plots[tex tplot]([[1.94,1.75,`c`],[2,2.2,`hypotenuse`]],color=brown):\nt4 := plo ts[textplot]([.63,.24,`q`],font=[SYMBOL,12],color=navy):\nplots[displa y]([p1,p2,t1,t2,t3,t4],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 34 "Code for 45 degree right triang le " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 695 "darkgreen := COLOR(RGB,0,.7,0):\np1 := plot([[[0,0], [4,4]],[[4,4],[4,0]],[[4,0],[0,0]]],\n color=[brown,darkgreen,red], thickness=2):\np2 := plot([[cos(t),sin(t),t=0..Pi/4],\n [[3.8,0],[ 3.8,.2],[4,.2]]],color=navy):\nt1 := plots[textplot]([4.2,1.8,`1`],col or=darkgreen):\nt2 := plots[textplot]([2,-.2,`1`],color=red):\nt3 := p lots[textplot]([1.82,2.3,`2`],font=[HELVETICA,10],color=brown):\nt4 := plots[textplot]([[1.7,2.34,`/`],[1.81,2.6,`_`]],\n font=[HEL VETICA,13],color=brown):\nt5 := plots[textplot]([1.65,2.25,`v`],font=[ HELVETICA,8],color=brown):\nt6 := plots[textplot]([[.6,.24,`45`],[.77, .4,`o`]],color=navy):\nplots[display]([p1,p2,t1,t2,t3,t4,t5,t6],axes=n one,scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 34 "Code for 60 degree right triangle " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 998 "darkgreen := COLOR(RGB,0,.7,0):\npurple := COLOR(RGB,.4,0,.9):\np 1 := plot([[[0,0],[1,sqrt(3)]],[[1,sqrt(3)],[2,0]],\n [[0,0],[1,0]], [[1,0],[2,0]]],\n color=[brown,darkgreen,red,purple],thickness=2) :\np2 := plot([[0.4*cos(t),0.4*sin(t),t=0..Pi/3],\n [2+0.4*cos(t), 0.4*sin(t),t=2*Pi/3..Pi],\n [[.9,0],[.9,.1],[1,.1]]],color=navy ):\np3 := plot([[1,0],[1,sqrt(3)]],color=black):\nt1 := plots[textplot ]([1.57,.93,`2`],color=darkgreen):\nt2 := plots[textplot]([.43,.93,`2` ],color=brown):\nt3 := plots[textplot]([.5,-.08,`1`],color=red):\nt4 : = plots[textplot]([1.5,-.08,`1`],color=purple):\nt5 := plots[textplot] ([1.13,.72,`3`],color=black):\nt6 := plots[textplot]([[1.08,.73,`/`],[ 1.13,.84,`_`]],\n font=[HELVETICA,13],color=black):\nt7 := pl ots[textplot]([1.06,.7,`v`],font=[HELVETICA,8],color=black):\nt8 := pl ots[textplot]([[.19,.14,`60`],[.26,.19,`o`],\n [1.76,.14,`60`],[1.83 ,.19,`o`]],color=navy):\nplots[display]([p1,p2,p3,t1,t2,t3,t4,t5,t6,t7 ,t8],\n axes=none,scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "Code for 1st circle pi cture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 687 "darkgreen := COLOR(RGB,0,.7,0):\ncs := evalf(cos(Pi/ 3)):\nsn := evalf(sin(Pi/3)):\np1 := plot([[cos(t),sin(t),t=0..Pi/3], \n[cos(t),sin(t),t=Pi/3..2*Pi]],color=[blue,red],thickness=2):\np2 := \+ plot([[[0,0],[cs,sn]],[[cs,0],[cs,sn]],[[0,0],[cs,0]]],\n \+ color=[brown,darkgreen,coral],thickness=2):\np3 := plot([0.25*cos(t),0 .25*sin(t),t=0..Pi/3],color=navy):\nt1 := plots[textplot]([[.85,.96,`P ( cos ,sin )`],\n [.22,.54,`1`],[1.15,-.04,`x`],[-.05,1.15,`y` ],\n [-.06,.09,`O`]],color=black):\nt2 := plots[textplot]([[.15,. 1,`q`],[.857,.961,`q`],\n [1.08,.961,`q`]],font=[SYMBOL ,11],color=black):\nplots[display]([p1,p2,p3,t1,t2],tickmarks=[0,0],sc aling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "Code for 2nd circle picture " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 717 "dark green := COLOR(RGB,0,.7,0):\ncs := evalf(cos(Pi/3)):\nsn := evalf(sin( Pi/3)):\np1 := plot([[cos(t),sin(t),t=0..2*Pi/3],\n [cos(t),sin(t) ,t=2*Pi/3..2*Pi]],color=[blue,red],thickness=2):\np2 := plot([[[0,0],[ -cs,sn]],[[-cs,0],[-cs,sn]],[[0,0],[-cs,0]]],\n color=[brown,dark green,coral],thickness=2):\np3 := plot([0.25*cos(t),0.25*sin(t),t=0..2 *Pi/3],color=navy):\nt1 := plots[textplot]([[-.85,.96,`P( cos ,sin \+ )`],\n [-.22,.54,`1`],[1.15,-.04,`x`],[-.05,1.15,`y`],\n [-. 05,-.06,`O`],[-.52,-.06,`Q`]],color=black):\nt2 := plots[textplot]([[. 085,.13,`q`],[-.842,.962,`q`],\n [-.61,.962,`q`]],font= [SYMBOL,11],color=black):\nplots[display]([p1,p2,p3,t1,t2],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 44 "Code for 2nd 45,45,90 degree right triangle " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 949 "dark green := COLOR(RGB,0,.7,0):\np1 := plot([[[0,0],[4,4]],[[4,4],[4,0]],[ [4,0],[0,0]]],\n color=[brown,darkgreen,red],thickness=2):\np2 := p lot([[cos(t),sin(t),t=0..Pi/4],\n [[3.8,0],[3.8,.2],[4,.2]]],color =navy):\nt1 := plots[textplot]([1.82,2.3,`1`],font=[HELVETICA,10],colo r=brown):\nt2 := plots[textplot]([[2.,-.3,`2`],[2.,-.7,`2`]],color=red ):\nt3 := plots[textplot]([[1.84,-.26,`/`],[2,.07,`_`],[1.94,-.3,`_`], [2,-.3,`_`]],\n font=[HELVETICA,13],color=red):\nt4 := plots[ textplot]([1.78,-.37,`v`],font=[HELVETICA,8],color=red):\nt5 := plots[ textplot]([[4.4,2.1,`2`],[4.4,1.7,`2`]],color=darkgreen):\nt6 := plots [textplot]([[4.27,2.14,`/`],[4.4,2.48,`_`],[4.34,2.11,`_`],[4.4,2.11,` _`]],\n font=[HELVETICA,13],color=darkgreen):\nt7 := plots[te xtplot]([4.2,2.03,`v`],font=[HELVETICA,8],color=darkgreen):\nt8 := plo ts[textplot]([[.6,.24,`45`],[.77,.4,`o`]],color=navy):\nplots[display] ([p1,p2,t||(1..8)],axes=none,scaling=constrained);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 44 "Code for 2nd 30,60,90 degree right triangle " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 808 "darkgreen := COLOR(RGB,0,.7,0):\npurple := COLOR(RGB,.4,0,.9):\np 1 := plot([[[0,0],[1,sqrt(3)]],[[1,sqrt(3)],[1,0]],[[0,0],[1,0]]],\n \+ color=[brown,darkgreen,red,purple],thickness=2):\np2 := plot([[0.4 *cos(t),0.4*sin(t),t=0..Pi/3],[[.9,0],[.9,.1],[1,.1]]],color=navy):\nt 1 := plots[textplot]([.43,.93,`1`],color=brown):\nt2 := plots[textplot ]([[.5,-.08,`1`],[.5,-.2,`2`],[.5,-.08,`_`]],color=red):\nt3 := plots[ textplot]([[1.15,.8,`3`],[1.15,.67,`2`]],color=darkgreen):\nt4 := plot s[textplot]([[1.1,.82,`/`],[1.15,.93,`_`],[1.13,.81,`_`],[1.15,.81,`_` ]],\n font=[HELVETICA,13],color=darkgreen):\nt5 := plots[text plot]([1.07,.78,`v`],font=[HELVETICA,8],color=darkgreen):\nt6 := plots [textplot]([[.2,.14,`60`],[.28,.19,`o`]],color=navy):\nplots[display]( [p1,p2,t||(1..6)],\n axes=none,scaling=constrained);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 0 "" 0 "" {TEXT -1 32 "Code for radian measure picture \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 970 "darkgreen := COLOR(RGB,0,.7,0):\nthet := 2.3:\ncs := evalf(co s(thet)):\nsn := evalf(sin(thet)):\nr1 := 1.3: r2 := 1.15:\np1 := plot ([[cos(t),sin(t),t=0..thet],[cos(t),sin(t),t=thet..2*Pi]],\n \+ color=[blue,red],thickness=2):\np2 := plot([[r2*cos(t),r2*sin(t),t=0.. thet/2-.1],[r2*cos(t),r2*sin(t),t=thet/2+.1..thet]],\n color= black):\np3 := plot([[[0,0],[cs,sn]],[[0,0],[1,0]]],color=[brown,coral ],thickness=2):\np4 := plot([[[1,0],[r1,0]],[[cs,sn],[r1*cs,r1*sn]]],l inestyle=2,color=black):\np5 := plot([0.25*cos(t),0.25*sin(t),t=0..the t],color=navy):\np6 := plot([[[r2*cs+.03,r2*sn+.08],[r2*cs,r2*sn],[r2* cs+.09,r2*sn+.02]],\n [[r2-.04,.08],[r2,0],[r2+.04,.08]]],co lor=black):\nt1 := plots[textplot]([[-.36,.28,`r`],[.5,-.1,`r`],[r2*co s(thet/2),r2*sin(thet/2),`s`],\n [-.06,-.09,`O`]],color= black):\nt2 := plots[textplot]([.04,.14,`q`],font=[SYMBOL,11],color=na vy):\nplots[display]([p||(1..6),t1,t2],tickmarks=[0,0],scaling=constra ined,axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 35 "Code for graphs of sine and cosine " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 408 "p 1:=plot([cos(t*Pi),sin(t*Pi)],t=-1..3.2,color=[blue,red],\n xtickm arks=[-1=`-p`,-.5=`-p/2`,0=`0`,.5=`p/2`,\n 1=`p`,1.5=`3p/2`,2=`2 p`,2.5=`5p/2`,3=`3p`],\n ytickmarks=3,thickness=2,labels=[``,``], labels=[``,``]):\np2 := plots[textplot]([3.2,-.12,`q`],font=[SYMBOL,10 ],color=black):\np3 := plots[textplot]([-.05,1.3,`y`],font=[HELVETICA, 10],color=black):\nplots[display]([p1,p2,p3],font=[SYMBOL,10]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 34 "Code for wrapping fun ction picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 543 "p1 := plot([[0,-3.5],[0,3.5]],color=blue,thickn ess=2):\np2 := plot([[-1,-3.5],[-1,3.5]],color=black):\np3 := plot([co s(t)-1,sin(t),t=0..2*Pi],color=red):\np4 := plot([[0,0]],style=point,s ymbol=circle,color=black):\nt1 := plots[textplot]([[.4,-.1,`x`],[-1.1, 3.5,`y`],[.3,.3,`(1,0)`]],font=[HELVETICA,10]):\nt2 := plots[textplot] ([-.2,3.4,`q`],font=[SYMBOL,11]):\npi := evalf(Pi):\nplots[display]([p ||(1..4),t1,t2],scaling=constrained,\n ytickmarks=[-pi=`-p`,-pi/2=`-p /2`,pi/2=`p/2`,pi=`p`],\n xtickmarks=0,view=[-2.1..0.4,-3.5..3.5],fon t=[SYMBOL,11]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 36 "Co de for wrapping function animation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1113 "numframes := 60:\nthet := \+ evalf(Pi/numframes):\nfor i from 0 to 2*numframes do\n tt := thet*i; \n if i=0 then\n p1 := plot([[1,0],[1,2.5]],color=blue,thicknes s=2):\n elif i0 then\n p2 := plot([cos(t),sin(t),t=0..tt],color=blue,thick ness=2);\n else\n p2 := plot([[1,0]],color=blue,thickness=2);\n end if;\n p3 := plot([cos(t),sin(t),t=tt..2*Pi]);\n p4 := plot( [[0,0],[cos(tt),sin(tt)]],color=black);\n p5 := plot([[cos(tt),sin(t t)]],style=point,symbol=circle,color=black);\n frm[i] := plots[displ ay]([p||(1..5)],view=[-2..2,-2.5..2.5]);\nend do:\nplots[display]([seq (frm[i],i=0..2*numframes)],insequence=true,view=[-2..2,-2.5..2.5],\n \+ scaling=constrained,tickmarks=[0,0]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 39 "Code for graph of the \+ tangent function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 459 "p1 := plot(tan(t*Pi),t=-1.1..3.3,-5..5,disco nt=true,color=red,thickness=2):\np2 := plot([[[-1/2,-5],[-1/2,5]],[[1/ 2,-5],[1/2,5]],\n [[3/2,-5],[3/2,5]],[[5/2,-5],[5/2,5]]],linestyle=3,c olor=black):\nt1 := plots[textplot]([3.3,-.3,`q`],font=[SYMBOL,11]):\n t2 := plots[textplot]([-.1,5,`y`],font=[HELVETICA,10]):\nplots[display ]([p1,p2,t1,t2],xtickmarks=[-1=`-p`,-.5=`-p/2`,.5=`p/2`,1=`p`,\n 1.5 =`3p/2`,2=`2p`,2.5=`5p/2`,3=`3p`],\n font=[SYMBOL,11],labels=[``,``] );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 46 "Code for circle picture in equation example 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 825 "darkgree n := COLOR(RGB,0,.7,0):\ncs := evalf(cos(Pi/3)):\nsn := evalf(sin(Pi/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,-1.05],[-cs,1.2]]], \n color=[brown$2,darkgreen],thickness=2):\np3 := plot([0.25*cos( t),0.25*sin(t),t=0..2*Pi/3],color=navy):\np4 := plot([[[-cs,sn],[-cs,- sn]]$3],style=point,color=black,symbol=[circle,diamond,cross]):\nt1 := plots[textplot]([[-.22,.54,`1`],[-.1,-.06,`O`],[-.57,-.06,`Q`],[-.57, .92,`P`],\n [-.57,-.92,`R`],[-.22,-.54,`1`],[1.15,-.04,`x`],[-.05 ,1.15,`y`],\n [-.73,.25,`x = -`],[-.57,.33,`1`],[-.57,.31,`_`],[- .57,.18,`2`]],color=black):\nt2 := plots[textplot]([[.085,.13,`q`]],fo nt=[SYMBOL,11],color=black):\nplots[display]([p||(1..4),t1,t2],tickmar ks=[0,0],view=[-1.05..1.18,-1.05..1.2],scaling=constrained);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 46 "Code for circle picture in equation example 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 895 "darkgreen := COLOR(RGB,0,.7,0):\nc s := evalf(cos(Pi/6)):\nsn := evalf(sin(Pi/6)):\np1 := plot([cos(t),si n(t),t=0..2*Pi],color=red,thickness=2):\np2 := plot([[[0,0],[cs,sn]],[ [0,0],[-cs,sn]],[[-1.05,sn],[1.2,sn]]],\n color=[brown$2,darkgree n],thickness=2):\np3 := plot([0.38*cos(t),0.38*sin(t),t=0..Pi/6],color =navy):\np4 := plot([[[-cs,sn],[cs,sn]]$3],style=point,color=black,sym bol=[circle,diamond,cross]):\np5 := plot([[[cs,0],[cs,sn]],[[-cs,0],[- cs,sn]]],color=black,linestyle=2):\nt1 := plots[textplot]([[.43,.34,`1 `],[-.1,-.06,`O`],[.9,.6,`P`],[.87,-.07,`S`],\n [-.87,-.07,`T`],[ -.9,.6,`R`],[-.43,.34,`1`],[1.15,-.04,`x`],[-.05,1.15,`y`],\n [-. 45,.7,`y =`],[-.31,.78,`1`],[-.31,.75,`_`],[-.31,.62,`2`]],color=black ):\nt2 := plots[textplot]([[.28,.08,`q`]],font=[SYMBOL,11],color=black ):\nplots[display]([p||(1..5),t1,t2],tickmarks=[0,0],view=[-1.05..1.18 ,-1.05..1.2],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 46 "Code for circle picture in equation example 3 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 875 "darkgreen := COLOR(RGB,0,.7,0):\nsc := evalf(cos(Pi/4)):\np1 := p lot([[cos(t),sin(t),t=0..2*Pi],[t,-t,t=-1.05..1.2]],\n col or=[red,darkgreen],thickness=2):\np2 := plot([[[-sc,0],[-sc,sc]],[[sc, 0],[sc,-sc]]],color=black,linestyle=2):\np3 := plot([0.25*cos(t),0.25* sin(t),t=0..3*Pi/4],color=navy):\np4 := plot([[[-sc,sc],[sc,-sc]]$3],s tyle=point,color=black,symbol=[circle,diamond,cross]):\np5 := plot([[[ sc,0],[sc,-sc]],[[-sc,0],[-sc,-sc]]],color=black,linestyle=2):\nt1 := \+ plots[textplot]([[.34,-.46,`1`],[-.06,-.06,`O`],[-.8,.71,`P`],[.72,.09 ,`T`],\n [-.72,-.07,`S`],[.82,-.71,`R`],[-.32,.46,`1`],[1.15,-.04 ,`x`],[-.05,1.15,`y`],\n [-.75,1,`y = -x`],[.68,-1.05,`gradient = -1`]],color=black):\nt2 := plots[textplot]([[.08,.15,`q`]],font=[SYMB OL,11],color=black):\nplots[display]([p||(1..4),t1,t2],tickmarks=[0,0] ,view=[-1.05..1.18,-1.05..1.2],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 }