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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 44 "The secant function is periodic with period " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 1 ":" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sec(theta+2*Pi) = sec*theta;" "6#/ -%$secG6#,&%&thetaG\"\"\"*&\"\"#F)%#PiGF)F)*&F%F)F(F)" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 42 "The following picture shows the graph of " }{XPPEDIT 18 0 "y = sec*theta;" "6#/%\"yG*&%$secG\"\"\"%&thetaG F'" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 5 "range" }{TEXT -1 37 " of the secant function is the set \{ " }{TEXT 266 1 "y" }{TEXT -1 3 " | " }{XPPEDIT 18 0 "y \+ in R" "6#-%#inG6$%\"yG%\"RG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y >=1" " 6#1\"\"\"%\"yG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "y <= -1" "6#1%\"yG, $\"\"\"!\"\"" }{TEXT -1 5 "\} = (" }{XPPEDIT 18 0 "-infinity,-1" "6$,$ %)infinityG!\"\",$\"\"\"F%" }{TEXT -1 1 "]" }{XPPEDIT 18 0 "`union`(`` ,``);" "6#-%&unionG6$%!GF&" }{TEXT -1 1 "[" }{XPPEDIT 18 0 "1,infinity " "6$\"\"\"%)infinityG" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "The cosecant function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 8 "cosecant" }{TEXT -1 24 " function is defined by " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "csc*theta = 1/(sin*theta);" "6#/ *&%$cscG\"\"\"%&thetaGF&*&F&F&*&%$sinGF&F'F&!\"\"" }{TEXT -1 2 ", " }} {PARA 257 "" 0 "" {TEXT 262 7 "_______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "for any real number \+ " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 11 " (or angle " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 12 ") such that " } {XPPEDIT 18 0 "sin*theta <> 0;" "6#0*&%$sinG\"\"\"%&thetaGF&\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 46 "The cosecant function is periodic with period " } {XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 1 ":" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "csc(theta+2*Pi) = csc *theta;" "6#/-%$cscG6#,&%&thetaG\"\"\"*&\"\"#F)%#PiGF)F)*&F%F)F(F)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "The following picture s hows the graph of " }{XPPEDIT 18 0 "y = csc*theta;" "6#/%\"yG*&%$cscG \"\"\"%&thetaGF'" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 5 "ra nge" }{TEXT -1 39 " of the cosecant function is the set \{ " }{TEXT 269 1 "y" }{TEXT -1 3 " | " }{XPPEDIT 18 0 "y in R" "6#-%#inG6$%\"yG% \"RG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y >=1" "6#1\"\"\"%\"yG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "y <= -1" "6#1%\"yG,$\"\"\"!\"\"" }{TEXT -1 5 "\} = (" }{XPPEDIT 18 0 "-infinity,-1" "6$,$%)infinityG!\"\",$\" \"\"F%" }{TEXT -1 1 "]" }{XPPEDIT 18 0 "`union`(``,``);" "6#-%&unionG6 $%!GF&" }{TEXT -1 1 "[" }{XPPEDIT 18 0 "1,infinity" "6$\"\"\"%)infinit yG" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "The cotangent function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 9 "cotangent" } {TEXT -1 24 " function is defined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cot*theta = cos*theta/(sin*theta);" "6#/*&%$cotG\" \"\"%&thetaGF&*(%$cosGF&F'F&*&%$sinGF&F'F&!\"\"" }{TEXT -1 2 ", " }} {PARA 257 "" 0 "" {TEXT 264 9 " ________" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "for any real numbe r " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 11 " (or angle " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 12 ") such that " } {XPPEDIT 18 0 "sin*theta <> 0;" "6#0*&%$sinG\"\"\"%&thetaGF&\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "If both " }{XPPEDIT 18 0 "sin*theta <> 0;" "6#0*&%$sinG\"\"\"%&thetaGF&\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos*theta <> 0;" "6#0*&%$cosG\"\"\"%&thetaGF&\"\"! " }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "cot*theta = 1/(tan*theta);" "6 #/*&%$cotG\"\"\"%&thetaGF&*&F&F&*&%$tanGF&F'F&!\"\"" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The cota ngent function is periodic with period " }{XPPEDIT 18 0 "Pi" "6#%#PiG " }{TEXT -1 1 ":" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c ot(theta+Pi) = cot*theta;" "6#/-%$cotG6#,&%&thetaG\"\"\"%#PiGF)*&F%F)F (F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "The following pic ture shows the graph of " }{XPPEDIT 18 0 "y = cot*theta;" "6#/%\"yG*& %$cotG\"\"\"%&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 "sin*theta = 0;" "6#/*&%$sinG \"\"\"%&thetaGF&\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 6 "domain" }{TEXT -1 39 " of the cotangent function is the set \{" }{XPPEDIT 18 0 "theta" " 6#%&thetaG" }{TEXT -1 3 " | " }{XPPEDIT 18 0 "theta in R" "6#-%#inG6$% &thetaG%\"RG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta <> k*Pi;" "6#0%&t hetaG*&%\"kG\"\"\"%#PiGF'" }{TEXT -1 8 ", where " }{TEXT 273 1 "k" } {TEXT -1 18 " is an integer \}. " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 5 "range" }{TEXT -1 58 " of the cotangent function is the s et of all real numbers " }{TEXT 274 1 "R" }{TEXT -1 2 ". " }}{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 35 "Special values and basic identities" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "S pecial values" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[theta, cos*theta, \+ sin*theta, tan*theta, sec*theta, csc*theta, cot*theta], [0, 1, 0, 0, 1 , `not defined`, `not defined`], [Pi/6, sqrt(3)/2, 1/2, sqrt(3)/3, 2*s qrt(3)/3, 2, sqrt(3)], [Pi/4, sqrt(2)/2, sqrt(2)/2, 1, sqrt(2), sqrt(2 ), 1], [Pi/3, 1/2, sqrt(3)/2, sqrt(3), 2, 2*sqrt(3)/3, sqrt(3)/3], [Pi /2, 0, 1, `not defined`, `not defined`, 1, 0], [2*Pi/3, -1/2, sqrt(3)/ 2, -sqrt(3), -2, 2*sqrt(3)/3, -sqrt(3)/3], [3*Pi/4, -sqrt(2)/2, sqrt(2 )/2, -1, -sqrt(2), sqrt(2), -1], [5*Pi/6, -sqrt(3)/2, 1/2, -sqrt(3)/3, -2*sqrt(3)/3, 2, -sqrt(3)], [Pi, -1, 0, 0, -1, `not defined`, `not de fined`], [7*Pi/6, -sqrt(3)/2, -1/2, sqrt(3)/3, -2*sqrt(3)/3, -2, sqrt( 3)], [5*Pi/4, -sqrt(2)/2, -sqrt(2)/2, 1, -sqrt(2), -sqrt(2), 1], [4*Pi /3, -1/2, -sqrt(3)/2, sqrt(3), -2, -2*sqrt(3)/3, sqrt(3)/3], [3*Pi/2, \+ 0, -1, `not defined`, `not defined`, -1, 0], [5*Pi/3, 1/2, -sqrt(3)/2, -sqrt(3), 2, -2*sqrt(3)/3, -sqrt(3)/3], [7*Pi/4, sqrt(2)/2, -sqrt(2)/ 2, -1, sqrt(2), -sqrt(2), -1], [11*Pi/6, sqrt(3)/2, -1/2, -sqrt(3)/3, \+ 2*sqrt(3)/3, -2, -sqrt(3)], [2*Pi, 1, 0, 0, 1, `not defined`, `not def ined`], [theta, cos*theta, sin*theta, tan*theta, sec*theta, csc*theta, cot*theta]]);" "6#-%'matrixG6#757)%&thetaG*&%$cosG\"\"\"F(F+*&%$sinGF +F(F+*&%$tanGF+F(F+*&%$secGF+F(F+*&%$cscGF+F(F+*&%$cotGF+F(F+7)\"\"!F+ F7F7F+%,not~definedGF87)*&%#PiGF+\"\"'!\"\"*&-%%sqrtG6#\"\"$F+\"\"#F=* &F+F+FCF=*&-F@6#FBF+FBF=*(FCF+-F@6#FBF+FBF=FC-F@6#FB7)*&F;F+\"\"%F=*&- F@6#FCF+FCF=*&-F@6#FCF+FCF=F+-F@6#FC-F@6#FCF+7)*&F;F+FBF=*&F+F+FCF=*&- F@6#FBF+FCF=-F@6#FBFC*(FCF+-F@6#FBF+FBF=*&-F@6#FBF+FBF=7)*&F;F+FCF=F7F +F8F8F+F77)*(FCF+F;F+FBF=,$*&F+F+FCF=F=*&-F@6#FBF+FCF=,$-F@6#FBF=,$FCF =*(FCF+-F@6#FBF+FBF=,$*&-F@6#FBF+FBF=F=7)*(FBF+F;F+FOF=,$*&-F@6#FCF+FC F=F=*&-F@6#FCF+FCF=,$F+F=,$-F@6#FCF=-F@6#FC,$F+F=7)*(\"\"&F+F;F+F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Basic identities" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 95 "The circle o f radius 1 unit with its centre at the origin in the coordinate plane \+ has equation " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$% \"yGF'F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "For any rea l number " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 11 ", the poi nt" }{XPPEDIT 18 0 " ``(cos*theta, sin*theta)" "6#-%!G6$*&%$cosG\"\"\" %&thetaGF(*&%$sinGF(F)F(" }{TEXT -1 21 " lies on this circle." }} {PARA 0 "" 0 "" {TEXT -1 27 "Hence we have the identity:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos^2*theta+sin^2*theta = 1;" " 6#/,&*&%$cosG\"\"#%&thetaG\"\"\"F)*&%$sinGF'F(F)F)F)" }{TEXT -1 14 " \+ ------- (i)." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 289 10 "______ ____" }{TEXT -1 18 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "Dividing both sides of equation (i) by " }{XPPEDIT 18 0 "cos^2*theta; " "6#*&%$cosG\"\"#%&thetaG\"\"\"" }{TEXT -1 7 " gives:" }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+tan^2*theta = sec^2*theta;" "6 #/,&\"\"\"F%*&%$tanG\"\"#%&thetaGF%F%*&%$secGF(F)F%" }{TEXT -1 16 " - ------ (ii). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 290 10 "_____ _____" }{TEXT -1 18 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "Dividing both sides of equation (i) by " }{XPPEDIT 18 0 "sin^2*the ta;" "6#*&%$sinG\"\"#%&thetaG\"\"\"" }{TEXT -1 7 " gives:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cot^2*theta+1 = csc^2*theta ;" "6#/,&*&%$cotG\"\"#%&thetaG\"\"\"F)F)F)*&%$cscGF'F(F)" }{TEXT -1 16 " ------- (iii)." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 291 10 "__________" }{TEXT -1 18 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{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 282 8 "Question" } {TEXT -1 13 ": Given that " }{XPPEDIT 18 0 "sin*theta=-4/5" "6#/*&%$si nG\"\"\"%&thetaGF&,$*&\"\"%F&\"\"&!\"\"F," }{TEXT -1 4 " and" } {XPPEDIT 18 0 "``(cos*theta,sin*theta);" "6#-%!G6$*&%$cosG\"\"\"%&thet aGF(*&%$sinGF(F)F(" }{TEXT -1 25 " is in quadrant IV, find " } {XPPEDIT 18 0 "cos*theta" "6#*&%$cosG\"\"\"%&thetaGF%" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "tan*theta" "6#*&%$tanG\"\"\"%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sec*theta" "6#*&%$secG\"\"\"%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "csc*theta" "6#*&%$cscG\"\"\"%&thetaGF%" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "cot*theta" "6#*&%$cotG\"\"\"%&theta GF%" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 335 "tt := arcsin(-4/5): \np1 := plot([[cos(t),sin(t),t=0..tt],[cos(t),sin(t),t=tt..2*Pi]],\n c olor=[blue,red],thickness=2):\np2 := plot([[0,0],[cos(tt),sin(tt)],[co s(tt),0]],color=black):\np3 := plot([[[cos(tt),sin(tt)]]$3],color=blac k,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$$\"3M)*G,++++g!#= $!3%fK!**********zF*7$$\"3=y>[$ei/;'F*$!3kO#e\"o(*4xyF*7$$\"3/?%HT&\\. )H'F*$!3i%yu]-Xvw(F*7$$\"3S\\9j4.U]kF*$!3Ghh^?4ZTwF*7$$\"3d)=N\\Nn7g'F *$!3&yCOaSU:^(F*7$$\"3CY7is'*z[nF*$!3q!yzRBv#ztF*7$$\"3s@A&)R_@$)oF*$! 3m9^\"[MdSD(F*7$$\"3rzW*HQY*>qF*$!3'Q3ywF*$!3r,%4.!y z1kF*7$$\"3so/<%H!>0yF*$!3'*f[I=-K^iF*7$$\"3;e'pn\"\\qCzF*$!3eh)oab-\" *4'F*7$$\"3j$pgILl1.)F*$!3gSN9:P*)efF*7$$\"3;)y9?jJM:)F*$!3Wtr*\\,!y*y &F*7$$\"3at,Pqqka#)F*$!3'R5VlLPXk&F*7$$\"3=eBwRw7p$)F*$!3I\">!z,IOtaF* 7$$\"3m!3/ur-vY)F*$!3Fp'G4!e!*>`F*7$$\"3Kurc\"z(>s&)F*$!3W\"eT,E2&\\^F *7$$\"3'*3z.**fqo')F*$!3!p'4a:@L&)\\F*7$$\"3s9Ht],0m()F*$!3zX6Iig57[F* 7$$\"3#QqvJ![R_))F*$!3O]/58`N^YF*7$$\"3I-$*)>1CA%*)F*$!3w4X#)3$HiZ%F*7 $$\"3L]X#Q4j=.*F*$!3'3y*f8v[#H%F*7$$\"3&['*Re0Ao5*F*$!3IQ,Hyj2JTF*7$$ \"3CF%HAFRX=*F*$!3=T!fw!)z_&RF*7$$\"3IRy9[\\Eh#*F*$!37JpV/48sPF*7$$\"3 xw/`'[#zK$*F*$!3#\\Q(o8c^\"f$F*7$$\"3aO,1#)ok)R*F*$!3f_*HX@qaT$F*7$$\" 3)pYp([e(yY*F*$!3W7$*4F3f=KF*7$$\"3AdEjiYcE&*F*$!3;KY>Xx[SIF*7$$\"3;#* zQR!Hbe*F*$!3[(Hb&=g9\\GF*7$$\"3MN/(e[Tcj*F*$!3[%=puZtZn#F*7$$\"3!>)ye (y(z'o*F*$!33(G_T#G8$[#F*7$$\"3K3wm6AXJ(*F*$!3k=jRq/#>I#F*7$$\"3==v5>` ]u(*F*$!3q$QO[FX;6#F*7$$\"3KJFJ78+8)*F*$!3ZVo#y,R[#>F*7$$\"3[.7$$\"3evC![')y/(**F*$!3g7.V&zB#ywF^y7$$\"3[GC#>--H) **F*$!3aq%=9rI_%eF^y7$$\"3Y<'*=pgV#***F*$!3g0aAC'>())QF^y7$$\"3[/rnB.( z***F*$!3g3Iv'G\"o9?F^y7$$\"\"\"\"\"!$FfzFfz-%'COLOURG6&%$RGBGFgzFgz$ \"*++++\"!\")-%*THICKNESSG6#\"\"#-F$6%7SF'7$$\"3I[(\\)>2AyrF*$!3]jH&*f fEipF*7$$\"3e$p*G\")zLg!)F*$!3`\\K)\\4&p=fF*7$$\"31[KN,F'=())F*$!3N?t[ bW69YF*7$$\"33V%z;qCsZ*F*$!31w\")pUy&4>$F*7$$\"3Y#[?ghfY&)*F*$!3a$\\'[ q+t)p\"F*7$$\"3QB1a.57'***F*$!3)oDYIEZ]y#F^y7$$\"3@#*=3x*>!G**F*$\"3I5 #>bYsw>\"F*7$$\"3TvlxZ)*fH'*F*$\"3?7I#fNVkp#F*7$$\"343*z\\6>$3\"*F*$\" 3QpJ()pXxFTF*7$$\"3'RN&3'[ho(F*7$$\"3,?-/`:eH^F*$\"3 u,-^1LZ[F*$\"3Y_u5#*H kBB\"GF*7$$!3_!eW%Rd]%3*F*$!37d))R-H#*zTF*7$$!3Oz-oshz*G)F*$!3CrC9h\") y#f&F*7$$!3_6L]rP=*Q(F*$!3\"RA(pz%\\zt'F*7$$!3eSv&fv/]D'F*$!3w.9C_yB-y F*7$$!3Ch-!osvd4&F*$!3,*pk?>ZUg)F*7$$!3S+ah;#\\Mr$F*$!3W*\\XX\"*\\\\G* F*7$$!3h'[)R#\\i;L#F*$!3/0+Bv)oVs*F*7$$!3[M)ynE$y`$)F^y$!3!)QEojg/l**F *7$$\"3)=7[,=FJY'F^y$!3'R**)eO@4z**F*7$$\"3#*>x$zge<=#F*$!3#RFs1q%4f(* F*7$$\"3.%f\"[l3P6OF*$!3E(H!Gqt7D$*F*7$$\"3ALHw\"yZ(*)\\F*$!3s8+6t];m' )F*7$$\"3X0IH>xwTiF*$!3+5[A![JG\"yF*7$$\"3qZ'3q?TlE(F*$!3i\"y$R`[.qoF* 7$$\"3!3'p>\"z(pm#)F*$!3>j " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "cos^2*th eta+sin^2*theta=1" "6#/,&*&%$cosG\"\"#%&thetaG\"\"\"F)*&%$sinGF'F(F)F) F)" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "cos^2*theta+(-4/5)^2=1" "6#/,&*&%$cosG\"\"#%&thetaG\"\"\"F)*$,$*&\"\"%F)\"\"&!\"\"F/F'F)F)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "c os^2*theta=1-16/25" "6#/*&%$cosG\"\"#%&thetaG\"\"\",&F(F(*&\"#;F(\"#D! \"\"F-" }{XPPEDIT 18 0 "``=9/25" "6#/%!G*&\"\"*\"\"\"\"#D!\"\"" } {TEXT -1 10 ", so that " }{XPPEDIT 18 0 "cos*theta=``" "6#/*&%$cosG\" \"\"%&thetaGF&%!G" }{TEXT 284 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3/ 5" "6#*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "th eta" "6#%&thetaG" }{TEXT -1 32 " is in quadrant IV, we see that " } {XPPEDIT 18 0 "cos*theta" "6#*&%$cosG\"\"\"%&thetaGF%" }{TEXT -1 21 " \+ is positive so that " }{XPPEDIT 18 0 "cos*theta=3/5" "6#/*&%$cosG\"\" \"%&thetaGF&*&\"\"$F&\"\"&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "tan*theta=sin*theta/(cos*theta)" "6 #/*&%$tanG\"\"\"%&thetaGF&*(%$sinGF&F'F&*&%$cosGF&F'F&!\"\"" } {XPPEDIT 18 0 "``=``(-4/5)/``(3/5)" "6#/%!G*&-F$6#,$*&\"\"%\"\"\"\"\"& !\"\"F-F+-F$6#*&\"\"$F+F,F-F-" }{XPPEDIT 18 0 "``=-4/3" "6#/%!G,$*&\" \"%\"\"\"\"\"$!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 " Also " }{XPPEDIT 18 0 "sec*theta=1/(cos*theta)" "6#/*&%$secG\"\"\"%&th etaGF&*&F&F&*&%$cosGF&F'F&!\"\"" }{XPPEDIT 18 0 "``=5/3" "6#/%!G*&\"\" &\"\"\"\"\"$!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "csc*theta=1/(sin*t heta)" "6#/*&%$cscG\"\"\"%&thetaGF&*&F&F&*&%$sinGF&F'F&!\"\"" } {XPPEDIT 18 0 "``=-5/4" "6#/%!G,$*&\"\"&\"\"\"\"\"%!\"\"F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cot*theta=1/(tan*theta)" "6#/*&%$cotG\"\"\"% &thetaGF&*&F&F&*&%$tanGF&F'F&!\"\"" }{XPPEDIT 18 0 "``=-3/4" "6#/%!G,$ *&\"\"$\"\"\"\"\"%!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 20 ": It turns \+ out that " }{XPPEDIT 18 0 "theta=arcsin(-4/5)" "6#/%&thetaG-%'arcsinG6 #,$*&\"\"%\"\"\"\"\"&!\"\"F-" }{TEXT -1 56 ", so we can check these re sults using Maple as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "theta := arcsin(-4/5):\ncos ('theta')=simplify(cos(theta));\ntan('theta')=simplify(tan(theta));\ns ec('theta')=simplify(sec(theta));\ncsc('theta')=simplify(csc(theta)); \ncot('theta')=simplify(cot(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%$cosG6#%&thetaG#\"\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% $tanG6#%&thetaG#!\"%\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$secG6 #%&thetaG#\"\"&\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cscG6#%&th etaG#!\"&\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cotG6#%&thetaG#! \"$\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 279 8 "Question" }{TEXT -1 13 ": Given that " }{XPPEDIT 18 0 "tan*theta = -3/4;" "6#/*&%$tanG\"\" \"%&thetaGF&,$*&\"\"$F&\"\"%!\"\"F," }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(cos*theta,sin*theta);" "6#-%!G6$*&%$cosG\"\"\"%&thetaGF(*&%$sinG F(F)F(" }{TEXT -1 25 " is in quadrant II, find " }{XPPEDIT 18 0 "cos*t heta" "6#*&%$cosG\"\"\"%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "si n*theta;" "6#*&%$sinG\"\"\"%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sec*theta" "6#*&%$secG\"\"\"%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "csc*theta" "6#*&%$cscG\"\"\"%&thetaGF%" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "cot*theta" "6#*&%$cotG\"\"\"%&thetaGF%" }{TEXT -1 3 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 8 "Solu tion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 338 "tt := arctan(-3/4)+Pi:\np1 := plot([[cos (t),sin(t),t=0..tt],[cos(t),sin(t),t=tt..2*Pi]],\n color=[blue,red],th ickness=2):\np2 := plot([[0,0],[cos(tt),sin(tt)],[cos(tt),0]],color=bl ack):\np3 := plot([[[cos(tt),sin(tt)]]$3],color=black,style=point,symb ol=[circle,cross,diamond]):\nplots[display]([p||(1..3)],scaling=constr ained,tickmarks=[0,0]);" }}{PARA 13 "" 1 "" {GLPLOT2D 228 186 186 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"37.M[\\*y^)**!#=$ \"3ng.w-$QCW&!#>7$$\"37gB>(**)>[**F/$\"3')zZX2<`;5F/7$$\"3!=Z'4M\\%*z) *F/$\"3h+Xxm4*[a\"F/7$$\"3e#G!e.\"=Hy*F/$\"3;tWig%F/$\"3]8zObV'f())F/7$$\"3f=#F/$\"3#G[,veb!f(*F/7$$\"33M S->,!Rn\"F/$\"3[Iv#ec2*e)*F/7$$\"3Dj:eH`NW6F/$\"37y&>Rw1V$**F/7$$\"3CM $Rb^,7B'F2$\"3sBe6#=n0)**F/7$$\"3%z7-$QYfr6F2$\"3=$e[%fOJ****F/7$$!3g6 !=V`&>\\WF2$\"3121]Eu4!***F/7$$!3gpzPMVC)[*F2$\"3m)HY3%[)[&**F/7$$!3q` hTXg9%[\"F/$\"3cE-%GF_#*))*F/7$$!3c([aG_Sb'>F/$\"3FwOc0*H\\!)*F/7$$!3Q 5%)\\J$>k[#F/$\"3E8mhBZ&fo*F/7$$!3#f9rl\"32qHF/$\"3#zf>Qw_([&*F/7$$!3y 9k![=#enMF/$\"3]D)o-]X&z$*F/7$$!30TE#[rUZ%RF/$\"3+o7>h)p!*=*F/7$$!3k@q \"G/3JV%F/$\"3mt\\(=mzO'*)F/7$$!3EsZ,TPN\"*[F/$\"3TViC*zy?s)F/7$$!3m!Q *3*3\"RY`F/$\"3[Ja#G$[!3X)F/7$$!3Kp\"p%z^%Gy&F/$\"3qV\\4\\;Ne\")F/7$$! 3-$*)Rpgc)phF/$\"3qZ-IDRupyF/7$$!3#42e2[xcf'F/$\"3jB)*>&>^k^(F/7$$!35& =S`$yYfpF/$\"3/\"zdcgL4=(F/7$$!30Ob`*R[&GtF/$\"3$\\aW'f-&Q!oF/7$$!3!QW dRe3Fm(F/$\"3z'=]\">o2s$)*******fF/-%'COLO URG6&%$RGBGF+F+$\"*++++\"!\")-%*THICKNESSG6#\"\"#-F$6%7SFcz7$$!3-'*QkW =Dn%)F/$\"3)GMy^30.K&F/7$$!3n_n]>B/F))F/$\"3%=fB=s*G*p%F/7$$!3+IZ&p/hs <*F/$\"3!e>'eAz8sRF/7$$!3g*\\Z8:?%p%*F/$\"3'>k3@\\WS@$F/7$$!3+r;;DS8)p *F/$\"3_yM0.C[QCF/7$$!3'f8LsXiN&)*F/$\"3Wn)Hqm#30qHkN*F27$$!37oN%e)>5****F/$\"3^aJI@(H,M\"F27$$!3sTQSzM!y(**F/ $!3iYY$3+7\"fmF27$$!3]%QopRR#*))*F/$!3=iO#F/7$$!3(ou#QSA1W&*F/$!3c@V9N!4^)HF/7$$!3/ky9jRen#*F/$!3'e^ PmAzlv$F/7$$!3%[]M%p(GA%*)F/$!3Aj\\o/*>iZ%F/7$$!3MB`NZ=R)f)F/$!3^(QY0e \\c5&F/7$$!34Ynq\"RH@8)F/$!3Uz72&\\i'>eF/7$$!3QWe6dys!p(F/$!3YpQk$37;R 'F/7$$!3)o*RGaa$p7(F/$!3jUdnO_v9qF/7$$!3/LS?Vpv&e'F/$!3_x@L\")[9DvF/7$ $!3I`f%y())=]fF/$!3<=!GY;'*\\W23`F/$!3IBQ#3_C\\Z)F/7$$!3 '*o9_R/)Rg%F/$!3mS\"oD*e7x))F/7$$!3@@JaQTiIRF/$!3QC^LPz6&>*F/7$$!3?g1Y 8i**zJF/$!3g41k*R34[*F/7$$!3a$[\"=(y]'yBF/$!3Bf:kc?)Hr*F/7$$!331VwY%*R n;F/$!3QD*p>&F/$!3HC%e!4J\\V&)F/7$$\"3CZ7988:KeF/ $!3:(y2$*RxJ7)F/7$$\"3IimR'*Q&)fkF/$!3:rd$Gt(\\LwF/7$$\"3DlIm&>$*R.(F/ $!3-9o#=0\\z5(F/7$$\"33WEUj*[$*e(F/$!3()f+:&4l;^'F/7$$\"31ui<0Ziw!)F/$ !3!f`'>84X'*eF/7$$\"3Y\"*[2&QkM_)F/$!39VN*fzv(H_F/7$$\"3'))33^r*Q7*)F/ $!3*[Z2<2S``%F/7$$\"3XWZ7cp/?#*F/$!3MduLUpyrQF/7$$\"3%zy/d'3\">^*F/$!3 [4^&RADg3$F/7$$\"3]OBMr&*Q;(*F/$!3Uk#**3:$pkBF/7$$\"3mO+MB8Au)*F/$!3uk :R,31\"e\"F/7$$\"3+`R;<-?m**F/$!3Au2x3^%\\@)F27$F($\"36YKhSr8/#)!#F-Fi z6&F[[lF\\[lF+F+F_[l-F$6$7%7$F+F+7$$!3U+++++++!)F/$\"3w**************f F/7$F][mF+-Fiz6&F[[lF*F*F*-F$6&7#F\\[m-%'SYMBOLG6#%'CIRCLEGFb[m-%&STYL EG6#%&POINTG-F$6&Ff[m-Fh[m6#%&CROSSGFb[mF[\\m-F$6&Ff[m-Fh[m6#%(DIAMOND GFb[mF[\\m-%*AXESTICKSG6$F*F*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG 6%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" "Cu rve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "S ince " }{XPPEDIT 18 0 "1+tan^2*theta = sec^2*theta;" "6#/,&\"\"\"F%*&% $tanG\"\"#%&thetaGF%F%*&%$secGF(F)F%" }{TEXT -1 10 ", we have " } {XPPEDIT 18 0 "1+(-3/4)^2 = sec^2*theta;" "6#/,&\"\"\"F%*$,$*&\"\"$F% \"\"%!\"\"F+\"\"#F%*&%$secGF,%&thetaGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "sec^2*theta = 1+9/16;" "6#/*& %$secG\"\"#%&thetaG\"\"\",&F(F(*&\"\"*F(\"#;!\"\"F(" }{XPPEDIT 18 0 "` ` = 25/16;" "6#/%!G*&\"#D\"\"\"\"#;!\"\"" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "sec*theta = ``;" "6#/*&%$secG\"\"\"%&thetaGF&%!G" } {TEXT 281 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "5/4;" "6#*&\"\"&\"\"\" \"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 32 " is in quadrant II, we see that " }{XPPEDIT 18 0 "sec*the ta;" "6#*&%$secG\"\"\"%&thetaGF%" }{TEXT -1 21 " is negative so that \+ " }{XPPEDIT 18 0 "sec*theta = -5/4;" "6#/*&%$secG\"\"\"%&thetaGF&,$*& \"\"&F&\"\"%!\"\"F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "The n " }{XPPEDIT 18 0 "cos*theta=1/(sec*theta)" "6#/*&%$cosG\"\"\"%&theta GF&*&F&F&*&%$secGF&F'F&!\"\"" }{XPPEDIT 18 0 "``=-4/5" "6#/%!G,$*&\"\" %\"\"\"\"\"&!\"\"F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin*theta=tan *theta*cos*theta" "6#/*&%$sinG\"\"\"%&thetaGF&**%$tanGF&F'F&%$cosGF&F' F&" }{XPPEDIT 18 0 "`` = ``(-3/4)*``(-4/5);" "6#/%!G*&-F$6#,$*&\"\"$\" \"\"\"\"%!\"\"F-F+-F$6#,$*&F,F+\"\"&F-F-F+" }{XPPEDIT 18 0 "`` = 3/5; " "6#/%!G*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Also " }{XPPEDIT 18 0 "csc*theta=1/(sin*theta)" "6#/*&%$cs cG\"\"\"%&thetaGF&*&F&F&*&%$sinGF&F'F&!\"\"" }{XPPEDIT 18 0 "`` = 5/3; " "6#/%!G*&\"\"&\"\"\"\"\"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " cot*theta=1/(tan*theta)" "6#/*&%$cotG\"\"\"%&thetaGF&*&F&F&*&%$tanGF&F 'F&!\"\"" }{XPPEDIT 18 0 "`` = -4/3;" "6#/%!G,$*&\"\"%\"\"\"\"\"$!\"\" F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 6 "Note 1" }{TEXT -1 131 ": Another (and perhaps simpler) \+ way to solve this problem is to enlarge the reference triangle so that its hypotenuse has length 5." }}{PARA 0 "" 0 "" {TEXT -1 29 "The shor ter sides then have \"" }{TEXT 259 14 "signed lengths" }{TEXT -1 142 " \" of 3 and -4. More precisely, the terminal side of the reference ang le meets a circle with its centre at the origin and radius 5 at the po int" }{XPPEDIT 18 0 "``(-4,3)" "6#-%!G6$,$\"\"%!\"\"\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 160 "By using an obvious generalisati on of the standard trigonometric ratios involving such \"signed length s\" the various trigonometric function values can be found. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 6 "Note 2" }{TEXT -1 20 ": It turns out that " }{XPPEDIT 18 0 "theta = arctan(-3/4)+Pi; " "6#/%&thetaG,&-%'arctanG6#,$*&\"\"$\"\"\"\"\"%!\"\"F.F,%#PiGF," } {TEXT -1 56 ", so we can check these results using Maple as follows. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "theta := arctan(-3/4)+Pi:\ncos('theta')=simplify(cos(theta)); \nsin('theta')=simplify(sin(theta));\nsec('theta')=simplify(sec(theta) );\ncsc('theta')=simplify(csc(theta));\ncot('theta')=simplify(cot(thet a));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%&thetaG#!\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%&thetaG#\"\"$\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$secG6#%&thetaG#!\"&\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cscG6#%&thetaG#\"\"&\"\"$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$cotG6#%&thetaG#!\"%\"\"$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 11 "Given that " }{XPPEDIT 18 0 "sin*the ta = 3/5;" "6#/*&%$sinG\"\"\"%&thetaGF&*&\"\"$F&\"\"&!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos*theta < 0;" "6#2*&%$cosG\"\"\"%&thetaGF& \"\"!" }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "cos*theta" "6#*&%$cosG\" \"\"%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tan*theta" "6#*&%$tan G\"\"\"%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sec*theta" "6#*&%$ secG\"\"\"%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "csc*theta" "6#* &%$cscG\"\"\"%&thetaGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cot*theta " "6#*&%$cotG\"\"\"%&thetaGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 8 ": Since " }{XPPEDIT 18 0 "0 " 0 "" {MPLTEXT 1 0 337 "tt := Pi-arcsin(3/5):\np1 := plot([[cos(t),sin(t),t= 0..tt],[cos(t),sin(t),t=tt..2*Pi]],\n color=[blue,red],thickness=2):\n p2 := plot([[0,0],[cos(tt),sin(tt)],[cos(tt),0]],color=black):\np3 := \+ plot([[[cos(tt),sin(tt)]]$3],color=black,style=point,symbol=[circle,cr oss,diamond]):\nplots[display]([p||(1..3)],scaling=constrained,tickmar ks=[0,0]);" }}{PARA 13 "" 1 "" {GLPLOT2D 261 196 196 {PLOTDATA 2 "6,-% 'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"37.M[\\*y^)**!#=$\"3ng.w-$QCW&!#>7 $$\"37gB>(**)>[**F/$\"3')zZX2<`;5F/7$$\"3!=Z'4M\\%*z)*F/$\"3h+Xxm4*[a \"F/7$$\"3e#G!e.\"=Hy*F/$\"3;tWig%F/$\"3]8z ObV'f())F/7$$\"3f=#F/$\"3#G[,veb!f(*F/7$$\"33MS->,!Rn\"F/$\"3[ Iv#ec2*e)*F/7$$\"3Dj:eH`NW6F/$\"37y&>Rw1V$**F/7$$\"3CM$Rb^,7B'F2$\"3sB e6#=n0)**F/7$$\"3%z7-$QYfr6F2$\"3=$e[%fOJ****F/7$$!3g6!=V`&>\\WF2$\"31 21]Eu4!***F/7$$!3gpzPMVC)[*F2$\"3m)HY3%[)[&**F/7$$!3q`hTXg9%[\"F/$\"3c E-%GF_#*))*F/7$$!3c([aG_Sb'>F/$\"3FwOc0*H\\!)*F/7$$!3Q5%)\\J$>k[#F/$\" 3E8mhBZ&fo*F/7$$!3#f9rl\"32qHF/$\"3#zf>Qw_([&*F/7$$!3y9k![=#enMF/$\"3] D)o-]X&z$*F/7$$!30TE#[rUZ%RF/$\"3+o7>h)p!*=*F/7$$!3k@q\"G/3JV%F/$\"3mt \\(=mzO'*)F/7$$!3EsZ,TPN\"*[F/$\"3TViC*zy?s)F/7$$!3m!Q*3*3\"RY`F/$\"3[ Ja#G$[!3X)F/7$$!3Kp\"p%z^%Gy&F/$\"3qV\\4\\;Ne\")F/7$$!3-$*)Rpgc)phF/$ \"3qZ-IDRupyF/7$$!3#42e2[xcf'F/$\"3jB)*>&>^k^(F/7$$!35&=S`$yYfpF/$\"3/ \"zdcgL4=(F/7$$!30Ob`*R[&GtF/$\"3$\\aW'f-&Q!oF/7$$!3!QWdRe3Fm(F/$\"3z' =]\">o2s$)*******fF/-%'COLOURG6&%$RGBGF+F+ $\"*++++\"!\")-%*THICKNESSG6#\"\"#-F$6%7SFcz7$$!3-'*QkW=Dn%)F/$\"3)GMy ^30.K&F/7$$!3n_n]>B/F))F/$\"3%=fB=s*G*p%F/7$$!3+IZ&p/hs<*F/$\"3!e>'eAz 8sRF/7$$!3g*\\Z8:?%p%*F/$\"3'>k3@\\WS@$F/7$$!3+r;;DS8)p*F/$\"3_yM0.C[Q CF/7$$!3'f8LsXiN&)*F/$\"3Wn)Hqm#30qHkN*F 27$$!37oN%e)>5****F/$\"3^aJI@(H,M\"F27$$!3sTQSzM!y(**F/$!3iYY$3+7\"fmF 27$$!3]%QopRR#*))*F/$!3=iO#F/7 $$!3(ou#QSA1W&*F/$!3c@V9N!4^)HF/7$$!3/ky9jRen#*F/$!3'e^PmAzlv$F/7$$!3% []M%p(GA%*)F/$!3Aj\\o/*>iZ%F/7$$!3MB`NZ=R)f)F/$!3^(QY0e\\c5&F/7$$!34Yn q\"RH@8)F/$!3Uz72&\\i'>eF/7$$!3QWe6dys!p(F/$!3YpQk$37;R'F/7$$!3)o*RGaa $p7(F/$!3jUdnO_v9qF/7$$!3/LS?Vpv&e'F/$!3_x@L\")[9DvF/7$$!3I`f%y())=]fF /$!3<=!GY;'*\\W23`F/$!3IBQ#3_C\\Z)F/7$$!3'*o9_R/)Rg%F/$! 3mS\"oD*e7x))F/7$$!3@@JaQTiIRF/$!3QC^LPz6&>*F/7$$!3?g1Y8i**zJF/$!3g41k *R34[*F/7$$!3a$[\"=(y]'yBF/$!3Bf:kc?)Hr*F/7$$!331VwY%*Rn;F/$!3QD*p>&F/$!3HC%e!4J\\V&)F/7$$\"3CZ7988:KeF/$!3:(y2$*RxJ7)F/ 7$$\"3IimR'*Q&)fkF/$!3:rd$Gt(\\LwF/7$$\"3DlIm&>$*R.(F/$!3-9o#=0\\z5(F/ 7$$\"33WEUj*[$*e(F/$!3()f+:&4l;^'F/7$$\"31ui<0Ziw!)F/$!3!f`'>84X'*eF/7 $$\"3Y\"*[2&QkM_)F/$!39VN*fzv(H_F/7$$\"3'))33^r*Q7*)F/$!3*[Z2<2S``%F/7 $$\"3XWZ7cp/?#*F/$!3MduLUpyrQF/7$$\"3%zy/d'3\">^*F/$!3[4^&RADg3$F/7$$ \"3]OBMr&*Q;(*F/$!3Uk#**3:$pkBF/7$$\"3mO+MB8Au)*F/$!3uk:R,31\"e\"F/7$$ \"3+`R;<-?m**F/$!3Au2x3^%\\@)F27$F($\"36YKhSr8/#)!#F-Fiz6&F[[lF\\[lF+F +F_[l-F$6$7%7$F+F+7$$!3U+++++++!)F/$\"3w**************fF/7$F][mF+-Fiz6 &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-%(SC ALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!6\"F`]m-%%FONTG6#%(DEFAULTG-%* AXESTICKSG6$F*F*-%%VIEWG6$Fe]mFe]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 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "cos^2*theta+sin^2*theta=1" "6#/,&*&%$cosG\"\"#%&thetaG \"\"\"F)*&%$sinGF'F(F)F)F)" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 " cos^2*theta+(3/5)^2 = 1;" "6#/,&*&%$cosG\"\"#%&thetaG\"\"\"F)*$*&\"\"$ F)\"\"&!\"\"F'F)F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Henc e " }{XPPEDIT 18 0 "cos^2*theta = 1-9/25;" "6#/*&%$cosG\"\"#%&thetaG\" \"\",&F(F(*&\"\"*F(\"#D!\"\"F-" }{XPPEDIT 18 0 "`` = 16/25;" "6#/%!G*& \"#;\"\"\"\"#D!\"\"" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "cos*the ta=``" "6#/*&%$cosG\"\"\"%&thetaGF&%!G" }{TEXT 285 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4/5;" "6#*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 32 " is in quadrant II, \+ we see that " }{XPPEDIT 18 0 "cos*theta" "6#*&%$cosG\"\"\"%&thetaGF%" }{TEXT -1 21 " is positive so that " }{XPPEDIT 18 0 "cos*theta = -4/5; " "6#/*&%$cosG\"\"\"%&thetaGF&,$*&\"\"%F&\"\"&!\"\"F," }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "tan*theta=sin*t heta/(cos*theta)" "6#/*&%$tanG\"\"\"%&thetaGF&*(%$sinGF&F'F&*&%$cosGF& F'F&!\"\"" }{XPPEDIT 18 0 "`` = ``(3/5)/``(4/5);" "6#/%!G*&-F$6#*&\"\" $\"\"\"\"\"&!\"\"F*-F$6#*&\"\"%F*F+F,F," }{XPPEDIT 18 0 "`` = -3/4;" " 6#/%!G,$*&\"\"$\"\"\"\"\"%!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "sec*theta=1/(cos*theta)" "6#/*&%$se cG\"\"\"%&thetaGF&*&F&F&*&%$cosGF&F'F&!\"\"" }{XPPEDIT 18 0 "`` = -5/4 ;" "6#/%!G,$*&\"\"&\"\"\"\"\"%!\"\"F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "csc*theta=1/(sin*theta)" "6#/*&%$cscG\"\"\"%&thetaGF&*&F&F&*&%$sinG F&F'F&!\"\"" }{XPPEDIT 18 0 "`` = 5/3;" "6#/%!G*&\"\"&\"\"\"\"\"$!\"\" " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cot*theta=1/(tan*theta)" "6#/*&% $cotG\"\"\"%&thetaGF&*&F&F&*&%$tanGF&F'F&!\"\"" }{XPPEDIT 18 0 "`` = - 4/3;" "6#/%!G,$*&\"\"%\"\"\"\"\"$!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Summary: " } {XPPEDIT 18 0 "cos*theta = -4/5;" "6#/*&%$cosG\"\"\"%&thetaGF&,$*&\"\" %F&\"\"&!\"\"F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tan*theta = -3/4;" " 6#/*&%$tanG\"\"\"%&thetaGF&,$*&\"\"$F&\"\"%!\"\"F," }{TEXT -1 2 ", " } {XPPEDIT 18 0 "sec*theta = -5/4;" "6#/*&%$secG\"\"\"%&thetaGF&,$*&\"\" &F&\"\"%!\"\"F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "csc*theta=5/3" "6#/* &%$cscG\"\"\"%&thetaGF&*&\"\"&F&\"\"$!\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "cot*theta = -4/3;" "6#/*&%$cotG\"\"\"%&thetaGF&,$*&\"\" %F&\"\"$!\"\"F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 20 ": It turns out that \+ " }{XPPEDIT 18 0 "theta = arcsin(3/5);" "6#/%&thetaG-%'arcsinG6#*&\"\" $\"\"\"\"\"&!\"\"" }{TEXT -1 56 ", so we can check these results using Maple as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "theta := Pi-arcsin(3/5):\ncos('theta')=sim plify(cos(theta));\ntan('theta')=simplify(tan(theta));\nsec('theta')=s implify(sec(theta));\ncsc('theta')=simplify(csc(theta));\ncot('theta') =simplify(cot(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%& thetaG#!\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$tanG6#%&thetaG #!\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$secG6#%&thetaG#!\"& \"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cscG6#%&thetaG#\"\"&\"\"$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cotG6#%&thetaG#!\"%\"\"$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 33 "_________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 11 "Given that " } {XPPEDIT 18 0 "cos*theta = -12/13;" "6#/*&%$cosG\"\"\"%&thetaGF&,$*&\" #7F&\"#8!\"\"F," }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin*theta < 0;" " 6#2*&%$sinG\"\"\"%&thetaGF&\"\"!" }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "cos*theta" "6#*&%$cosG\"\"\"%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tan*theta" "6#*&%$tanG\"\"\"%&thetaGF%" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "sec*theta" "6#*&%$secG\"\"\"%&thetaGF%" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "csc*theta" "6#*&%$cscG\"\"\"%&thetaGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cot*theta" "6#*&%$cotG\"\"\"%&thetaGF%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 8 ": Si nce " }{XPPEDIT 18 0 "cos*theta < 0;" "6#2*&%$cosG\"\"\"%&thetaGF&\"\" !" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin*theta < 0;" "6#2*&%$sinG\" \"\"%&thetaGF&\"\"!" }{TEXT -1 11 ", the point" }{XPPEDIT 18 0 " ``(co s*theta,sin*theta) " "6#-%!G6$*&%$cosG\"\"\"%&thetaGF(*&%$sinGF(F)F(" }{TEXT -1 21 " is in quadrant III. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "tt := arccos(12/13)+ Pi:\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):\np3 := plot([[[cos(tt),sin(tt)]]$3],color= black,style=point,symbol=[circle,cross,diamond]):\nplots[display]([p|| (1..3)],scaling=constrained,tickmarks=[0,0]);" }}{PARA 13 "" 1 "" {GLPLOT2D 261 196 196 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F* 7$$\"3'3:4Uo0.(**!#=$\"3i3mN`*y1q(!#>7$$\"3q'*>k%zzi*)*F/$\"3f1<$4nRlV \"F/7$$\"3ALtTZ8*)f(*F/$\"3%oGcU6#>y@F/7$$\"3AgX*HS[lc*F/$\"3^'ze')oTA \"HF/7$$\"3b;pT'=\"R>$*F/$\"3w&4$\\1y9EOF/7$$\"3L6lkU>pU!*F/$\"3yOu..z ipUF/7$$\"3#o1\\5H_%4()F/$\"3l9-:`5!Q\"\\F/7$$\"3X`$\\y8\\mJ)F/$\"3Y\" R=UKyFb&F/7$$\"3;(=UdlR&yyF/$\"3qDD-A5iehF/7$$\"3u,$H_rn>Q(F/$\"3Mi0'* yW&eu'F/7$$\"3Wv=H(3@$3pF/$\"3plHHD@:IsF/7$$\"3$Gj7]r*GPjF/$\"3Fom2L;b NxF/7$$\"3\"*fnee%Ros&F/$\"3m$pg$)[ix>)F/7$$\"3c8!=0$eu1^F/$\"3!)ow@43 u(f)F/7$$\"3_(yMFXe%>XF/$\"31FSagPX?*)F/7$$\"3!R#3Z[a#\\z$F/$\"3a')[V[ z%>D*F/7$$\"3!f2vgD9@;$F/$\"3'QispW()o[*F/7$$\"3/>.)z`x!3CF/$\"3Z[4+Q$ Gdq*F/7$$\"3Q'G7sw!4GF/$\"3' \\&zum]@4)*F/7$$!3%4Gs:3o@q#F/$\"3uBt#o*\\*zi*F/7$$!3Q,]kL<\"*[LF/$\"3 MQx*=8oDU*F/7$$!3txY+BM+ISF/$\"3YO(yTF)*>:*F/7$$!32;/0G]'3r%F/$\"3>qOX x,(3#))F/7$$!3'=QP[n]3N&F/$\"3ys'*GL=)zW)F/7$$!3b=Q\"3\"=;UfF/$\"3I2$* *>r`I/)F/7$$!3m3^EF)HGc'F/$\"3C$Hx&=%[^a(F/7$$!3wAEz\\M>&3(F/$\"3s%4z! **Q\"p0(F/7$$!3u19.K3)Gg(F/$\"3k`$Q_Xge\\'F/7$$!3N,LyNLzL!)F/$\"3)4H!Q jenafF/7$$!35*R]SML3Y)F/$\"3_z[Qgk]I`F/7$$!36LJO-r\\=))F/$\"3/*4@s!eI: ZF/7$$!3'e8)*RY'fW\"*F/$\"318o/)*4xYSF/7$$!3bw@68c^9%*F/$\"3u@Z\"es#[r LF/7$$!3,aK4ymsV'*F/$\"3s&R34euak#F/7$$!3ts(ecm`;\")*F/$\"3=%4+Hv'pJ>F /7$$!3Wr$3cJ<)G**F/$\"3]+;AzV/\">\"F/7$$!3w'\\\"3.W()*)**F/$\"3#3P\\\" RL**)\\%F27$$!3crCh2_F(***F/$!3q6XAYbFMBF27$$!33.&zcp[$[**F/$!3KnA,Gc1 :5F/7$$!3igT$f*o(G&)*F/$!3=yx*=vR!4$4g\"R%F/7$$!3'fgZ__CKu)F/$!3mL-zW\\X`[F/7$$!3v>3ErC /W%)F/$!3-D:2C^1d`F/7$$!3ci89qmh8\")F/$!3/+&4vwWa%eF/7$$!3cyqK#\\\\nv( F/$!3w6^o8hK6jF/7$$!3=,4=@Z*>S(F/$!3Da\\@[M(Qs'F/7$$!3Yp)HeK)e6qF/$!3i [6HK60IrF/7$$!3c*fjUlNWe'F/$!3u#pIZ-,j_(F/7$$!3\"\\Iy!oiSOhF/$!3G\"3. \"*[ae*yF/7$$!3gJ!f)3L(Rl&F/$!3aG%3.#o=[#)F/7$$!3,Fm\\_UF7_F/$!3\"[yp%F/$!3DUfONZ#y#))F/7$$!3g*o$4u%pY;%F/$!3GVR7v&3:4 *F/7$$!3F\"ov2\\:$*F/7$$!3k1_g/)os9$F/$!3a2VeZF#=\\*F/ 7$$!3aBF^>1/aDF/$!3g$3p>/W$o'*F/7$$!3abFs!*>vV?F/$!3++*[VGE*)y*F/7$$!3 taL5OA>V9F/$!3'Qv7y#=J&*)*F/7$$!37I?%yDS/2*F2$!3#*y$flfy(e**F/7$$!3w%p lcn7#eJF2$!3Et>9-;,&***F/7$$\"3i(H@rBLA[#F2$!3W\"y$Q%y=p***F/7$$\"3Q\" zzZc^(e$)F2$!3-zNa,W+l**F/7$$\"3S\\/&*\\;-t8F/$!3X#H/\")4#H0**F/7$$\"3 ;/`+?\\'y%>F/$!3_iKJqmX3)*F/7$$\"3u$f>kPIz`#F/$!3y#fUgX&es'*F/7$$\"3n` +&*>P3WIF/$!3%HyPuA;a_*F/7$$\"35Ii-37?\"e$F/$!3'*oz+8_vO$*F/7$$\"3*yIi FY()Q7%F/$!3)GKqHqz+6*F/7$$\"3X\")=])o45k%F/$!3;/=dVI#y&))F/7$$\"36W5J \"*\\uE^F/$!3%fG?W_Iee)F/7$$\"3)exu'[IRZcF/$!3Cib&fg$p_#)F/7$$\"3YZEfJ 0(o4'F/$!3^&4_ADAk#zF/7$$\"3wgw+\"**))fb'F/$!3%Hda!*=$4^vF/7$$\"3#*=o( Hx\\@&pF/$!3sC%p)[(=!)=(F/7$$\"3#H0]ewZAO(F/$!3]'f#R@3PnnF/7$$\"3jn*fK oS\\s(F/$!3!=MIYz>-N'F/7$$\"3;c56f,\")y!)F/$!3[MviOiX$*eF/7$$\"31#>Nd \"[r)R)F/$!3d8na:J&yU&F/7$$\"3?q?Jt:*[q)F/$!3N&)))pKf(=#\\F/7$$\"3-]-] b`+r*)F/$!3e(=/Ypk#=WF/7$$\"3**\\*QA%*F/$!3%**y=#y@'*\\LF/7$$\"3s/\"R&\\Jo'e*F/$!3E)4RFIg_%GF/7$$\"3H-I J0!f>u*F/$!3_t-!p9TqD#F/7$$\"3USVK;tI])*F/$!3-:#)\\/#)yBrW&>#)**F/$!3'>[%RO)zY'fF27$F($\"36YK hSr8/#)!#F-Fiz6&F[[lF\\[lF+F+F_[l-F$6$7%7$F+F+7$$!3GJ#p2Bp2B*F/$!3OYQ: YQ:YQF/7$F\\[mF+-Fiz6&F[[lF*F*F*-F$6&7#F[[m-%'SYMBOLG6#%'CIRCLEGFa[m-% &STYLEG6#%&POINTG-F$6&Fe[m-Fg[m6#%&CROSSGFa[mFj[m-F$6&Fe[m-Fg[m6#%(DIA MONDGFa[mFj[m-%+AXESLABELSG6%Q!6\"F[]m-%%FONTG6#%(DEFAULTG-%(SCALINGG6 #%,CONSTRAINEDG-%*AXESTICKSG6$F*F*-%%VIEWG6$F`]mF`]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 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "cos^2*theta+sin^2*theta=1" "6#/,&*&%$cosG\" \"#%&thetaG\"\"\"F)*&%$sinGF'F(F)F)F)" }{TEXT -1 10 ", we have " } {XPPEDIT 18 0 "(-12/13)^2+sin*theta^2 = 1;" "6#/,&*$,$*&\"#7\"\"\"\"#8 !\"\"F+\"\"#F)*&%$sinGF)*$%&thetaGF,F)F)F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "sin^2*theta = 1-144/169;" "6#/*&%$sinG\"\"#%&thetaG\"\"\",&F(F(*&\"$W\"F(\"$p\"!\"\"F-" } {XPPEDIT 18 0 "`` = 25/169;" "6#/%!G*&\"#D\"\"\"\"$p\"!\"\"" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "sin*theta = ``;" "6#/*&%$sinG\"\"\" %&thetaGF&%!G" }{TEXT 286 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "5/13; " "6#*&\"\"&\"\"\"\"#8!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "theta" " 6#%&thetaG" }{TEXT -1 17 " is in quadrant I" }{TEXT 287 0 "" }{TEXT -1 16 "II, we see that " }{XPPEDIT 18 0 "sin*theta;" "6#*&%$sinG\"\"\" %&thetaGF%" }{TEXT -1 21 " is positive so that " }{XPPEDIT 18 0 "sin*t heta = -5/13;" "6#/*&%$sinG\"\"\"%&thetaGF&,$*&\"\"&F&\"#8!\"\"F," } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "ta n*theta=sin*theta/(cos*theta)" "6#/*&%$tanG\"\"\"%&thetaGF&*(%$sinGF&F 'F&*&%$cosGF&F'F&!\"\"" }{XPPEDIT 18 0 "`` = ``(-5/13)/``(-12/13);" "6 #/%!G*&-F$6#,$*&\"\"&\"\"\"\"#8!\"\"F-F+-F$6#,$*&\"#7F+F,F-F-F-" } {XPPEDIT 18 0 "`` = 5/12;" "6#/%!G*&\"\"&\"\"\"\"#7!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "sec*theta=1/ (cos*theta)" "6#/*&%$secG\"\"\"%&thetaGF&*&F&F&*&%$cosGF&F'F&!\"\"" } {XPPEDIT 18 0 "`` = -13/12;" "6#/%!G,$*&\"#8\"\"\"\"#7!\"\"F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "csc*theta=1/(sin*theta)" "6#/*&%$cscG\"\"\"% &thetaGF&*&F&F&*&%$sinGF&F'F&!\"\"" }{XPPEDIT 18 0 "`` = -13/5;" "6#/% !G,$*&\"#8\"\"\"\"\"&!\"\"F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cot* theta=1/(tan*theta)" "6#/*&%$cotG\"\"\"%&thetaGF&*&F&F&*&%$tanGF&F'F&! \"\"" }{XPPEDIT 18 0 "`` = 12/5;" "6#/%!G*&\"#7\"\"\"\"\"&!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Summary: " }{XPPEDIT 18 0 "sin*theta = -5/13;" "6#/*&%$sin G\"\"\"%&thetaGF&,$*&\"\"&F&\"#8!\"\"F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tan*theta = 5/12;" "6#/*&%$tanG\"\"\"%&thetaGF&*&\"\"&F&\"#7!\" \"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sec*theta = -13/12;" "6#/*&%$secG \"\"\"%&thetaGF&,$*&\"#8F&\"#7!\"\"F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "csc*theta = -13/5;" "6#/*&%$cscG\"\"\"%&thetaGF&,$*&\"#8F&\"\"&!\" \"F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cot*theta = 12/5;" "6#/*&%$cotG \"\"\"%&thetaGF&*&\"#7F&\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 20 ": I t turns out that " }{XPPEDIT 18 0 "theta = arccos(12/13)+Pi;" "6#/%&th etaG,&-%'arccosG6#*&\"#7\"\"\"\"#8!\"\"F+%#PiGF+" }{TEXT -1 56 ", so w e can check these results using Maple as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "theta := ar ccos(12/13)+Pi:\nsin('theta')=simplify(sin(theta));\ntan('theta')=simp lify(tan(theta));\nsec('theta')=simplify(sec(theta));\ncsc('theta')=si mplify(csc(theta));\ncot('theta')=simplify(cot(theta));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$sinG6#%&thetaG#!\"&\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$tanG6#%&thetaG#\"\"&\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$secG6#%&thetaG#!#8\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cscG6#%&thetaG#!#8\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cotG6#%&thetaG#\"#7\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 33 "__ _______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 11 "Given that " }{XPPEDIT 18 0 "tan*theta = -4/3;" "6#/*& %$tanG\"\"\"%&thetaGF&,$*&\"\"%F&\"\"$!\"\"F," }{TEXT -1 5 " and " } {XPPEDIT 18 0 "sin*theta < 0;" "6#2*&%$sinG\"\"\"%&thetaGF&\"\"!" } {TEXT -1 7 ", find " }{XPPEDIT 18 0 "cos*theta" "6#*&%$cosG\"\"\"%&the taGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin*theta;" "6#*&%$sinG\"\"\"% &thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sec*theta" "6#*&%$secG\"\" \"%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "csc*theta" "6#*&%$cscG \"\"\"%&thetaGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cot*theta" "6#*& %$cotG\"\"\"%&thetaGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 8 ": Since " }{XPPEDIT 18 0 "tan*theta < 0;" "6#2*& %$tanG\"\"\"%&thetaGF&\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin*t heta < 0;" "6#2*&%$sinG\"\"\"%&thetaGF&\"\"!" }{TEXT -1 18 ", it follo ws that " }{XPPEDIT 18 0 "0< cos*theta" "6#2\"\"!*&%$cosG\"\"\"%&theta GF'" }{TEXT -1 13 " so the point" }{XPPEDIT 18 0 " ``(cos*theta,sin*th eta) " "6#-%!G6$*&%$cosG\"\"\"%&thetaGF(*&%$sinGF(F)F(" }{TEXT -1 20 " is in quadrant IV. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 335 "tt := arctan(-4/3):\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(tt)],[cos(tt),0]],color =black):\np3 := plot([[[cos(tt),sin(tt)]]$3],color=black,style=point,s ymbol=[circle,cross,diamond]):\nplots[display]([p||(1..3)],scaling=con strained,tickmarks=[0,0]);" }}{PARA 13 "" 1 "" {GLPLOT2D 261 196 196 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$\"3S)*G\"3++++'!#=$!3!fK!R********zF*7 $$\"3[=DD%ei/;'F*$!3q=cbn(*4xyF*7$$\"3!=^u[&\\.)H'F*$!3U`1ZC]anxF*7$$ \"3W\\\"[.J?/X'F*$!3yq6\"*>4ZTwF*7$$\"3pvNibtE,mF*$!3)GGJ[SU:^(F*7$$\" 3YVqF*$!30Y\"yGHA=7(F*7$$\"3K[\\bI;oerF*$!3I;E:6ZN#)pF*7$$\"3')fv`0] @%H(F*$!3O)[;7DX1%oF*7$$\"3j3T&[(4qIuF*$!33bP$)3f8#p'F*7$$\"3e#y%=pcT[ vF*$!3C&fA\"[;2flF*7$$\"3o2$**)=>3ywF*$!3&yhL(*z(z1kF*7$$\"37?di%H!>0y F*$!3)zWOx@?8D'F*7$$\"3;%4,s\"\\qCzF*$!3m'G3\\b-\"*4'F*7$$\"3#)=4ZL`mI !)F*$!3]*o!f9P*)efF*7$$\"33B.SK;V`\")F*$!3]XUX9+y*y&F*7$$\"3Gf_tqqka#) F*$!3g'>4gLPXk&F*7$$\"3;>U5Sw7p$)F*$!3+luE,IOtaF*7$$\"3o(*es`F*7$$\"3Oxv'=z(>s&)F*$!3M5:kfs]\\^F*7$$\"3s'\\=$**fqo')F*$! 3GoI0:@L&)\\F*7$$\"3wSM*4:]gw)F*$!3Kal#=11@\"[F*7$$\"39u$=M![R_))F*$!3 C/'QEJb8l%F*7$$\"3@ML@iSAU*)F*$!3?dpP3$HiZ%F*7$$\"3rA*HS4j=.*F*$!3'[ln J^([#H%F*7$$\"3J&oHg0Ao5*F*$!3'R!>(yPw58%F*7$$\"3EwGSs#RX=*F*$!3gviD2) z_&RF*7$$\"35!=0$[\\Eh#*F*$!3+M10/48sPF*7$$\"3jfFn'[#zK$*F*$!37bwJ8c^ \"f$F*7$$\"3]?&)=#)ok)R*F*$!3>lm<9-Z:MF*7$$\"3A3K))[e(yY*F*$!3uJZwE3f= KF*7$$\"3(R&RtiYcE&*F*$!3FVs([u([SIF*7$$\"3QjnZR!Hbe*F*$!3QTmD=g9\\GF* 7$$\"3`R&[f[Tcj*F*$!3a>y=xMxuEF*7$$\"3Ez]l(y(z'o*F*$!3G\\,*Q#G8$[#F*7$ $\"3()o_s6AXJ(*F*$!3=fD:q/#>I#F*7$$\"3Upf:>`]u(*F*$!305@hu_k6@F*7$$\"3 /QHN78+8)*F*$!3$*p=iF*7$$\"3Y'3uK!zZ\\)*F*$!3n97l`5_Gm@o*3))*F*$!3fb'GQ3\"zQ:F*7$$\"3Q^=sWwC4**F*$!3R+ZoUnL-m]6F*7$$\"3q+7U*4Xs*!#>7$$\"3]S)3[' )y/(**F*$!3CARg%zB#ywF^y7$$\"3U9h#>--H)**F*$!39J))y52BXeF^y7$$\"3Q[7>p gV#***F*$!3Cei!Qi>())QF^y7$$\"3+VvnB.(z***F*$!3#4xNlG\"o9?F^y7$$\"\"\" \"\"!$FfzFfz-%'COLOURG6&%$RGBGFgzFgz$\"*++++\"!\")-%*THICKNESSG6#\"\"# -F$6%7SF'7$$\"33*zI0s?#yrF*$!3S(y]#ffEipF*7$$\"3APu&=)zLg!)F*$!3Mr+@%4 &p=fF*7$$\"3m4gy,F'=())F*$!3'4AbYX9Th%F*7$$\"3su=(>qCsZ*F*$!3#pkH=%y&4 >$F*7$$\"3sHD<;'fY&)*F*$!3!>V/'p+t)p\"F*7$$\"3W**\\c.57'***F*$!3akqHas /&y#F^y7$$\"3)=_zp(*>!G**F*$\"3-.yOmCn(>\"F*7$$\"3`==bZ)*fH'*F*$\"3epc scLW'p#F*7$$\"36uXk9\">$3\"*F*$\"3:9JhqXxFTF*7$$\"3c>0l&[ho(F*7$$\"3=\"3:Cb\"eH^F*$\"3#*ys#*Hh8%e)F*7$$\"3ie6.J\"oYy$F*$\"3 %4ooDh[hD*F*7$$\"3+Rp)*HWR)[#F*$\"3OV=4%\\Zao*F*7$$\"3qEEu%*Ra')))F^y$ \"3]h$3SSO/'**F*7$$!3mb&=k3$fl[F^y$\"3S*fFz)f:))**F*7$$!3_debt6!H2#F*$ \"3A'pV'Q^z#y*F*7$$!3AEr3a.5MMF*$\"3Y<&Gvwb=R*F*7$$!3em.`^1LZ[F*$\"3Ww *3%z8iY()F*7$$!3AX&)=6O 8F*7$$!3%)\\z)y,(R'f*F*$!31tBPVOK7GF*7$$!3UeuGRd]%3*F*$!3'*Q.u-H#*zTF* 7$$!3!>k#[shz*G)F*$!3!eTN9;)y#f&F*7$$!3?K)y7x$=*Q(F*$!3@*RV*z%\\zt'F*7 $$!3e]WrbZ+biF*$!3w'GOC&yB-yF*7$$!3A7!\\lsvd4&F*$!3())\\8A>ZUg)F*7$$!3 FSTO;#\\Mr$F*$!31*)fk9*\\\\G*F*7$$!38)*[:#\\i;L#F*$!3w7%)Gv)oVs*F*7$$! 3;BC\\kKy`$)F^y$!3!o!=qjg/l**F*7$$\"3asDB#=FJY'F^y$!3w%\\vl8#4z**F*7$$ \"3%*e073'e<=#F*$!3s(RJ1q%4f(*F*7$$\"3vapjl3P6OF*$!3$)H,Aqt7D$*F*7$$\" 3?N*))=yZ(*)\\F*$!33lu.t];m')F*7$$\"3sn,R>xwTiF*$!3V&=Z,[JG\"yF*7$$\"3 u23327amsF*$!3+cuJ`[.qoF*7$$\"31*fV7z(pm#)F*$!3mWKL*ptoi&F*7$$\"3-x.9) *Rg$)*)F*$!37#4'o*R\"f#R%F*7$$\"3S4X:My2Y&*F*$!3ML#>t2d'yHF*7$$\"3;n[- \"fDv()*F*$!3g,Q3`YGg:F*7$Fdz$\"36YKhSr8/#)!#F-Fiz6&F[[lF\\[lFgzFgzF_[ l-F$6$7%7$FgzFgz7$$\"3w**************fF*$!3U+++++++!)F*7$F\\[mFgz-Fiz6 &F[[lFfzFfzFfz-F$6&7#F[[m-%'SYMBOLG6#%'CIRCLEGFa[m-%&STYLEG6#%&POINTG- F$6&Fe[m-Fg[m6#%&CROSSGFa[mFj[m-F$6&Fe[m-Fg[m6#%(DIAMONDGFa[mFj[m-%+AX ESLABELSG6%Q!6\"F[]m-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%* AXESTICKSG6$FfzFfz-%%VIEWG6$F`]mF`]m" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "1+tan^2*theta = sec^2*theta; " "6#/,&\"\"\"F%*&%$tanG\"\"#%&thetaGF%F%*&%$secGF(F)F%" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "1+(-4/3)^2 = sec^2*theta;" "6#/,&\"\"\"F% *$,$*&\"\"%F%\"\"$!\"\"F+\"\"#F%*&%$secGF,%&thetaGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "sec^2*theta = 1+ 16/9;" "6#/*&%$secG\"\"#%&thetaG\"\"\",&F(F(*&\"#;F(\"\"*!\"\"F(" } {XPPEDIT 18 0 "`` = 25/9;" "6#/%!G*&\"#D\"\"\"\"\"*!\"\"" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "sec*theta = ``;" "6#/*&%$secG\"\"\"%&t hetaGF&%!G" }{TEXT 288 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "5/3;" "6# *&\"\"&\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "theta" "6#% &thetaG" }{TEXT -1 32 " is in quadrant IV, we see that " }{XPPEDIT 18 0 "sec*theta;" "6#*&%$secG\"\"\"%&thetaGF%" }{TEXT -1 21 " is positive so that " }{XPPEDIT 18 0 "sec*theta = 5/3;" "6#/*&%$secG\"\"\"%&theta GF&*&\"\"&F&\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 " Then " }{XPPEDIT 18 0 "cos*theta=1/(sec*theta)" "6#/*&%$cosG\"\"\"%&th etaGF&*&F&F&*&%$secGF&F'F&!\"\"" }{XPPEDIT 18 0 "`` = 3/5;" "6#/%!G*& \"\"$\"\"\"\"\"&!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin*theta=t an*theta*cos*theta" "6#/*&%$sinG\"\"\"%&thetaGF&**%$tanGF&F'F&%$cosGF& F'F&" }{XPPEDIT 18 0 "`` = ``(-4/3)*``(3/5);" "6#/%!G*&-F$6#,$*&\"\"% \"\"\"\"\"$!\"\"F-F+-F$6#*&F,F+\"\"&F-F+" }{XPPEDIT 18 0 "`` = -4/5;" "6#/%!G,$*&\"\"%\"\"\"\"\"&!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 5 "Also " }{XPPEDIT 18 0 "csc*theta=1/(sin*theta)" "6#/*&%$ cscG\"\"\"%&thetaGF&*&F&F&*&%$sinGF&F'F&!\"\"" }{XPPEDIT 18 0 "`` = -5 /4;" "6#/%!G,$*&\"\"&\"\"\"\"\"%!\"\"F*" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "cot*theta=1/(tan*theta)" "6#/*&%$cotG\"\"\"%&thetaGF&*& F&F&*&%$tanGF&F'F&!\"\"" }{XPPEDIT 18 0 "`` = -3/4;" "6#/%!G,$*&\"\"$ \"\"\"\"\"%!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 9 "Summary: " }{XPPEDIT 18 0 "cos*theta = -3 /5;" "6#/*&%$cosG\"\"\"%&thetaGF&,$*&\"\"$F&\"\"&!\"\"F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin*theta = -4/5;" "6#/*&%$sinG\"\"\"%&thetaGF&,$ *&\"\"%F&\"\"&!\"\"F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sec*theta = 5/ 3;" "6#/*&%$secG\"\"\"%&thetaGF&*&\"\"&F&\"\"$!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "csc*theta = -5/4;" "6#/*&%$cscG\"\"\"%&thetaGF&,$*&\" \"&F&\"\"%!\"\"F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cot*theta = -3/4; " "6#/*&%$cotG\"\"\"%&thetaGF&,$*&\"\"$F&\"\"%!\"\"F," }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note " }{TEXT -1 20 ": It turns out that " }{XPPEDIT 18 0 "theta = arctan(- 4/3);" "6#/%&thetaG-%'arctanG6#,$*&\"\"%\"\"\"\"\"$!\"\"F-" }{TEXT -1 56 ", so we can check these results using Maple as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "t heta := arctan(-4/3):\nsin('theta')=simplify(sin(theta));\ncos('theta' )=simplify(cos(theta));\nsec('theta')=simplify(sec(theta));\ncsc('thet a')=simplify(csc(theta));\ncot('theta')=simplify(cot(theta));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%&thetaG#!\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%&thetaG#\"\"$\"\"&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$secG6#%&thetaG#\"\"&\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$cscG6#%&thetaG#!\"&\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cotG6#%&thetaG#!\"$\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 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 258 "" 0 "" {TEXT -1 38 "Code for graph of the secant function " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 460 "p1 := plo t(sec(t*Pi),t=-1.1..3.3,-5..5,discont=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,color=black):\nt1 := plots[textplot ]([3.3,-.3,`q`],font=[SYMBOL,11]):\nt2 := 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 258 "" 0 "" {TEXT -1 40 "Code for graph of the cos ecant function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 432 "p1 := plot(csc(t*Pi),t=-1.6..2.8,-5..5,discon t=true,color=red,thickness=2):\np2 := plot([[[-1,-5],[-1,5]],[[1,-5],[ 1,5]],[[2,-5],[2,5]]],linestyle=3,color=black):\nt1 := plots[textplot] ([2.8,-.3,`q`],font=[SYMBOL,11]):\nt2 := plots[textplot]([-.1,5,`y`],f ont=[HELVETICA,10]):\nplots[display]([p1,p2,t1,t2],xtickmarks=[-1.5=`- 3p/2`,-1=`-p`,-.5=`-p/2`,.5=`p/2`,1=`p`,\n 1.5=`3p/2`,2=`2p`,2.5=`5p /2`],\n font=[SYMBOL,11],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 41 "Code for graph of th e cotangent function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 432 "p1 := plot(cot(t*Pi),t=-1.6..2.8,-5..5,d iscont=true,color=red,thickness=2):\np2 := plot([[[-1,-5],[-1,5]],[[1, -5],[1,5]],[[2,-5],[2,5]]],linestyle=3,color=black):\nt1 := plots[text plot]([2.8,-.3,`q`],font=[SYMBOL,11]):\nt2 := plots[textplot]([-.1,5,` y`],font=[HELVETICA,10]):\nplots[display]([p1,p2,t1,t2],xtickmarks=[-1 .5=`-3p/2`,-1=`-p`,-.5=`-p/2`,.5=`p/2`,1=`p`,\n 1.5=`3p/2`,2=`2p`,2. 5=`5p/2`],\n font=[SYMBOL,11],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 34 "Code for additio n formula picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1989 "purple := COLOR(RGB,.5,0,.9):\ndarkred := \+ COLOR(RGB,.85,0,0):\ndarkgreen := COLOR(RGB,0,.85,0):\ndarkcyan := COL OR(RGB,0,.83,.83): \ncsA := evalf(cos(2*Pi/9)):\nsnA := evalf(sin(2*Pi /9)):\ncsB := evalf(cos(5*Pi/36)):\nsnB := evalf(sin(5*Pi/36)):\ncAB : = evalf(cos(13*Pi/36)):\nsAB := evalf(sin(13*Pi/36)):\ncAcB := csA*csB :\ncAsB := csA*snB:\np1 := plot([[[0,0],[cAB,sAB]],[[cAB,0],[cAB,cAsB] ]],\n color=black,linestyle=[1,3]):\np2 := plot([[cAcB,cAsB],[c AB,sAB]],color=red,thickness=2):\np3 := plot([[0,0],[cAcB,cAsB]],color =darkred,thickness=2):\np4 := plot([[cAcB,0],[cAcB,cAsB]],color=aquama rine,thickness=2):\np5 := plot([[cAB,cAsB],[cAcB,cAsB]],color=cyan,thi ckness=2):\np6 := plot([[cAB,cAsB],[cAB,sAB]],color=purple,thickness=2 ):\np7 := plot([[0,0],[cAcB,0]],color=magenta,thickness=2):\nt1 := plo ts[textplot]([[.06,.07,`a`],[.12,.03,`b`],\n [.45,.78,`b`],[.57,.3,` b`]],font=[SYMBOL,11],color=navy):\nt2 := plots[textplot]([.63,.6,`sin `],color=red):\nt3 := plots[textplot]([.665,.6,`a`],font=[SYMBOL,11],c olor=red):\nt4 := plots[textplot]([.3,.19,`cos`],color=darkred):\nt5 : = plots[textplot]([.335,.19,`a`],font=[SYMBOL,11],color=darkred):\nt6 \+ := plots[textplot]([.8,.17,`cos sin`],color=darkgreen):\nt7 := plots [textplot]([.835,.17,`a b`],font=[SYMBOL,11],color=darkgreen):\nt8 := plots[textplot]([.54,.37,`sin sin`],color=darkcyan):\nt9 := plot s[textplot]([.571,.37,`a b`],font=[SYMBOL,11],color=darkcyan):\nt1 0 := plots[textplot]([.32,.46,`sin cos`],color=purple):\nt11 := plot s[textplot]([.35,.46,`a b`],font=[SYMBOL,11],color=purple):\nt12 \+ := plots[textplot]([.36,-.03,`cos cos`],color=magenta):\nt13 := plot s[textplot]([.395,-.03,`a b`],font=[SYMBOL,11],color=magenta):\nt 14 := plots[textplot]([[.17,.45,`1`],[.4,.91,`P`],[.72,.33,`Q`],\n [ .72,.035,`R`],[-.02,.035,`O`],[.4,.035,`S`],\n [.4,.33,`T`],[.81,-.0 2,`x`],[-.02,.92,`y`]],color=black):\nplots[display]([p1,p2,p3,p4,p5,p 6,p7,t1,t2,t3,t4,t5,t6,t7,t8,t9,t10,t11,\n t12,t13,t14],scaling=cons trained,tickmarks=[0,0]);" }}}{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 }