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1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal " -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 37 "Double-angle and half-angle formu las " }{TEXT 265 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, N anaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 23.3.200 7" }}{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 22 "Double-angle formulas " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "alpha = beta;" "6#/%&alphaG%%betaG" } {XPPEDIT 18 0 "``=theta" "6#/%!G%&thetaG" }{TEXT -1 16 " in the formul a " }{XPPEDIT 18 0 "sin(alpha+beta) = sin*alpha*cos*beta+cos*alpha*sin *beta;" "6#/-%$sinG6#,&%&alphaG\"\"\"%%betaGF),&**F%F)F(F)%$cosGF)F*F) F)**F-F)F(F)F%F)F*F)F)" }{TEXT -1 8 " gives: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(t heta+theta) = sin*theta*cos*theta+cos*theta*sin*theta;" "6#/-%$sinG6#, &%&thetaG\"\"\"F(F),&**F%F)F(F)%$cosGF)F(F)F)**F,F)F(F)F%F)F(F)F)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*2*theta = 2*sin*theta*cos*thet a;" "6#/*(%$sinG\"\"\"\"\"#F&%&thetaGF&*,F'F&F%F&F(F&%$cosGF&F(F&" } {TEXT -1 14 " ------- (i). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {TEXT 266 11 "___________" }{TEXT -1 17 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "alpha = beta;" "6#/%&alphaG%%b etaG" }{XPPEDIT 18 0 "``=theta" "6#/%!G%&thetaG" }{TEXT -1 16 " in the formula " }{XPPEDIT 18 0 "cos(alpha+beta) = cos*alpha*cos*beta-sin*al pha*sin*beta;" "6#/-%$cosG6#,&%&alphaG\"\"\"%%betaGF),&**F%F)F(F)F%F)F *F)F)**%$sinGF)F(F)F.F)F*F)!\"\"" }{TEXT -1 8 " gives: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " cos(theta+theta) = cos*theta*cos*theta-sin*theta*sin*theta;" "6#/-%$co sG6#,&%&thetaG\"\"\"F(F),&**F%F)F(F)F%F)F(F)F)**%$sinGF)F(F)F-F)F(F)! \"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*2*theta = cos^2*theta-s in^2*theta;" "6#/*(%$cosG\"\"\"\"\"#F&%&thetaGF&,&*&F%F'F(F&F&*&%$sinG F'F(F&!\"\"" }{TEXT -1 15 " ------- (ii). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 267 13 "_____________" }{TEXT -1 17 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "Subs tituting " }{XPPEDIT 18 0 "sin^2*theta = 1-cos^2*theta;" "6#/*&%$sinG \"\"#%&thetaG\"\"\",&F(F(*&%$cosGF&F'F(!\"\"" }{TEXT -1 17 " in (ii) \+ gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*2*thet a = cos^2*theta-(1-cos^2*theta);" "6#/*(%$cosG\"\"\"\"\"#F&%&thetaGF&, &*&F%F'F(F&F&,&F&F&*&F%F'F(F&!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*2*theta = 2*cos^2*theta-1;" "6#/*(%$cosG\"\"\"\"\"#F&%&theta GF&,&*(F'F&*$F%F'F&F(F&F&F&!\"\"" }{TEXT -1 16 " ------- (iii). " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 268 12 "____________" }{TEXT -1 18 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "cos^2*theta = 1-sin ^2*theta;" "6#/*&%$cosG\"\"#%&thetaG\"\"\",&F(F(*&%$sinGF&F'F(!\"\"" } {TEXT -1 15 " in (ii) gives" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "cos*2*theta = 1-sin^2*theta-sin^2*theta;" "6#/*(%$cosG \"\"\"\"\"#F&%&thetaGF&,(F&F&*&%$sinGF'F(F&!\"\"*&F+F'F(F&F," }{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*2*theta = 1-2*sin^2*theta;" "6#/* (%$cosG\"\"\"\"\"#F&%&thetaGF&,&F&F&*(F'F&*$%$sinGF'F&F(F&!\"\"" } {TEXT -1 15 " ------- (iv). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {TEXT 269 12 "____________" }{TEXT -1 19 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Collectin g together the formula from (i), (ii), (iii) and (iv) we have the foll owing " }{TEXT 259 21 "double-angle formulas" }{TEXT -1 35 " for the s ine and cosine functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([sin*2*theta = 2* sin*theta*cos*theta, ``],[cos*2*theta = cos^2*theta-sin^2*theta, ``],[ `` = 2*cos^2*theta-1, ``],[`` = 1-2*sin^2*theta, ``]);" "6#-%*PIECEWIS EG6&7$/*(%$sinG\"\"\"\"\"#F*%&thetaGF**,F+F*F)F*F,F*%$cosGF*F,F*%!G7$/ *(F.F*F+F*F,F*,&*&F.F+F,F*F**&F)F+F,F*!\"\"F/7$/F/,&*(F+F**$F.F+F*F,F* F*F*F6F/7$/F/,&F*F**(F+F**$F)F+F*F,F*F6F/" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 262 13 "_____________" }{TEXT -1 10 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Since " }{XPPEDIT 18 0 "2*sin*x*cos*x = sin*2*x;" "6#/*,\"\"#\" \"\"%$sinGF&%\"xGF&%$cosGF&F(F&*(F'F&F%F&F(F&" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "cos^2*x-sin^2*x = cos*2*x;" "6#/,&*&%$cosG\"\"#%\"xG\" \"\"F)*&%$sinGF'F(F)!\"\"*(F&F)F'F)F(F)" }{TEXT -1 17 ", the graphs of " }{XPPEDIT 18 0 "y = 2*sin*x*cos*x;" "6#/%\"yG*,\"\"#\"\"\"%$sinGF' %\"xGF'%$cosGF'F)F'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y = cos^2*x -sin^2*x;" "6#/%\"yG,&*&%$cosG\"\"#%\"xG\"\"\"F**&%$sinGF(F)F*!\"\"" } {TEXT -1 31 " are \"sine curves\" with period " }{XPPEDIT 18 0 "Pi" "6 #%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot([2*sin(x)*cos(x),cos(x)^2-sin( x)^2],x=-Pi..2*Pi,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 515 193 193 {PLOTDATA 2 "6'-%'CURVESG6$7gt7$$!3*****4tk#fTJ!#<$\"3/UE[]'efD\"! #D7$$!3'yGjGJM-4$F*$\"3-fO!Gah`-\"!#=7$$!3?wlTyf()QIF*$\"3N`7%><9*R?F3 7$$!3_k)pRkJMX#)F37$$!37p-i%y$RcDF*$\"3$H#fHIq04#*F37$$!3]$ QaV*e!e]#F*$\"3AOrAk(ecb*F37$$!3)y\\)3/!=_X#F*$\"3:DdSUq_/)*F37$$!3&[b b*eS#*HCF*$\"3g3$z_(eZ\"*)*F37$$!3F7E#Q6IYS#F*$\"3Cpq`([;J&**F37$$!3op '*oohLzBF*$\"39B7>m6H*)**F37$$!3mEnbBA/aBF*$\"37N'4^N2*****F37$$!3dM(H h\\o)GBF*$\"3Q99x<&p])**F37$$!3#Hu-(oZp.BF*$\"3hd6G6i#\\%**F37$$!3G^dF T5_yAF*$\"3EZm0s\"z&z)*F37$$!3?f([QJZLD#F*$\"3CQ;66S>*y*F37$$!3YvZ**e) **H?#F*$\"3E<3+i%)GM&*F37$$!3u\"zSTS_E:#F*$\"3A**Hm')>z#=*F37$$!3)y%e* HL&Hf?F*$\"3;5APgX@)G)F37$$!3//4&=EQf'>F*$\"3?3fUW/`0rF37$$!3W9H8ACFp= F*$\"35e*p=*QC@cF37$$!3kC\\T#e1Ex\"F*$\"3TB6SW\\]FRF37$$!3+nmpKYjs;F*$ \"3Z,;tUIrA?F37$$!3Q4%yHoiEd\"F*$\"3ERK'*)p.Ft$!#?7$$!3?#yqXM6IZ\"F*$! 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([.^F&)o%F*$!3MkJ+mZh))**F37$Fgcm$!2W7>DY$******F*7$$\"3vlqT%=Zzt%F*$! 3w;Ja3%Qp)**F37$$\"3)pCB%fO$Hw%F*$!3!zuXd?[*[**F37$$\"3AG%HW8?zy%F*$!3 3?i:2x6'))*F37$F\\dm$!3W<)fky.')z*F37$$\"3!>(zWf&zG'[F*$!3o$RBGxg/b*F3 7$Fadm$!3i?E4[e*p?*F37$Ffdm$!3[WMWJc2e#)F37$F[em$!3)Qj2i+$p&)pF37$F`em $!3yfx1km)Qd&F37$Feem$!3+2')4R$Rw(RF37$Fjem$!37\"y4^:&)***HF37$F_fm$!3 m@jI%oq'*)>F37$Fifm$!3?f@yX.&pd*Fb[l7$Fcgm$\"3)G2;7\\'oq%)F[t7$Fghm$\" 3[bG(y')R'R>F37$Faim$\"3N#f %y#>`*F37$F_[n$\"3i^(pV:c5z*F37$$\"3^C^+kDR1iF*$\"3oxgG]\"*G#))*F37$Fd [n$\"35H-jyoiZ**F37$$\"3Uv$[xr(ediF*$\"3)))Q*GI\")*o)**F37$Fi[n$\"2m** *************F*-F^\\n6&F`\\nFd\\nFa\\nFd\\n-%+AXESLABELSG6$Q\"x6\"Q!6 \"-%*THICKNESSG6#\"\"#-%%VIEWG6$;$!+aEfTJ!\"*$\"+3`=$G'Ff`p%(DEFAULTG " 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "alpha = beta;" "6#/%&alphaG%%betaG" }{XPPEDIT 18 0 "``=theta" "6 #/%!G%&thetaG" }{TEXT -1 16 " in the formula " }{XPPEDIT 18 0 "tan(alp ha+beta) = (tan*alpha+tan*beta)/(1-tan*alpha*tan*beta);" "6#/-%$tanG6# ,&%&alphaG\"\"\"%%betaGF)*&,&*&F%F)F(F)F)*&F%F)F*F)F)F),&F)F)**F%F)F(F )F%F)F*F)!\"\"F1" }{TEXT -1 21 " gives the following " }{TEXT 259 20 " double-angle formula" }{TEXT -1 26 " for the tangent function." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "tan*2*theta = 2*tan*theta/(1-tan^2*theta);" "6#/*(%$tan G\"\"\"\"\"#F&%&thetaGF&**F'F&F%F&F(F&,&F&F&*&F%F'F(F&!\"\"F," }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 263 11 "___________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "2*tan*x/(1-tan^2*x) = tan*2*x;" "6 #/**\"\"#\"\"\"%$tanGF&%\"xGF&,&F&F&*&F'F%F(F&!\"\"F+*(F'F&F%F&F(F&" } {TEXT -1 16 ", the graph of " }{XPPEDIT 18 0 "y = 2*tan*x/(1-tan^2*x) ;" "6#/%\"yG**\"\"#\"\"\"%$tanGF'%\"xGF',&F'F'*&F(F&F)F'!\"\"F," } {TEXT -1 47 " has the same shape as the graph with equation " } {XPPEDIT 18 0 "y = tan*x;" "6#/%\"yG*&%$tanG\"\"\"%\"xGF'" }{TEXT -1 36 ", but is \"squashed\" parallel to the " }{TEXT 270 1 "x" }{TEXT -1 48 " axis so that the resulting function has period " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 20 "The asymptotes are: " }{XPPEDIT 18 0 "x=-Pi/4,x=Pi/4,x= 3*Pi/4,x=5*Pi/4" "6&/%\"xG,$*&%#PiG\"\"\"\"\"%!\"\"F*/F$*&F'F(F)F*/F$* (\"\"$F(F'F(F)F*/F$*(\"\"&F(F'F(F)F*" }{TEXT -1 5 " etc." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "p1 : = plot(2*tan(x)/(1-tan(x)^2),x=-Pi/2..3*Pi/2,y=-5..5,discont=true):\np 2 := plots[implicitplot](\{x=-Pi/4,x=Pi/4,x=3*Pi/4,x=5*Pi/4\},\n x=-P i..3*Pi,y=-5..5,linestyle=3,color=black):\nplots[display]([p1,p2]);" } }{PARA 13 "" 1 "" {GLPLOT2D 437 348 348 {PLOTDATA 2 "6)-%'CURVESG6)7gn 7$$!3-+++Cjzq:!#<$\"3/bBK&>$z*e&!#E7$$!3MO4Gzon`:F*$\"3BPNdu!G_U$!#>7$ 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*****z\"F*Fher7$F\\fr7$F\\br$!3g***********\\n\"F*7$7$F\\br$!3!******* *******R\"F*F`fr7$Fdfr7$F\\br$!3q***********\\F\"F*7$7$F\\br$!2y****** *********F*Fhfr7$F\\gr7$F\\br$!3c&***********\\()FC7$7$F\\br$!3m)***** ********fFCF`gr7$Fdgr7$F\\br$!3W'***********\\ZFC7$7$F\\br$!3W)******* ******>FCFhgr7$F\\hr7$F\\br$!3Ei***********\\(F37$7$F\\br$\"3w,++++++? FCF`hr7$Fdhr7$F\\br$\"3+/+++++]KFC7$7$F\\br$\"3)>+++++++'FCFhhr7$F\\ir 7$F\\br$\"3A/+++++]sFC7$7$F\\br$\"3A+++++++5F*F`ir7$Fdir7$F\\br$\"3X++ ++++D6F*7$7$F\\br$\"3M+++++++9F*Fhir7$F\\jr7$F\\br$\"3e++++++D:F*7$7$F \\br$\"3E+++++++=F*F`jr7$Fdjr7$F\\br$\"3\\++++++D>F*7$7$F\\br$\"3;++++ +++AF*Fhjr7$F\\[s7$F\\br$\"3=++++++DBF*7$7$F\\br$\"33+++++++EF*F`[s7$F d[s7$F\\br$\"34++++++DFF*7$7$F\\br$\"\"$FgarFh[s7$F\\\\s7$F\\br$\"3+++ ++++DJF*7$7$F\\br$\"3!**************R$F*F`\\s7$Fd\\s7$F\\br$\"3O++++++ DNF*7$7$F\\br$\"3#)*************z$F*Fh\\s7$F\\]s7$F\\br$\"3#)********* **\\#RF*7$7$F\\br$\"3;+++++++UF*F`]s7$Fd]s7$F\\br$\"32,+++++DVF*7$7$F \\br$\"3_+++++++YF*Fh]s7$F\\^s7$F\\br$\"3U,+++++DZF*7$7$F\\br$\"3)3+++ ++++&F*F`^s-F`ar6&FbarFgarFgarFgar-%*LINESTYLEG6#F^\\s-F$6V7$7$$\"3RTs )p\"3*p#RF*F^br7$F`_s$!3b.+++++DYF*7$7$F`_sFebrFb_s7$7$F`_s$!3_+++++++ YF*7$F`_s$!3?.+++++DUF*7$7$F`_sF]crF[`s7$F_`s7$F`_s$!3H.+++++DQF*7$7$F `_s$!3Q*************z$F*Fa`s7$Fe`s7$F`_s$!3#Q+++++]U$F*7$7$F`_sF]drFi` s7$7$F`_s$!3!**************R$F*7$F`_s$!3Y.+++++DIF*7$7$F`_s$!3c******* *******HF*Fbas7$Ffas7$F`_s$!3+/+++++DEF*7$7$F`_sF]erFjas7$F^bs7$F`_s$! 3k.+++++DAF*7$7$F`_sFeerF`bs7$Fdbs7$F`_s$!3&R+++++]#=F*7$7$F`_sF]frFfb s7$Fjbs7$F`_s$!3//+++++D9F*7$7$F`_sFefrF\\cs7$F`cs7$F`_s$!3\"R+++++]- \"F*7$7$F`_s$!2))***************F*Fbcs7$Ffcs7$F`_s$!3)*R+++++]iFC7$7$F `_sFegrFjcs7$7$F`_s$!3w**************fFC7$F`_s$!3uR+++++]AFC7$7$F`_sF] hrFcds7$Fgds7$F`_s$\"3[g**********\\!\\%>cBF*F^br7$F`js$!3K-+++++v\\F*7$7$F`jsFebrFb js7$Ffjs7$F`js$!3'>+++++]d%F*7$7$F`jsF]crFhjs7$F\\[t7$F`js$!3g,+++++vT F*7$7$F`jsFf`sF^[t7$Fb[t7$F`js$!39-+++++vPF*7$7$F`jsF]drFd[t7$Fh[t7$F` js$!3y,+++++vLF*7$7$F`jsFgasFj[t7$F^\\t7$F`js$!3K-+++++vHF*7$7$F`jsF]e rF`\\t7$Fd\\t7$F`js$!3S-+++++vDF*7$7$F`jsFeerFf\\t7$Fj\\t7$F`js$!3/-++ +++v@F*7$7$F`js$!3g*************z\"F*F\\]t7$7$F`jsF]fr7$F`js$!3O-+++++ v \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Hal f-angle formulas " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 7 "Making " }{XPPEDIT 18 0 "cos^2*theta;" "6# *&%$cosG\"\"#%&thetaG\"\"\"" }{TEXT -1 28 " the subject of the formula " }{XPPEDIT 18 0 "cos*2*theta = 1-2*cos^2*theta;" "6#/*(%$cosG\"\"\" \"\"#F&%&thetaGF&,&F&F&*(F'F&*$F%F'F&F(F&!\"\"" }{TEXT -1 10 ", we hav e " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "cos^2*theta = \+ (1+cos*2*theta)/2;" "6#/*&%$cosG\"\"#%&thetaG\"\"\"*&,&F(F(*(F%F(F&F(F 'F(F(F(F&!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta =`` " "6#/* &%$cosG\"\"\"%&thetaGF&%!G" }{TEXT 274 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt((1+cos*2*theta)/2);" "6#-%%sqrtG6#*&,&\"\"\"F(*(%$ cosGF(\"\"#F(%&thetaGF(F(F(F+!\"\"" }{TEXT -1 13 " ------- (i)." }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 264 14 "______________" } {TEXT -1 16 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Making " }{XPPEDIT 18 0 "sin^2*theta;" "6# *&%$sinG\"\"#%&thetaG\"\"\"" }{TEXT -1 28 " the subject of the formula " }{XPPEDIT 18 0 "cos*2*theta = 2*sin^2*theta-1;" "6#/*(%$cosG\"\"\" \"\"#F&%&thetaGF&,&*(F'F&*$%$sinGF'F&F(F&F&F&!\"\"" }{TEXT -1 10 ", we have " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "sin^2*thet a = (1-cos*2*theta)/2;" "6#/*&%$sinG\"\"#%&thetaG\"\"\"*&,&F(F(*(%$cos GF(F&F(F'F(!\"\"F(F&F-" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 " or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*theta=``" "6#/*&%$sinG\"\"\"%&thetaGF&%!G" }{TEXT 275 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt((1-cos*2*theta)/2);" "6#-%%sqrtG6#*&,&\"\"\"F(*(%$ cosGF(\"\"#F(%&thetaGF(!\"\"F(F+F-" }{TEXT -1 14 " ------- (ii)." }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 271 14 "______________" } {TEXT -1 16 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 40 "The formulas (i) and (ii) can be called \+ " }{TEXT 259 19 "half-angle formulas" }{TEXT -1 73 ", because the can \+ be used to express the sine and cosine of a half-angle " }{XPPEDIT 18 0 "theta/2;" "6#*&%&thetaG\"\"\"\"\"#!\"\"" }{TEXT -1 17 " in terms of the " }{XPPEDIT 18 0 "cos*theta;" "6#*&%$cosG\"\"\"%&thetaGF%" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Replacing " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "theta/2" "6#*&%&thetaG\"\"\"\"\"#!\"\"" }{TEXT -1 37 " in the formulas (i) and \+ (ii) gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(t heta/2) = ``" "6#/-%$cosG6#*&%&thetaG\"\"\"\"\"#!\"\"%!G" }{TEXT 278 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt((1+cos*theta)/2" "6#-%%sqrt G6#*&,&\"\"\"F(*&%$cosGF(%&thetaGF(F(F(\"\"#!\"\"" }{TEXT -1 16 " ---- --- (iii), " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(theta/2) =``" "6#/-%$sinG6#*&%&thet aG\"\"\"\"\"#!\"\"%!G" }{TEXT 276 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt((1-cos*theta)/2)" "6#-%%sqrtG6#*&,&\"\"\"F(*&%$cosGF(%&thetaGF (!\"\"F(\"\"#F," }{TEXT -1 15 " ------- (iv). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 277 14 "______________" }{TEXT -1 19 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(theta/2) = sin(theta/2)/cos(theta/2);" "6#/-%$tanG6#*&%&thetaG\"\"\"\"\"#!\"\"*& -%$sinG6#*&F(F)F*F+F)-%$cosG6#*&F(F)F*F+F+" }{XPPEDIT 18 0 "``=``" "6# /%!GF$" }{TEXT 279 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt((1-cos*t heta)/2)/sqrt((1+cos*theta)/2)" "6#*&-%%sqrtG6#*&,&\"\"\"F)*&%$cosGF)% &thetaGF)!\"\"F)\"\"#F-F)-F%6#*&,&F)F)*&F+F)F,F)F)F)F.F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 12 "which gives " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(theta/2)=``" "6#/-%$tanG6#*&%&the taG\"\"\"\"\"#!\"\"%!G" }{TEXT 280 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt((1-cos*theta)/(1+cos*theta))" "6#-%%sqrtG6#*&,&\"\"\"F(*&%$cos GF(%&thetaGF(!\"\"F(,&F(F(*&F*F(F+F(F(F," }{TEXT -1 14 " ------- (v). \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 281 14 "______________" } {TEXT -1 19 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 52 "There are two modifications of (v) which \+ are useful." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(th eta/2)=``" "6#/-%$tanG6#*&%&thetaG\"\"\"\"\"#!\"\"%!G" }{TEXT 282 1 "+ " }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt((1-cos*theta)/(1+cos*theta)) = \+ ``;" "6#/-%%sqrtG6#*&,&\"\"\"F)*&%$cosGF)%&thetaGF)!\"\"F),&F)F)*&F+F) F,F)F)F-%!G" }{TEXT -1 1 " " }{TEXT 284 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt((1-cos*theta)*(1+cos*theta)/((1+cos*theta)*(1+cos* theta))" "6#-%%sqrtG6#*(,&\"\"\"F(*&%$cosGF(%&thetaGF(!\"\"F(,&F(F(*&F *F(F+F(F(F(*&,&F(F(*&F*F(F+F(F(F(,&F(F(*&F*F(F+F(F(F(F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``" "6#/%!GF$" }{TEXT 283 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt((1-cos^2*theta)/(1+cos*theta)^2)" "6#-%%sqrtG6#*&, &\"\"\"F(*&%$cosG\"\"#%&thetaGF(!\"\"F(*$,&F(F(*&F*F(F,F(F(F+F-" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``" "6#/%!GF$" }{TEXT 285 1 "+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(sin^2*theta)/(1+cos*theta)^2)" "6# *&-%%sqrtG6#*&%$sinG\"\"#%&thetaG\"\"\"F+*$,&F+F+*&%$cosGF+F*F+F+F)!\" \"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``" "6#/%!GF$" }{TEXT 286 1 "+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*theta/(1+cos*theta)" "6#*(%$sinG\" \"\"%&thetaGF%,&F%F%*&%$cosGF%F&F%F%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Now consider the \+ functions " }{XPPEDIT 18 0 "f(theta)=tan(theta/2)" "6#/-%\"fG6#%&theta G-%$tanG6#*&F'\"\"\"\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g( theta)=sin*theta/(1+cos*theta)" "6#/-%\"gG6#%&thetaG*(%$sinG\"\"\"F'F* ,&F*F**&%$cosGF*F'F*F*!\"\"" }{TEXT -1 42 ". Because the tangent funct ion has period " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 15 ", the fun ction " }{XPPEDIT 18 0 "f(theta)=tan(theta/2)" "6#/-%\"fG6#%&thetaG-%$ tanG6#*&F'\"\"\"\"\"#!\"\"" }{TEXT -1 12 " has period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 8 ". Since " }{XPPEDIT 18 0 "sin*theta" "6#*&%$sinG\"\"\"%&thetaGF%" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "cos*theta" "6#*&%$cosG\"\"\"%&thetaGF%" }{TEXT -1 36 " \+ are periodic functions with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"# \"\"\"%#PiGF%" }{TEXT -1 8 ", so is " }{XPPEDIT 18 0 "g(theta)=sin*the ta/(1+cos*theta)" "6#/-%\"gG6#%&thetaG*(%$sinG\"\"\"F'F*,&F*F**&%$cosG F*F'F*F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "The dis cussion above shows that " }{XPPEDIT 18 0 "f(theta)" "6#-%\"fG6#%&thet aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(theta)" "6#-%\"gG6#%&thetaG " }{TEXT -1 86 " have the same magnitude. We now show that these two f unctions have the same sign for " }{XPPEDIT 18 0 "0 < theta;" "6#2\"\" !%&thetaG" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 43 "Because the two function s both have period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" } {TEXT -1 20 ", this implies that " }{XPPEDIT 18 0 "f(theta)=g(theta)" "6#/-%\"fG6#%&thetaG-%\"gG6#F'" }{TEXT -1 24 " for all real values of \+ " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 35 " where both functi ons are defined. " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "1+cos*theta" "6#,&\"\"\"F$*&%$cosGF$%&thetaGF$F$" }{TEXT -1 35 " is n ever negative, so the sign of " }{XPPEDIT 18 0 "g(theta) = sin*theta/( 1+cos*theta)" "6#/-%\"gG6#%&thetaG*(%$sinG\"\"\"F'F*,&F*F**&%$cosGF*F' F*F*!\"\"" }{TEXT -1 28 " is the same as the sign of " }{XPPEDIT 18 0 "sin*theta" "6#*&%$sinG\"\"\"%&thetaGF%" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "g(theta)" "6#-%\"gG6#%&thet aG" }{TEXT -1 17 " is positive for " }{XPPEDIT 18 0 "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 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT 288 8 "Question" }{TEXT -1 67 ": Use an ap propriate half-angle formula to find the exact value of " }{XPPEDIT 18 0 "sin(Pi/8);" "6#-%$sinG6#*&%#PiG\"\"\"\"\")!\"\"" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "cos(Pi/8)" "6#-%$cosG6#*&%#PiG\"\"\"\"\")!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 289 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 12 "The formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin( theta/2) = ``;" "6#/-%$sinG6#*&%&thetaG\"\"\"\"\"#!\"\"%!G" }{TEXT 290 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt((1-cos*theta)/2);" "6#- %%sqrtG6#*&,&\"\"\"F(*&%$cosGF(%&thetaGF(!\"\"F(\"\"#F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "can be used with " }{XPPEDIT 18 0 "t heta=Pi/4" "6#/%&thetaG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "sin(Pi/8)" "6#-%$s inG6#*&%#PiG\"\"\"\"\")!\"\"" }{TEXT -1 30 " is clearly positive, we h ave " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi/8)=sqr t((1-cos(Pi/4))/2)" "6#/-%$sinG6#*&%#PiG\"\"\"\"\")!\"\"-%%sqrtG6#*&,& F)F)-%$cosG6#*&F(F)\"\"%F+F+F)\"\"#F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =sqrt(1-sqrt(2)/2)/2) " "6#/%!G*&-%%sqrtG6#,&\"\"\"F**&-F'6#\"\"#F*F.! \"\"F/F*F.F/" }{XPPEDIT 18 0 "``=sqrt((2-sqrt(2))/4)" "6#/%!G-%%sqrtG6 #*&,&\"\"#\"\"\"-F&6#F*!\"\"F+\"\"%F." }{XPPEDIT 18 0 "``=sqrt(2-sqrt( 2))/2" "6#/%!G*&-%%sqrtG6#,&\"\"#\"\"\"-F'6#F*!\"\"F+F*F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "Similarly, since " }{XPPEDIT 18 0 "cos(Pi/8);" "6#-%$cosG6#*&%#PiG\"\"\"\"\")!\"\"" }{TEXT -1 24 " is \+ a positive, we have " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(Pi/8) = sqrt((1+cos(Pi/4))/2);" "6#/-%$cosG6#*&%#PiG\"\"\"\"\") !\"\"-%%sqrtG6#*&,&F)F)-F%6#*&F(F)\"\"%F+F)F)\"\"#F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = sqrt(1+sqrt(2)/2)/2;" "6#/%!G*&-%%sqrtG6#,&\"\"\"F **&-F'6#\"\"#F*F.!\"\"F*F*F.F/" }{XPPEDIT 18 0 "`` = sqrt((2+sqrt(2))/ 4);" "6#/%!G-%%sqrtG6#*&,&\"\"#\"\"\"-F&6#F*F+F+\"\"%!\"\"" }{XPPEDIT 18 0 "`` = sqrt(2+sqrt(2))/2;" "6#/%!G*&-%%sqrtG6#,&\"\"#\"\"\"-F'6#F* F+F+F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "sin(Pi/8) = sqrt(2- sqrt(2))/2" "6#/-%$sinG6#*&%#PiG\"\"\"\"\")!\"\"*&-%%sqrtG6#,&\"\"#F)- F.6#F1F+F)F1F+" }{TEXT -1 1 " " }{TEXT 291 1 "~" }{TEXT -1 18 " 0.3826 834324 and " }{XPPEDIT 18 0 "cos(Pi/8) = sqrt(2+sqrt(2))/2;" "6#/-%$co sG6#*&%#PiG\"\"\"\"\")!\"\"*&-%%sqrtG6#,&\"\"#F)-F.6#F1F)F)F1F+" } {TEXT -1 1 " " }{TEXT 292 1 "~" }{TEXT -1 15 " 0.9238795325. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sin(Pi/8);\n``=evalf(%);\n``;\n'sqrt((1-cos(Pi/4))/2)';\n``=eval(% );\nevalf(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#,$*&\" \")!\"\"%#PiG\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+DV$o# Q!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%sqrtG6#,&#\"\"\"\"\"#F(*&#F(F)F(-%$cosG6#,$*&\"\"%! \"\"%#PiGF(F(F(F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"#!\" \",&F'\"\"\"*$F'#F*F'F(F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+DV$ o#Q!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 " . . . and now we calculate " }{XPPEDIT 18 0 "cos(Pi/8)" "6#-%$cos G6#*&%#PiG\"\"\"\"\")!\"\"" }{TEXT -1 19 " in two ways . . . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "cos(Pi/8);\n``=evalf(%);\n``;\n'sqrt((1+cos(Pi/4))/2)';\n``=eval(% );\nevalf(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$cosG6#,$*&\" \")!\"\"%#PiG\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+D`zQ# *!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%sqrtG6#,&#\"\"\"\"\"#F(*&F'F(-%$cosG6#,$*&\"\"%!\"\" %#PiGF(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"#!\"\",&F '\"\"\"*$F'#F*F'F*F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+D`zQ#*!# 5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" } }{PARA 0 "" 0 "" {TEXT 272 8 "Question" }{TEXT -1 13 ": Given that " } {XPPEDIT 18 0 "sin*theta=4/5" "6#/*&%$sinG\"\"\"%&thetaGF&*&\"\"%F&\" \"&!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Pi/2<=theta" "6#1*&%#PiG \"\"\"\"\"#!\"\"%&thetaG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 170 "theta := Pi-arcsin(4/5):\n'cos(the ta)'=simplify(cos(theta));\n'sin(2*theta)'=expand(sin(2*theta));\n'cos (2*theta)'=expand(cos(2*theta));\n'tan(2*theta)'=expand(tan(2*theta)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%&thetaG#!\"$\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#,$*&\"\"#\"\"\"%&thetaGF*F*# !#C\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$*&\"\"#\"\"\"%& thetaGF*F*#!\"(\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$tanG6#,$*& \"\"#\"\"\"%&thetaGF*F*#\"#C\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT 295 8 "Question" } {TEXT -1 23 ": Verify the identity: " }{XPPEDIT 18 0 "(cos*theta+sin*t heta)^2=1+sin*2*theta" "6#/*$,&*&%$cosG\"\"\"%&thetaGF(F(*&%$sinGF(F)F (F(\"\"#,&F(F(*(F+F(F,F(F)F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 8 "Solution" }{TEXT -1 2 ": \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(cos*theta+sin*th eta)^2=cos^2*theta+2*sin*theta*cos*theta+sin^2*theta" "6#/*$,&*&%$cosG \"\"\"%&thetaGF(F(*&%$sinGF(F)F(F(\"\"#,(*&F'F,F)F(F(*,F,F(F+F(F)F(F'F (F)F(F(*&F+F,F)F(F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``=``(cos^2*theta+sin^2*theta)+``(2*sin*theta*cos*th eta)" "6#/%!G,&-F$6#,&*&%$cosG\"\"#%&thetaG\"\"\"F-*&%$sinGF+F,F-F-F-- F$6#*,F+F-F/F-F,F-F*F-F,F-F-" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1+sin*2*theta" "6#/%!G,&\"\"\"F&*(%$ sinGF&\"\"#F&%&thetaGF&F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "(using the identities " }{XPPEDIT 18 0 "cos^2*theta+sin^2*theta =1" "6#/,&*&%$cosG\"\"#%&thetaG\"\"\"F)*&%$sinGF'F(F)F)F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "2*sin*theta*cos*theta=sin*2*theta" "6#/*,\" \"#\"\"\"%$sinGF&%&thetaGF&%$cosGF&F(F&*(F'F&F%F&F(F&" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "(cos(theta)+sin(theta))^2;\n``=simplify(%);\n``=combi ne(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&-%$cosG6#%&thetaG \"\"\"-%$sinGF(F*\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(\" \"#\"\"\"-%$cosG6#%&thetaGF(-%$sinGF+F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%$sinG6#,$*&\"\"#\"\"\"%&thetaGF,F,F,F,F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The graph s of the left and right sides coincide. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 544 "pi := evalf(Pi):\nf \+ := theta->(cos(theta)+sin(theta))^2:'f(theta)'=f(theta);\ng := theta-> 1+sin(2*theta):'g(theta)'=g(theta);\np1:=plot([f(theta),g(theta)],thet a=-Pi/2..7.3,color=[red,green],thickness=[1,2],\n thickness=2,discon t=true):\nt1:=plots[textplot]([7.5,-.2,`q`],font=[SYMBOL,11]):\nplots[ display]([p1,t1],xtickmarks=[-pi/2=`-p/2`,-pi/4=`-p/4`,pi/4=`p/4`,\n \+ pi/2=`p/2`,3*pi/4=`3p/4`,pi=`p`,5*pi/4=`5p/4`,3*pi/2=`3p/2`,7*pi/4=`7 p/4`,2*pi=`2p`,\n 9*pi/4=`9p/4`],ytickmarks=3,font=[SYMBOL,11],\n \+ labels=[``,``],view=[-1.58..7.5,-.2..2.1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%&thetaG*$),&-%$cosGF&\"\"\"-%$sinGF&F-\"\"#F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%&thetaG,&-%$sinG6#,$*&\" \"#\"\"\"F'F/F/F/F/F/" }}{PARA 13 "" 1 "" {GLPLOT2D 603 186 186 {PLOTDATA 2 "6)-%'CURVESG6%7er7$$!3+++lBjzq:!#<$\"2k1-r$********F*7$$! 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Ffdn%%3p/4G/$\"+aEfTJFfdn%\"pG/$\"+=3*p#RFfdn%%5p/4G/$\"+\")*)Q7ZFfdn% %3p/2G/$\"+Wry(\\&Ffdn%%7p/4G/$\"+3`=$G'Ffdn%#2pG/$\"+sMeoqFfdn%%9p/4G \"\"$-%+AXESLABELSG6%%!GFegn-F[dn6#%(DEFAULTGFjcn-%%VIEWG6$;$!$e\"Fgcn Fccn;Ffcn$\"#@Fecn" 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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT 303 8 " Question" }{TEXT -1 23 ": Verify the identity: " }{XPPEDIT 18 0 "cos*2 *theta/(cos^2*theta) = 1-tan^2*theta;" "6#/**%$cosG\"\"\"\"\"#F&%&thet aGF&*&F%F'F(F&!\"\",&F&F&*&%$tanGF'F(F&F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 304 8 "Solution" }{TEXT -1 2 ": " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*2*the ta/(cos^2*theta) =(cos^2*theta-sin^2*theta)/(cos^2*theta)" "6#/**%$cos G\"\"\"\"\"#F&%&thetaGF&*&F%F'F(F&!\"\"*&,&*&F%F'F(F&F&*&%$sinGF'F(F&F *F&*&F%F'F(F&F*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1-tan^2*theta" "6#/%!G,&\"\"\"F&*&%$tanG\"\"#%&theta GF&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT 297 8 "Question" }{TEXT -1 23 ": Verify the identity: " } {XPPEDIT 18 0 "(cos*2*alpha+cos*2*beta)/(sin*alpha+cos*beta)=2*cos*bet a-2*sin*alpha)" "6#/*&,&*(%$cosG\"\"\"\"\"#F(%&alphaGF(F(*(F'F(F)F(%%b etaGF(F(F(,&*&%$sinGF(F*F(F(*&F'F(F,F(F(!\"\",&*(F)F(F'F(F,F(F(*(F)F(F /F(F*F(F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 8 "Solution" }{TEXT -1 2 ": " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(cos*2*alpha+cos*2*beta)/(sin*alpha+cos *beta)=(1-2*sin^2*alpha+2*cos^2*beta-1)/(sin*alpha+cos*beta)" "6#/*&,& *(%$cosG\"\"\"\"\"#F(%&alphaGF(F(*(F'F(F)F(%%betaGF(F(F(,&*&%$sinGF(F* F(F(*&F'F(F,F(F(!\"\"*&,*F(F(*(F)F(*$F/F)F(F*F(F1*(F)F(*$F'F)F(F,F(F(F (F1F(,&*&F/F(F*F(F(*&F'F(F,F(F(F1" }{TEXT -1 1 " " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(2*cos^2*beta-2*sin^2*alpha)/(sin* alpha+cos*beta)" "6#/%!G*&,&*(\"\"#\"\"\"*$%$cosGF(F)%%betaGF)F)*(F(F) *$%$sinGF(F)%&alphaGF)!\"\"F),&*&F/F)F0F)F)*&F+F)F,F)F)F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*(cos*beta+sin*alpha)*(cos*beta-sin*alpha)/(sin *alpha+cos*beta)" "6#/%!G**\"\"#\"\"\",&*&%$cosGF'%%betaGF'F'*&%$sinGF '%&alphaGF'F'F',&*&F*F'F+F'F'*&F-F'F.F'!\"\"F',&*&F-F'F.F'F'*&F*F'F+F' F'F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*(cos*beta-sin*alpha)" "6#/%!G *&\"\"#\"\"\",&*&%$cosGF'%%betaGF'F'*&%$sinGF'%&alphaGF'!\"\"F'" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*cos*beta-2*sin*alpha" "6#/%!G,&*( \"\"#\"\"\"%$cosGF(%%betaGF(F(*(F'F(%$sinGF(%&alphaGF(!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 0 "" 0 "" {TEXT 293 8 " Question" }{TEXT -1 23 ": Verify the identity: " }{XPPEDIT 18 0 "2*sin ^2*``(theta/2)=sin^2*theta/(1+cos*theta)" "6#/*(\"\"#\"\"\"*$%$sinGF%F &-%!G6#*&%&thetaGF&F%!\"\"F&*(F(F%F-F&,&F&F&*&%$cosGF&F-F&F&F." } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 36 "Squaring both sides of the identity:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(theta/2) =``" "6#/-%$sinG6#*&%&thetaG\"\"\"\" \"#!\"\"%!G" }{TEXT 299 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt((1- cos*theta)/2)" "6#-%%sqrtG6#*&,&\"\"\"F(*&%$cosGF(%&thetaGF(!\"\"F(\" \"#F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 10 "shows that" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin^2*``(theta/2)=(1- cos*theta)/2" "6#/*&%$sinG\"\"#-%!G6#*&%&thetaG\"\"\"F&!\"\"F,*&,&F,F, *&%$cosGF,F+F,F-F,F&F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "so that: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*sin^2 *``(theta/2) = (1-cos*theta)" "6#/*(\"\"#\"\"\"*$%$sinGF%F&-%!G6#*&%&t hetaGF&F%!\"\"F&,&F&F&*&%$cosGF&F-F&F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "On the other hand," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin^2*theta/(1+cos*theta)=(1-cos^2*theta)/(1+cos*th eta)" "6#/*(%$sinG\"\"#%&thetaG\"\"\",&F(F(*&%$cosGF(F'F(F(!\"\"*&,&F( F(*&F+F&F'F(F,F(,&F(F(*&F+F(F'F(F(F," }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= ((1-cos*theta)*(1+cos*theta))/(1+cos*theta)" "6#/%!G*(,&\"\"\"F'*&%$co sGF'%&thetaGF'!\"\"F',&F'F'*&F)F'F*F'F'F',&F'F'*&F)F'F*F'F'F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1-cos*theta;" "6#/%!G,&\"\"\"F&*&%$cosGF&%& thetaGF&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 152 "The ab ove analysis verifies the given original identity by showing that the \+ left and right sides of the original identity are both identically equ al to " }{XPPEDIT 18 0 "1-cos*theta" "6#,&\"\"\"F$*&%$cosGF$%&thetaGF$ !\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "An alternative verification can be obtained by letti ng " }{XPPEDIT 18 0 "phi=theta/2" "6#/%$phiG*&%&thetaG\"\"\"\"\"#!\"\" " }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "theta=2*phi" "6#/%&thetaG* &\"\"#\"\"\"%$phiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Th en " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin^2*theta/(1 +cos*theta)=sin^2*2*phi/(1+cos*2*phi)" "6#/*(%$sinG\"\"#%&thetaG\"\"\" ,&F(F(*&%$cosGF(F'F(F(!\"\"**F%F&F&F(%$phiGF(,&F(F(*(F+F(F&F(F.F(F(F, " }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` `=(2*sin*phi*cos*phi)^2/(1+cos^2*phi-sin^2*phi)" "6#/%!G*&*,\"\"#\"\" \"%$sinGF(%$phiGF(%$cosGF(F*F(F',(F(F(*&F+F'F*F(F(*&F)F'F*F(!\"\"F/" } {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=4 *sin^2*phi*cos^2*phi/(1-sin^2*phi+cos^2*phi)" "6#/%!G*.\"\"%\"\"\"*$%$ sinG\"\"#F'%$phiGF'%$cosGF*F+F',(F'F'*&F)F*F+F'!\"\"*&F,F*F+F'F'F/" } {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 4*sin^2*phi*cos^2*phi/(2*cos^2*phi)" "6#/%!G*.\"\"%\"\"\"*$%$sinG\"\" #F'%$phiGF'%$cosGF*F+F'*(F*F'*$F,F*F'F+F'!\"\"" }{TEXT -1 1 " " }} {PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 2*sin^2*phi" "6 #/%!G*(\"\"#\"\"\"*$%$sinGF&F'%$phiGF'" }{TEXT -1 1 " " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*sin^2*``(theta/2)" "6#/%!G* (\"\"#\"\"\"*$%$sinGF&F'-F$6#*&%&thetaGF'F&!\"\"F'" }{TEXT -1 2 ". " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "sin(2*phi)^2/(1+cos(2*phi)) ;\n``=expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$sinG6#,$*&\" \"#\"\"\"%$phiGF*F*F),&F*F*-%$cosGF&F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"#\"\"\")-%$sinG6#%$phiGF'F(F(" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{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 29 "Some trigonometric eq uations " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT 300 8 " Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 20 "Solve the eq uation " }{XPPEDIT 18 0 "sin*2*theta = sin*theta;" "6#/*(%$sinG\"\"\" \"\"#F&%&thetaGF&*&F%F&F(F&" }{TEXT -1 16 " for values of " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 11 " such that " } {XPPEDIT 18 0 "0 <= theta;" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``< 2*P i" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 8 "Solution" }{TEXT -1 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 31 "Using the double-angle formula " } {XPPEDIT 18 0 "sin*2*theta = 2*sin*theta*cos*theta;" "6#/*(%$sinG\"\" \"\"\"#F&%&thetaGF&*,F'F&F%F&F(F&%$cosGF&F(F&" }{TEXT -1 15 ", the equ ation " }{XPPEDIT 18 0 "sin*2*theta = sin*theta;" "6#/*(%$sinG\"\"\"\" \"#F&%&thetaGF&*&F%F&F(F&" }{TEXT -1 29 " can be written in the form: \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*sin*theta*cos*t heta = sin*theta;" "6#/*,\"\"#\"\"\"%$sinGF&%&thetaGF&%$cosGF&F(F&*&F' F&F(F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Collecting all the terms on the left side of the equ ation, we have: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2 *sin*theta*cos*theta-sin*theta = 0;" "6#/,&*,\"\"#\"\"\"%$sinGF'%&thet aGF'%$cosGF'F)F'F'*&F(F'F)F'!\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "The left side of the equation can be factored to giv e " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*theta*(2*co s*theta-1) = 0;" "6#/*(%$sinG\"\"\"%&thetaGF&,&*(\"\"#F&%$cosGF&F'F&F& F&!\"\"F&\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*theta = 0;" "6#/* &%$sinG\"\"\"%&thetaGF&\"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "cos*t heta = 1/2;" "6#/*&%$cosG\"\"\"%&thetaGF&*&F&F&\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta = 1/2;" "6#/*&%$cosG\"\"\"%&thetaGF&*&F& F&\"\"#!\"\"" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "theta = Pi/3;" "6#/ %&thetaG*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "th eta = 5*Pi/3;" "6#/%&thetaG*(\"\"&\"\"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0 <= theta;" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``< 2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 7 ", and " } {XPPEDIT 18 0 "sin*theta = 0;" "6#/*&%$sinG\"\"\"%&thetaGF&\"\"!" } {TEXT -1 6 " when " }{XPPEDIT 18 0 "theta = 0;" "6#/%&thetaG\"\"!" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "theta=Pi" "6#/%&thetaG%#PiG" }{TEXT -1 5 " 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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "This can be seen from the following picture of the unit circle with its ce ntre at the origin in which the points P, Q, R and S have respective c oordinates" }{XPPEDIT 18 0 "``(1,0),``(-1,0),``(1/2,sqrt(3)/2);" "6%-% !G6$\"\"\"\"\"!-F$6$,$F&!\"\"F'-F$6$*&F&F&\"\"#F+*&-%%sqrtG6#\"\"$F&F/ F+" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(1/2,-sqrt(3)/2);" "6#-%!G6$* &\"\"\"F'\"\"#!\"\",$*&-%%sqrtG6#\"\"$F'F(F)F)" }{TEXT -1 2 ". 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7Fi]nF_]nF]^nFb^nFg^n-%%TEXTG6%7$$\"#6Fhjm$\"#:!\"#Q\"P6\"-Fdjm6&F\\[l $F)Fd_nFi_nFi_n-F]_n6%7$$!#6FhjmFb_nQ\"QFf_nFg_n-F]_n6%7$$\"\"'FhjmF(Q \"RFf_nFg_n-F]_n6%7$Fc`nFg\\nQ\"SFf_nFg_n-F]_n6%7$$\"$D\"Fd_n$!\"'Fd_n Q\"xFf_nFg_n-F]_n6%7$F_anF]anQ\"yFf_nFg_n-%(SCALINGG6#%,CONSTRAINEDG-% +AXESLABELSG6%Q!Ff_nF]bn-%%FONTG6#%(DEFAULTG-%*AXESTICKSG6$F*F*-%%VIEW G6$;$!#7Fhjm$\"#8Fhjm;FibnF]an" 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" }}{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 21 "The set of solutions " }{XPPEDIT 18 0 "theta" "6#%&thet aG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "0 <= theta;" "6#1\"\"!%&theta G" }{XPPEDIT 18 0 "``< 2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 18 " for the equation " }{XPPEDIT 18 0 "sin*2*theta = sin*theta;" "6#/ *(%$sinG\"\"\"\"\"#F&%&thetaGF&*&F%F&F(F&" }{TEXT -1 4 " is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "\{0, Pi/3, Pi, 5*Pi/3\};" " 6#<&\"\"!*&%#PiG\"\"\"\"\"$!\"\"F&*(\"\"&F'F&F'F(F)" }{TEXT -1 1 "." } {TEXT 302 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The following picture shows the graphs of the functions \+ " }{XPPEDIT 18 0 "f(theta) = sin*2*theta;" "6#/-%\"fG6#%&thetaG*(%$sin G\"\"\"\"\"#F*F'F*" }{TEXT -1 11 " (drawn in " }{TEXT 260 3 "red" } {TEXT -1 6 ") and " }{XPPEDIT 18 0 "g(theta) = sin*theta;" "6#/-%\"gG6 #%&thetaG*&%$sinG\"\"\"F'F*" }{TEXT -1 11 " (drawn in " }{TEXT 256 4 " blue" }{TEXT -1 55 "), which give the left and right sides of the equa tion " }{XPPEDIT 18 0 "sin*2*theta = sin*theta;" "6#/*(%$sinG\"\"\"\" \"#F&%&thetaGF&*&F%F&F(F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 604 "f := theta -> sin(2* theta): 'f(theta)'=f(theta);\ng := theta -> sin(theta): 'g(theta)'=g(t heta);\npi := evalf(Pi):\np1 := plot([f(theta),g(theta)],theta=0..2*Pi ,color=[red,blue],thickness=2):\np2 := plot([[[0,0],[Pi,0],[Pi/3,sqrt( 3)/2],[5*Pi/3,-sqrt(3)/2]]$4],style=point,\n symbol=[cross,diamond,c ircle$2],symbolsize=[10$3,12],color=[green$3,black]):\nt1 := plots[tex tplot]([6.7,-.13,`q`],font=[SYMBOL,11],color=COLOR(RGB,.01,.01,.01)): \nplots[display]([p1,p2,t1],labels=[``,``],view=[0..6.7,-1.2..1.2],\nx tickmarks=[pi/3=`p/3`,2*pi/3=`2p/3`,pi=`p`,4*pi/3=`4p/3`,5*pi/3=`5p/3` ,2*pi=`2p`],\n font=[SYMBOL,11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"fG6#%&thetaG-%$sinG6#,$*&\"\"#\"\"\"F'F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%&thetaG-%$sinGF&" }}{PARA 13 "" 1 "" {GLPLOT2D 564 265 265 {PLOTDATA 2 "6--%'CURVESG6%7[s7$$\"\"!F)F(7$$\"3 ;`#Gi#zxZo!#>$\"3+9&o'z\"y_O\"!#=7$$\"3i]cC&eb&p8F0$\"3lw.#z2))\\q#F07 $$\"3q5)>63x`'>F0$\"3Pa%G-$4JIQF07$$\"3YqR*pd)>hDF0$\"3v=HM,sI,\\F07$$ \"3Y@%HT;i7B$F0$\"3i+T:by)>-'F07$$\"3Zs[E^dK,RF0$\"3^%R2Q$*yY.(F07$$\" 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%%2p/3G/$\"+aEfTJF]cn%\"pG/$\"+0-z)=%F]cn%%4p/3G/$\"+dx)fB&F]cn%%5p/3G /$\"+3`=$G'F]cn%#2pG%(DEFAULTG-%+AXESLABELSG6%%!GFgdn-Fcbn6#Fcdn-%%VIE WG6$;F(Fean;$!#7Fgan$F`anFgan" 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The \+ solutions of the equation " }{XPPEDIT 18 0 "sin*2*theta = sin*theta;" "6#/*(%$sinG\"\"\"\"\"#F&%&thetaGF&*&F%F&F(F&" }{TEXT -1 9 " are the \+ " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 62 " coordinates of th e points of intersection of the two graphs. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 10 ": Maple's \+ " }{TEXT 0 5 "solve" }{TEXT -1 20 " finds 4 solutions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(s in(2*theta)=sin(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%#PiG\"\"! ,$*&\"\"$!\"\"F#\"\"\"F),$*&F'F(F#F)F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT 305 8 "Question " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 20 "Solve the equation \+ " }{XPPEDIT 18 0 "cos*2*theta = cos*theta;" "6#/*(%$cosG\"\"\"\"\"#F&% &thetaGF&*&F%F&F(F&" }{TEXT -1 16 " for values of " }{XPPEDIT 18 0 "t heta" "6#%&thetaG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "0 <= the ta;" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``< 2*Pi" "6#2%!G*&\"\"#\"\"\" %#PiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 306 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 31 "Using the double-angle formula " }{XPPEDIT 18 0 "cos*2*theta = \+ 2*cos^2*theta-1;" "6#/*(%$cosG\"\"\"\"\"#F&%&thetaGF&,&*(F'F&*$F%F'F&F (F&F&F&!\"\"" }{TEXT -1 15 ", the equation " }{XPPEDIT 18 0 "cos*2*the ta = cos*theta;" "6#/*(%$cosG\"\"\"\"\"#F&%&thetaGF&*&F%F&F(F&" } {TEXT -1 29 " can be written in the form: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*cos^2*theta-1 = cos*theta;" "6#/,&*(\"\"#\" \"\"*$%$cosGF&F'%&thetaGF'F'F'!\"\"*&F)F'F*F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Collectin g all the terms on the left side of the equation, we have: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*cos^2*theta-cos*theta-1 = 0;" "6#/,(*(\"\"#\"\"\"*$%$cosGF&F'%&thetaGF'F'*&F)F'F*F'!\"\"F'F,\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "The left side of t he equation can be factored to give " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(2*cos*theta+1)*(cos*theta-1) = 0;" "6#/*&,&*(\"\"# \"\"\"%$cosGF(%&thetaGF(F(F(F(F(,&*&F)F(F*F(F(F(!\"\"F(\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta = -1/2;" "6#/*&%$cosG\"\"\"%&thetaG F&,$*&F&F&\"\"#!\"\"F+" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "cos*theta = 1;" "6#/*&%$cosG\"\"\"%&thetaGF&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos *theta = -1/2;" "6#/*&%$cosG\"\"\"%&thetaGF&,$*&F&F&\"\"#!\"\"F+" } {TEXT -1 6 " 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 5 " fo r " }{XPPEDIT 18 0 "0 <= theta;" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "`` < 2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "cos*theta = 1;" "6#/*&%$cosG\"\"\"%&thetaGF&F&" }{TEXT -1 6 " wh en " }{XPPEDIT 18 0 "theta = 0;" "6#/%&thetaG\"\"!" }{TEXT -1 6 " 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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 150 "This can be seen from \+ the following picture of the unit circle with its centre at the origin in which the points P,Q and R have respective coordinates" }{XPPEDIT 18 0 "``(1,0),``(-1/2,sqrt(3)/2)" "6$-%!G6$\"\"\"\"\"!-F$6$,$*&F&F&\" \"#!\"\"F-*&-%%sqrtG6#\"\"$F&F,F-" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(-1/2,-sqrt(3)/2)" "6#-%!G6$,$*&\"\"\"F(\"\"#!\"\"F*,$*&-%%sqrtG6#\" \"$F(F)F*F*" }{TEXT -1 2 ". 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is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "\{0, 2*Pi/3, 4*Pi/3\};" "6#<%\"\"!*(\"\"#\"\"\"%#PiGF'\"\"$!\"\"*( \"\"%F'F(F'F)F*" }{TEXT -1 1 "." }{TEXT 307 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The following picture sho ws the graphs of the functions " }{XPPEDIT 18 0 "f(theta) = cos*2*thet a;" "6#/-%\"fG6#%&thetaG*(%$cosG\"\"\"\"\"#F*F'F*" }{TEXT -1 11 " (dra wn in " }{TEXT 260 3 "red" }{TEXT -1 6 ") and " }{XPPEDIT 18 0 "g(thet a) = cos*theta;" "6#/-%\"gG6#%&thetaG*&%$cosG\"\"\"F'F*" }{TEXT -1 11 " (drawn in " }{TEXT 256 4 "blue" }{TEXT -1 55 "), which give the left and right sides of the equation " }{XPPEDIT 18 0 "cos*2*theta = cos*t heta;" "6#/*(%$cosG\"\"\"\"\"#F&%&thetaGF&*&F%F&F(F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 586 "f := theta -> cos(2*theta): 'f(theta)'=f(theta);\ng := theta -> cos(theta): 'g(theta)'=g(theta);\npi := evalf(Pi):\np1 := plot([f(the ta),g(theta)],theta=0..2*Pi,color=[red,blue],thickness=2):\np2 := plot ([[[0,1],[2*Pi/3,-.5],[4*Pi/3,-.5]]$4],style=point,\n symbol=[cross, diamond,circle$2],symbolsize=[10$3,12],color=[green$3,black]):\nt1 := \+ plots[textplot]([6.7,-.13,`q`],font=[SYMBOL,11],color=COLOR(RGB,.01,.0 1,.01)):\nplots[display]([p1,p2,t1],labels=[``,``],view=[0..6.7,-1.2.. 1.2],\nxtickmarks=[pi/3=`p/3`,2*pi/3=`2p/3`,pi=`p`,4*pi/3=`4p/3`,5*pi/ 3=`5p/3`,2*pi=`2p`],\n font=[SYMBOL,11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%&thetaG-%$cosG6#,$*&\"\"#\"\"\"F'F.F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%&thetaG-%$cosGF&" }}{PARA 13 "" 1 "" {GLPLOT2D 585 220 220 {PLOTDATA 2 "6--%'CURVESG6%7]s7$$\"\" !F)$\"\"\"F)7$$\"3Hjqb\"[W>r\"!#>$\"3k))Hyk!RT***!#=7$$\"3eET6j*))QU$F /$\"3!eFW#HJcw**F27$$\"3_*=rYWLe8&F/$\"3Czxm&z#HZ**F27$$\"3;`#Gi#zxZoF /$\"3#*p#fVPij!**F27$$\"3\"zBM*)omr-\"F2$\"3i>)[DzE(*y*F27$$\"3i]cC&eb &p8F2$\"3>0UU)4.si*F27$$\"3q5)>63x`'>F2$\"3;kR.,XNP#*F27$$\"3YqR*pd)>h 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N4hF27$Ff^m$\"3EO\\&RI+$*4(F27$F`_m$\"3KytI?$y,!zF27$Fj_m$\"3C%Qb^XPVn )F27$Fd`m$\"3`bo#[!4+D#*F27$F^am$\"3Yfw#e$))oa'*F27$Fcam$\"3**=**yvne+ )*F27$Fham$\"3n&\\^\"=b&p!**F27$Fbbm$\"37jT:e$R-^R%4#F[r$!3+++ +++++]F27$$\"3_!R'y/-z)=%F[rFe_n-F[dm6&F]dmF(F^dmF(-%'SYMBOLG6$%&CROSS G\"#5-%&STYLEG6#%&POINTG-F$6&Fa_nFj_n-F]`n6$%(DIAMONDGF``nFa`n-F$6&Fa_ nFj_n-F]`n6$%'CIRCLEGF``nFa`n-F$6&Fa_n-F[dm6&F]dmF)F)F)-F]`n6$F^an\"#7 Fa`n-%%TEXTG6&7$$\"#n!\"\"$!#8!\"#Q\"q6\"-%&COLORG6&F]dm$F+F_bnFebnFeb n-%%FONTG6$F]`n\"#6Ffbn-%*AXESTICKSG6$7(/$\"+^v>Z5!\"*%$p/3G/$\"+.^R%4 #Facn%%2p/3G/$\"+aEfTJFacn%\"pG/$\"+0-z)=%Facn%%4p/3G/$\"+dx)fB&Facn%% 5p/3G/$\"+3`=$G'Facn%#2pG%(DEFAULTG-%+AXESLABELSG6%%!GF[en-Fgbn6#Fgdn- %%VIEWG6$;F(Fjan;$!#7F\\bn$FeanF\\bn" 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The \+ solutions of the equation " }{XPPEDIT 18 0 "cos*2*theta=cos*theta" "6# /*(%$cosG\"\"\"\"\"#F&%&thetaGF&*&F%F&F(F&" }{TEXT -1 9 " are the " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 62 " coordinates of the p oints of intersection of the two graphs. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 10 ": Maple's " } {TEXT 0 5 "solve" }{TEXT -1 19 " finds 2 solutions." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(cos( 2*theta)=cos(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!,$*(\"\" #\"\"\"\"\"$!\"\"%#PiGF'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT 308 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 20 "Solve the equation " } {XPPEDIT 18 0 "2*cos*2*theta = 1;" "6#/**\"\"#\"\"\"%$cosGF&F%F&%&thet aGF&F&" }{TEXT -1 16 " for values of " }{XPPEDIT 18 0 "theta" "6#%&th etaG" }{TEXT -1 11 " such that " }{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 "" }}{PARA 0 "" 0 "" {TEXT 309 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "2*cos*2*theta = 1;" "6#/**\"\"#\"\"\"% $cosGF&F%F&%&thetaGF&F&" }{TEXT -1 18 " is equivalent to " }{XPPEDIT 18 0 "cos*2*theta = 1/2;" "6#/*(%$cosG\"\"\"\"\"#F&%&thetaGF&*&F&F&F'! \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "Then, using the d ouble-angle formula " }{XPPEDIT 18 0 "cos*2*theta = 2*cos^2*theta-1;" "6#/*(%$cosG\"\"\"\"\"#F&%&thetaGF&,&*(F'F&*$F%F'F&F(F&F&F&!\"\"" } {TEXT -1 15 ", the equation " }{XPPEDIT 18 0 "cos*2*theta = 1/2;" "6#/ *(%$cosG\"\"\"\"\"#F&%&thetaGF&*&F&F&F'!\"\"" }{TEXT -1 29 " can be wr itten in the form: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*cos^2*theta-1 = 1/2;" "6#/,&*(\"\"#\"\"\"*$%$cosGF&F'%&thetaGF'F' F'!\"\"*&F'F'F&F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*cos^2*theta = \+ 3/2;" "6#/*(\"\"#\"\"\"*$%$cosGF%F&%&thetaGF&*&\"\"$F&F%!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos^2*theta = 3/4" "6#/*&%$cosG\"\"#%&t hetaG\"\"\"*&\"\"$F(\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "cos*theta=``" "6#/*&%$cosG\"\"\"%&thetaGF&%!G" }{TEXT 311 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(3)/2" "6#*&-%%sqrtG6#\" \"$\"\"\"\"\"#!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta = sqrt( 3)/2;" "6#/*&%$cosG\"\"\"%&thetaGF&*&-%%sqrtG6#\"\"$F&\"\"#!\"\"" } {TEXT -1 6 " when " }{XPPEDIT 18 0 "theta = Pi/6;" "6#/%&thetaG*&%#PiG \"\"\"\"\"'!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "theta = 11*Pi/6; " "6#/%&thetaG*(\"#6\"\"\"%#PiGF'\"\"'!\"\"" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "0 <= theta;" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``< 2*P i" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 " cos*theta = -sqrt(3)/2;" "6#/*&%$cosG\"\"\"%&thetaGF&,$*&-%%sqrtG6#\" \"$F&\"\"#!\"\"F/" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "theta = 5*Pi/6 ;" "6#/%&thetaG*(\"\"&\"\"\"%#PiGF'\"\"'!\"\"" }{TEXT -1 5 " or " } {XPPEDIT 18 0 "theta=7*Pi/6" "6#/%&thetaG*(\"\"(\"\"\"%#PiGF'\"\"'!\" \"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0 <= theta;" "6#1\"\"!%&thetaG " }{XPPEDIT 18 0 "``< 2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "Th is can be seen from the following picture of the unit circle with its \+ centre at the origin in which the points P,Q, R and S have respective \+ coordinates " }{XPPEDIT 18 0 "``(sqrt(3)/2,1/2), ``(-sqrt(3)/2,1/2), ` `(-sqrt(3)/2,-1/2)" "6%-%!G6$*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"*&F+F+F, F--F$6$,$*&-F(6#F*F+F,F-F-*&F+F+F,F--F$6$,$*&-F(6#F*F+F,F-F-,$*&F+F+F, F-F-" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``(sqrt(3)/2,-1/2)" "6#-%!G6 $*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\",$*&F+F+F,F-F-" }{TEXT -1 2 ". 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*F*-%*LINESTYLEGFb[l-F$6%7$Fb]lF\\]lFf]lFh]l-F$6&7&Fh[lFf\\lF\\]lFb]l- Fjz6&F\\[lF+F][lF+-%'SYMBOLG6$%&CROSSG\"#5-%&STYLEG6#%&POINTG-F$6&F_^l F`^l-Fc^l6$%(DIAMONDGFf^lFg^l-F$6&F_^lF`^l-Fc^l6$%'CIRCLEGFf^lFg^l-F$6 &F_^lFf]l-Fc^l6$Fd_l\"#7Fg^l-%%TEXTG6%7$F($\"#a!\"#Q\"P6\"-%&COLORG6&F \\[l$F)F``lFf`lFf`l-F[`l6%7$$!\"\"F*F^`lQ\"QFb`lFc`l-F[`l6%7$Fj`l$!#aF ``lQ\"RFb`lFc`l-F[`l6%7$F(F`alQ\"SFb`lFc`l-F[`l6%7$$\"$D\"F``l$!\"'F`` lQ\"xFb`lFc`l-F[`l6%7$F\\blFjalQ\"yFb`lFc`l-%(SCALINGG6#%,CONSTRAINEDG -%*AXESTICKSG6$F*F*-%+AXESLABELSG6%Q!Fb`lF]cl-%%FONTG6#%(DEFAULTG-%%VI EWG6$;$!#7F[al$\"#8F[al;FfclFjal" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 43.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" }}{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "The set of solutions " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 6 " with " }{XPPEDIT 18 0 "0 <= theta;" "6#1\"\"!%&thetaG" } {XPPEDIT 18 0 "``< 2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 18 " f or the equation " }{XPPEDIT 18 0 "2*cos*2*theta = 1;" "6#/**\"\"#\"\" \"%$cosGF&F%F&%&thetaGF&F&" }{TEXT -1 4 " is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "\{Pi/6, 5*Pi/6, 7*Pi/6, 11*Pi/6\};" "6# <&*&%#PiG\"\"\"\"\"'!\"\"*(\"\"&F&F%F&F'F(*(\"\"(F&F%F&F'F(*(\"#6F&F%F &F'F(" }{TEXT -1 1 "." }{TEXT 310 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 56 "The following picture shows the graph s of the functions " }{XPPEDIT 18 0 "f(theta) = 2*cos*2*theta;" "6#/-% \"fG6#%&thetaG**\"\"#\"\"\"%$cosGF*F)F*F'F*" }{TEXT -1 11 " (drawn in \+ " }{TEXT 260 3 "red" }{TEXT -1 6 ") and " }{XPPEDIT 18 0 "g(theta) = 1 ;" "6#/-%\"gG6#%&thetaG\"\"\"" }{TEXT -1 11 " (drawn in " }{TEXT 256 4 "blue" }{TEXT -1 55 "), which give the left and right sides of the e quation " }{XPPEDIT 18 0 "2*cos*2*theta = 1;" "6#/**\"\"#\"\"\"%$cosGF &F%F&%&thetaGF&F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 590 "f := theta -> 2*cos(2*theta ): 'f(theta)'=f(theta);\ng := theta -> 1: 'g(theta)'=g(theta);\npi := \+ evalf(Pi):\np1 := plot([f(theta),g(theta)],theta=0..2*Pi,color=[red,bl ue],thickness=2):\np2 := plot([[[Pi/6,1],[5*Pi/6,1],[7*Pi/6,1],[11*Pi/ 6,1]]$4],style=point,\n symbol=[cross,diamond,circle$2],symbolsize=[ 10$3,12],color=[green$3,black]):\nt1 := plots[textplot]([6.7,-.13,`q`] ,font=[SYMBOL,11],color=COLOR(RGB,.01,.01,.01)):\nplots[display]([p1,p 2,t1],labels=[``,``],view=[0..6.7,-2.2..2.2],\nxtickmarks=[pi/3=`p/3`, 2*pi/3=`2p/3`,pi=`p`,4*pi/3=`4p/3`,5*pi/3=`5p/3`,2*pi=`2p`],\n font=[ SYMBOL,11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%&thetaG,$*& \"\"#\"\"\"-%$cosG6#,$*&F*F+F'F+F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%&thetaG\"\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 585 220 220 {PLOTDATA 2 "6--%'CURVESG6%7]s7$$\"\"!F)$\"\"#F)7$$\"3Hjqb\"[W>r\" 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{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The solutions of the equation \+ " }{XPPEDIT 18 0 "2*cos*2*theta = 1;" "6#/**\"\"#\"\"\"%$cosGF&F%F&%&t hetaGF&F&" }{TEXT -1 9 " are the " }{XPPEDIT 18 0 "theta" "6#%&thetaG " }{TEXT -1 62 " coordinates of the points of intersection of the two \+ graphs. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 312 30 "Alternative method of solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 " " {TEXT -1 12 "The equation" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2*cos*2*theta=1" "6#/**\"\"#\"\"\"%$cosGF&F%F&%&thetaGF &F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "is equivalent to \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*2*theta=1/2" "6#/*(%$cosG\"\"\"\"\"#F&%&thetaGF&*&F&F&F'!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 31 "In order to find solutions for " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 18 " in the interval [" } {XPPEDIT 18 0 "0,2*Pi;" "6$\"\"!*&\"\"#\"\"\"%#PiGF&" }{TEXT -1 25 "), we look for values of " }{XPPEDIT 18 0 "2*theta" "6#*&\"\"#\"\"\"%&th etaGF%" }{TEXT -1 18 " in the interval [" }{XPPEDIT 18 0 "0,4*Pi" "6$ \"\"!*&\"\"%\"\"\"%#PiGF&" }{TEXT -1 34 ") which satisfy the last equa tion." }}{PARA 0 "" 0 "" {TEXT -1 19 "These values are: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*theta = Pi/3,5*Pi/3,7*Pi/ 3;" "6%/*&\"\"#\"\"\"%&thetaGF&*&%#PiGF&\"\"$!\"\"*(\"\"&F&F)F&F*F+*( \"\"(F&F)F&F*F+" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "11*Pi/3;" "6#*(\"# 6\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta=Pi /6,5*Pi/6,7*Pi/6" "6%/%&thetaG*&%#PiG\"\"\"\"\"'!\"\"*(\"\"&F'F&F'F(F) *(\"\"(F'F&F'F(F)" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "11*Pi/6" "6#*(\" #6\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 10 ": Maple's " }{TEXT 0 5 "solve" }{TEXT -1 25 " fin ds only one solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(2*cos(2*theta)=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"'!\"\"%#PiG\"\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 41 "Use the half-angle formulas to calcul ate " }{XPPEDIT 18 0 "sin*15^o = sin(Pi/12);" "6#/*&%$sinG\"\"\")\"#:% \"oGF&-F%6#*&%#PiGF&\"#7!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos *15^o = cos(Pi/12);" "6#/*&%$cosG\"\"\")\"#:%\"oGF&-F%6#*&%#PiGF&\"#7! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "cos(Pi/12)=sqrt(2+sqrt(3))/2" "6#/-%$cosG6#*&%#PiG\"\" \"\"#7!\"\"*&-%%sqrtG6#,&\"\"#F)-F.6#\"\"$F)F)F1F+" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "sin(Pi/12)=sqrt(2-sqrt(3))/2" "6#/-%$sinG6#*&%#PiG\"\" \"\"#7!\"\"*&-%%sqrtG6#,&\"\"#F)-F.6#\"\"$F+F)F1F+" }{TEXT -1 2 " " } }}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 11 "Given that " }{XPPEDIT 18 0 "cos*theta = \+ 4/5;" "6#/*&%$cosG\"\"\"%&thetaGF&*&\"\"%F&\"\"&!\"\"" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "3*Pi/2 <= theta;" "6#1*(\"\"$\"\"\"%#PiGF&\"\"#! \"\"%&thetaG" }{XPPEDIT 18 0 "`` < 2*Pi;" "6#2%!G*&\"\"#\"\"\"%#PiGF' " }{TEXT -1 8 " , find " }{XPPEDIT 18 0 "sin*theta;" "6#*&%$sinG\"\"\" %&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin*2*theta" "6#*(%$sinG \"\"\"\"\"#F%%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cos*2*theta; " "6#*(%$cosG\"\"\"\"\"#F%%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tan*2*theta" "6#*(%$tanG\"\"\"\"\"#F%%&thetaGF%" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "sin(theta/2)" "6#-%$sinG6#*&%&thetaG\"\"\"\"\"#!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "cos(theta/2)" "6#-%$cosG6#*&%&thetaG\" \"\"\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "tan(theta/2)" "6#- %$tanG6#*&%&thetaG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*theta = -3/5;" "6#/*&%$sinG \"\"\"%&thetaGF&,$*&\"\"$F&\"\"&!\"\"F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin*2*theta = -24/25;" "6#/*(%$sinG\"\"\"\"\"#F&%&thetaGF&,$*&\" #CF&\"#D!\"\"F-" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cos*2*theta = 7/25; " "6#/*(%$cosG\"\"\"\"\"#F&%&thetaGF&*&\"\"(F&\"#D!\"\"" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "tan*2*theta = -24/7;" "6#/*(%$tanG\"\"\"\"\"#F&%&t hetaGF&,$*&\"#CF&\"\"(!\"\"F-" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(th eta/2) = 1/sqrt(10);" "6#/-%$sinG6#*&%&thetaG\"\"\"\"\"#!\"\"*&F)F)-%% sqrtG6#\"#5F+" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cos(theta/2) = -3/sqrt (10);" "6#/-%$cosG6#*&%&thetaG\"\"\"\"\"#!\"\",$*&\"\"$F)-%%sqrtG6#\"# 5F+F+" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tan(theta/2) = -1/3;" "6#/-%$t anG6#*&%&thetaG\"\"\"\"\"#!\"\",$*&F)F)\"\"$F+F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "theta := 2*Pi-arccos(4/5):\n'sin(theta)'=simplify(sin(theta));\n' sin(2*theta)'=expand(sin(2*theta));\n'cos(2*theta)'=expand(cos(2*theta ));\n'tan(2*theta)'=expand(tan(2*theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%&thetaG#!\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#,$*&\"\"#\"\"\"%&thetaGF*F*#!#C\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$*&\"\"#\"\"\"%&thetaGF*F*#\"\"( \"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$tanG6#,$*&\"\"#\"\"\"%&the taGF*F*#!#C\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 33 "___________ ______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "_______________________________ __" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 11 "Given that " } {XPPEDIT 18 0 "cos*theta = -1/3;" "6#/*&%$cosG\"\"\"%&thetaGF&,$*&F&F& \"\"$!\"\"F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Pi <= theta;" "6#1%# PiG%&thetaG" }{XPPEDIT 18 0 "`` < 3*Pi/2;" "6#2%!G*(\"\"$\"\"\"%#PiGF' \"\"#!\"\"" }{TEXT -1 8 " , find " }{XPPEDIT 18 0 "sin*theta" "6#*&%$s inG\"\"\"%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin*2*theta" "6# *(%$sinG\"\"\"\"\"#F%%&thetaGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cos* 2*theta;" "6#*(%$cosG\"\"\"\"\"#F%%&thetaGF%" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "tan*2*theta" "6#*(%$tanG\"\"\"\"\"#F%%&thetaGF%" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(theta/2)" "6#-%$sinG6#*&%&thetaG\" \"\"\"\"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cos(theta/2)" "6#-%$c osG6#*&%&thetaG\"\"\"\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t an(theta/2)" "6#-%$tanG6#*&%&thetaG\"\"\"\"\"#!\"\"" }{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 1 " \+ " }{XPPEDIT 18 0 "sin*theta=-2*sqrt(2)/3" "6#/*&%$sinG\"\"\"%&thetaGF& ,$*(\"\"#F&-%%sqrtG6#F*F&\"\"$!\"\"F/" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin*2*theta = 4*sqrt(2)/9;" "6#/*(%$sinG\"\"\"\"\"#F&%&thetaGF&*(\" \"%F&-%%sqrtG6#F'F&\"\"*!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cos*2* theta=-7/9" "6#/*(%$cosG\"\"\"\"\"#F&%&thetaGF&,$*&\"\"(F&\"\"*!\"\"F- " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tan*2*theta=-4*sqrt(2)/7" "6#/*(%$t anG\"\"\"\"\"#F&%&thetaGF&,$*(\"\"%F&-%%sqrtG6#F'F&\"\"(!\"\"F0" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(theta/2)=sqrt(2/3)" "6#/-%$sinG6#* &%&thetaG\"\"\"\"\"#!\"\"-%%sqrtG6#*&F*F)\"\"$F+" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "cos(theta/2) = -1/sqrt(3);" "6#/-%$cosG6#*&%&thetaG\"\" \"\"\"#!\"\",$*&F)F)-%%sqrtG6#\"\"$F+F+" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tan(theta/2) = -sqrt(2);" "6#/-%$tanG6#*&%&thetaG\"\"\"\"\"#!\" \",$-%%sqrtG6#F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "theta := arccos(1/3)+Pi:\n 'sin(theta)'=simplify(sin(theta));\n'sin(2*theta)'=expand(sin(2*theta) );\n'cos(2*theta)'=expand(cos(2*theta));\n'tan(2*theta)'=expand(tan(2* theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%&thetaG,$*(\"\" #\"\"\"\"\"$!\"\"F*#F+F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG 6#,$*&\"\"#\"\"\"%&thetaGF*F*,$*(\"\"%F*\"\"*!\"\"F)#F*F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$*&\"\"#\"\"\"%&thetaGF*F*#!\"(\" \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$tanG6#,$*&\"\"#\"\"\"%&thet aGF*F*,$*(\"\"%F*\"\"(!\"\"F)#F*F)F0" }}}{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 2 "Q4" }}{PARA 0 "" 0 " " {TEXT -1 33 "Verify the following identities: " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "cos*2*theta/(sin^2*theta)= cot^2*the ta-1" "6#/**%$cosG\"\"\"\"\"#F&%&thetaGF&*&%$sinGF'F(F&!\"\",&*&%$cotG F'F(F&F&F&F+" }{TEXT -1 17 " (b) " }{XPPEDIT 18 0 "(cos*th eta-sin*theta)^2 = 1-sin*2*theta;" "6#/*$,&*&%$cosG\"\"\"%&thetaGF(F(* &%$sinGF(F)F(!\"\"\"\"#,&F(F(*(F+F(F-F(F)F(F," }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 4 "(c) " }{XPPEDIT 18 0 "(1-cos^2*theta)/(1-s in^2*theta)=(1-cos*2*theta)/(1+cos*2*theta)" "6#/*&,&\"\"\"F&*&%$cosG \"\"#%&thetaGF&!\"\"F&,&F&F&*&%$sinGF)F*F&F+F+*&,&F&F&*(F(F&F)F&F*F&F+ F&,&F&F&*(F(F&F)F&F*F&F&F+" }{TEXT -1 9 " (d) " }{XPPEDIT 18 0 "2* cos^2*``(theta/2) = sin^2*theta/(1-cos*theta);" "6#/*(\"\"#\"\"\"*$%$c osGF%F&-%!G6#*&%&thetaGF&F%!\"\"F&*(%$sinGF%F-F&,&F&F&*&F(F&F-F&F.F." }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "__ _______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 " " {TEXT -1 19 "Solve the equation " }{XPPEDIT 18 0 "sin*2*theta = cos* theta;" "6#/*(%$sinG\"\"\"\"\"#F&%&thetaGF&*&%$cosGF&F(F&" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0 <= theta;" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "Illustrate the solution graphically as the " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 102 " coordinates of the \+ points of intersection of the graphs of the left and right sides of th e equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c os*theta*(2*sin*theta-1)=0" "6#/*(%$cosG\"\"\"%&thetaGF&,&*(\"\"#F&%$s inGF&F'F&F&F&!\"\"F&\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta = Pi /6,Pi/2,5*Pi/6;" "6%/%&thetaG*&%#PiG\"\"\"\"\"'!\"\"*&F&F'\"\"#F)*(\" \"&F'F&F'F(F)" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "3*Pi/2" "6#*(\"\"$\" \"\"%#PiGF%\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 611 "f := theta -> sin(2*the ta): 'f(theta)'=f(theta);\ng := theta -> cos(theta): 'g(theta)'=g(thet a);\npi := evalf(Pi):\np1 := plot([f(theta),g(theta)],theta=0..2*Pi,co lor=[red,blue],thickness=2):\np2 := plot([[[Pi/6,sqrt(3)/2],[Pi/2,0],[ 5*Pi/6,-sqrt(3)/2],[3*Pi/2,0]]$4],style=point,\n symbol=[cross,diamo nd,circle$2],symbolsize=[10$3,12],color=[green$3,black]):\nt1 := plots [textplot]([6.7,-.13,`q`],font=[SYMBOL,11],color=COLOR(RGB,.01,.01,.01 )):\nplots[display]([p1,p2,t1],labels=[``,``],view=[0..6.7,-1.2..1.2], \nxtickmarks=[pi/3=`p/3`,2*pi/3=`2p/3`,pi=`p`,4*pi/3=`4p/3`,5*pi/3=`5p /3`,2*pi=`2p`],\n font=[SYMBOL,11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%&thetaG-%$sinG6#,$*&\"\"#\"\"\"F'F.F." }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"gG6#%&thetaG-%$cosGF&" }}{PARA 13 "" 1 "" {GLPLOT2D 584 213 213 {PLOTDATA 2 "6--%'CURVESG6%7[s7$$\"\"!F)F(7$$\"3 ;`#Gi#zxZo!#>$\"3+9&o'z\"y_O\"!#=7$$\"3i]cC&eb&p8F0$\"3lw.#z2))\\q#F07 $$\"3q5)>63x`'>F0$\"3Pa%G-$4JIQF07$$\"3YqR*pd)>hDF0$\"3v=HM,sI,\\F07$$ \"3Y@%HT;i7B$F0$\"3i+T:by)>-'F07$$\"3Zs[E^dK,RF0$\"3^%R2Q$*yY.(F07$$\" 3+9+\"[&4$ed%F0$\"3_On6w\"Rm#zF07$$\"3ab^NehL]_F0$\"3)Gzl;wmXn)F07$$\" 3Ls#3kbN;#fF0$\"3yQ:r_lXi#*F07$$\"35*QhW&\\$Hf'F0$\"3?Z)e**eMOo*F07$$ \"3J-7h\">DT!pF0$\"3K#Q)Rrl4?)*F07$$\"3]:5wGaJ:sF0$\"3V'H%)R&>`=**F07$ $\"3c@fLZ0\"4P(F0$\"3_5lidZO`**F07$$\"3fF3\"fm0l_(F0$\"3u#*)*)Rcf&y**F 07$$\"3uMd[%y+@o(F0$\"3%)eJbw>4%***F07$$\"3zS11.fpPyF0$\"39;1/bp%***** F07$$\"3'y&R#fj0))*zF0$\"3a!)e&\\\\0e***F07$$\"3'QF(yo`\"*f\")F0$\"3iG TK\\nG\")**F07$$\"3%)*e];5D5K)F0$\"3W!f%)4z0k&**F07$$\"3#e!R^M[8#[)F0$ \"30GGX]%)=@**F07$$\"3yP0C+VN/))F0$\"3k^#=7c,*>)*F07$$\"3upr'fwtl7*F0$ \"3Su&o?eYyn*F07$$\"3!Rbb2V`Iz*F0$\"3G#Q@H0ytD*F07$$\"3!QRa&4L&f/\"!#< $\"3moX\"pipEn)F07$$\"3_y%f^`(Q76F\\s$\"3>>:bAipOzF07$$\"3YjXwg<#)y6F \\s$\"3s4p2$>931(F07$$\"3!or.w_drC\"F\\s$\"3Ek&H$f!p,.'F07$$\"36qGW%H$ \\:8F\\s$\"3sn*\\$o41()[F07$$\"3SH22vMov8F\\s$\"3#R8F07$$\" 396:YIMRr:F\\s$!3S'oz$\\q?%>\"!#?7$$\"3Z'z]kaJ%R;F\\s$!3_Lk6NuRo8F07$$ \"3!=3SCmpuq\"F\\s$!3)=/[Lh`&*p#F07$$\"33u?N%*p.tF\\s$!3unQ](e^%))pF07$$\"3#=uye<)G*4#F\\s$!3\"RK\"\\/$p (3()F07$$\"3!zz^8\\l#f@F\\s$!3gesRUTNM#*F07$$\"3Ua[#o!GC>AF\\s$!3!)QCk iSAF'*F07$$\"3W_![\\uETD#F\\s$!3wW()zVgO#z*F07$$\"3W]72$o5!*G#F\\s$!3 \"*4;egC')4**F07$$\"3A\\G8_EX1BF\\s$!319iB!zb0&**F07$$\"3,[W>@Y*QK#F\\ s$!3S;v[C;9z**F07$$\"3zYgD!fO8M#F\\s$!3qw;2\"=&e&***F07$$\"3-YwJf&y(eB F\\s$!3YlB._k')****F07$$\"3;#3ltl)*F07$ $\"3oNrpV8H#[#F\\s$!31g?u*etOo*F07$$\"3C24z$*y/]DF\\s$!3)*z`1DZyd#*F07 $$\"3EzY)QW/yh#F\\s$!3-IfJdx9i')F07$$\"3=xAg-ZK#o#F\\s$!3'\\7&Rvy8ZzF0 7$$\"35v)>8'\\%ou#F\\s$!3-B\"Qw5!)**4(F07$$\"3Vz?n#3lT\"GF\\s$!3q\"p7Z dV/4'F07$$\"3w$GCS?&[\")GF\\s$!3ym_c*>m1(\\F07$$\"3*Q3$\\!41L%HF\\s$!3 !*)zQ8-(fiQF07$$\"3Z%)='p(p70IF\\s$!3InO4'Rabp#F07$$\"3g`,ta\"4=2$F\\s $!3]9L\"zNV6R\"F07$$\"3sA%)\\K8\\QJF\\s$!37C)z1T-E?'Fgu7$$\"3#*3]^u`v2 KF\\s$\"3dXjc&4'R>8F07$$\"3c&fJlT>qF$F\\s$\"3mw3PG%Rbn#F07$$\"3e!=@(oR JPLF\\s$\"3vl\")R3KA:QF07$$\"39l2\"4_3wR$F\\s$\"3Yk\\GLU\\**[F07$$\"3_ ]()4*GGFY$F\\s$\"3OE\\xin:!*fF07$$\"3MOnGd![y_$F\\s$\"3kZ@!)ydNzpF07$$ \"3iF))z\\J7&f$F\\s$\"3uW(R5Hlp(yF07$$\"3#*=4JU#)RiOF\\s$\"3(3Jb>=(=K' )F07$$\"36m\\Q&z8#GPF\\s$\"3CH?x#Q2,A*F07$$\"3%G,f%[$HSz$F\\s$\"3/Xf2T O]['*F07$$\"3%p#\\M9$pe#QF\\s$\"3)[dRY'\\='z*F07$$\"3eS3B!G4x&QF\\s$\" 3GNGjq[:/**F07$$\"3i(ztJEHO(QF\\s$\"3%*oQKC\\5V**F07$$\"3Aan6Y#\\&*)QF \\s$\"3!*ofeuc(>(**F07$$\"3G6(f!H#pa!RF\\s$\"3%>d._&yt!***F07$$\"3KoE+ 7#*Q@RF\\s$\"3+fd*pWs$****F07$$\"3GM!y(Qc1RRF\\s$\"3]D=H:T3(***F07$$\" 3o+Mbl?ucRF\\s$\"3EpHn'[-B)**F07$$\"35n(GB\\=W(RF\\s$\"3g'3`O.Y]&**F07 $$\"31LT5>\\4#*RF\\s$\"3>s.Y=)[`\"**F07$$\"3)\\'[lsxWFSF\\s$\"3%\\O2U- Y))z*F07$$\"3y(f0ii+G1%F\\s$\"3&p@!Q$HwLj*F07$$\"3ZG'*z[GLETF\\s$\"3># Q:7YEd@*F07$$\"3;fORr]')*=%F\\s$\"3qwK,p`[\\')F07$$\"3R&p%*z[LbK%F\\s$ \"33u50Zf%z)pF07$$\"3'epgGO,qQ%F\\s$\"3!*4q!)eP.egF07$$\"3M'pExBp%[WF \\s$\"35M&*Ra0oO]F07$$\"3,z&[2')pc^%F\\s$\"31(pgJ?iO$QF07$$\"3oh/x$[qG e%F\\s$\"3C%ort#z\\hDF07$$\"3qQq'H.,hk%F\\s$\"3DO\")z)oy=K\"F07$$\"3e; 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" }}{PARA 0 "" 0 "" {TEXT -1 43 "Illustrate the solution graphically \+ as the " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 102 " coordinat es of the points of intersection of the graphs of the left and right s ides of the equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2*sin^2*theta+sin*theta-1 = 0;" "6#/,(*(\"\"#\"\"\"*$%$ sinGF&F'%&thetaGF'F'*&F)F'F*F'F'F'!\"\"\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "sin*theta=1/2" "6#/*&%$sinG\"\"\"%&thetaGF&*&F&F&\"\"#! \"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta = Pi/6,5*Pi/6;" "6$/%&thetaG*&%#PiG\" \"\"\"\"'!\"\"*(\"\"&F'F&F'F(F)" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "3* Pi/2" "6#*(\"\"$\"\"\"%#PiGF%\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 590 "f := theta -> cos(2*theta): 'f(theta)'=f(theta);\ng := theta -> sin(theta) : 'g(theta)'=g(theta);\npi := evalf(Pi):\np1 := plot([f(theta),g(theta )],theta=0..2*Pi,color=[red,blue],thickness=2):\np2 := plot([[[Pi/6,1/ 2],[5*Pi/6,1/2],[3*Pi/2,-1]]$4],style=point,\n symbol=[cross,diamond ,circle$2],symbolsize=[10$3,12],color=[green$3,black]):\nt1 := plots[t extplot]([6.7,-.13,`q`],font=[SYMBOL,11],color=COLOR(RGB,.01,.01,.01)) :\nplots[display]([p1,p2,t1],labels=[``,``],view=[0..6.7,-1.2..1.2],\n xtickmarks=[pi/3=`p/3`,2*pi/3=`2p/3`,pi=`p`,4*pi/3=`4p/3`,5*pi/3=`5p/3 `,2*pi=`2p`],\n font=[SYMBOL,11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"fG6#%&thetaG-%$cosG6#,$*&\"\"#\"\"\"F'F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%&thetaG-%$sinGF&" }}{PARA 13 "" 1 "" {GLPLOT2D 656 217 217 {PLOTDATA 2 "6--%'CURVESG6%7]s7$$\"\"!F)$\"\"\"F )7$$\"3Hjqb\"[W>r\"!#>$\"3k))Hyk!RT***!#=7$$\"3eET6j*))QU$F/$\"3!eFW#H Jcw**F27$$\"3_*=rYWLe8&F/$\"3Czxm&z#HZ**F27$$\"3;`#Gi#zxZoF/$\"3#*p#fV Pij!**F27$$\"3\"zBM*)omr-\"F2$\"3i>)[DzE(*y*F27$$\"3i]cC&eb&p8F2$\"3>0 UU)4.si*F27$$\"3q5)>63x`'>F2$\"3;kR.,XNP#*F27$$\"3YqR*pd)>hDF2$\"3F#H& y9%*[;()F27$$\"3Y@%HT;i7B$F2$\"3k#p9vzhM)zF27$$\"3Zs[E^dK,RF2$\"3aOG3$ Qqs5(F27$$\"3+9+\"[&4$ed%F2$\"3)\\gKF])e'4'F27$$\"3ab^NehL]_F2$\"3M#)G a%GF^(\\F27$$\"3Ls#3kbN;#fF2$\"3KMt_vR?pPF27$$\"35*QhW&\\$Hf'F2$\"3[zF .n+W&\\#F27$$\"3]:5wGaJ:sF2$\"3EJ](pzhQF\"F27$$\"3zS11.fpPyF2$\"3mDNfB /9dK!#?7$$\"3#e!R^M[8#[)F2$!3'f`c>G/ID\"F27$$\"3upr'fwtl7*F2$!3L5d/$y$ z$*Q*f`(p9F[r$!3#R_h,J,lz*F27$$\"3/]+3VNj.:F[r$!3rssEz(=* 4**F27$$\"3P:a#\\^t0_\"F[r$!3!ez[My&f\\**F27$$\"3q!yqn[8v`\"F[r$!3-Jjo IK&y(**F27$$\"3/YhheMXa:F[r$!3G4]v)oeY***F27$$\"396:YIMRr:F[r$!2M2X$pG ******F[r7$$\"3[K)e%fHS)e\"F[r$!3uB\"Q.x+Q***F27$$\"3\"Q:c%)[7ag\"F[r$ !3k*o&)z*R/w**F27$$\"39vMX$f!**F27$$\"39RaW/1Xt;F[r$!31Ow(=E\")**y*F27$$\"3!=3SCmpuq\" F[r$!3#*RDy$QG(G'*F27$$\"33u?N%*p.t*F27$$\"3LmSEEVgQ=F [r$!3!3H\"RXSa*f)F27$$\"3')4G5Xd9)*=F[r$!3r)GOf[8B$zF27$$\"3S`:%R;(od> F[r$!3)f\\Kr&)HF:(F27$$\"3^Z,\"*pw[G?F[r$!3E=^**zA([4'F27$$\"3#=uye<)G *4#F[r$!3whB999,:\\F27$$\"3!zz^8\\l#f@F[r$!3_z-$*Q(Qv$QF27$$\"3Ua[#o!G C>AF[r$!3GY2*RY8\\q#F27$$\"3W]72$o5!*G#F[r$!3PDuZAljR8F27$$\"3-YwJf&y( eBF[r$\"3'H..TC:\"o^Fhp7$$\"31\"R2:&\\`?CF[r$\"3iOqkL3E$G\"F27$$\"3oNr pV8H#[#F[r$\"3m&zQAs)G&\\#F27$$\"3C24z$*y/]DF[r$\"3%H^%GDUm!y$F27$$\"3 EzY)QW/yh#F[r$\"3'*)y[ia=n*\\F27$$\"3=xAg-ZK#o#F[r$\"3%3y]nXV)pgF27$$ \"35v)>8'\\%ou#F[r$\"3+)=!o7o.UqF27$$\"3Vz?n#3lT\"GF[r$\"3kulq9;OJzF27 $$\"3w$GCS?&[\")GF[r$\"3#ot*H0Q7x')F27$$\"3*Q3$\\!41L%HF[r$\"3b#p**oq+ RA*F27$$\"3Z%)='p(p70IF[r$\"3o%)fiE([)H'*F27$$\"3D>g%e1o%QIF[r$\"3o)[o fHf!)y*F27$$\"3g`,ta\"4=2$F[r$\"3z*o$peKw-**F27$$\"3)4As\"*pz%)3$F[r$ \"3Qhre9LjV**F27$$\"3$zG9OC]^5$F[r$\"3GT.7r2Xt**F27$$\"3Kbj0)y?=7$F[r$ \"3gTyV&[#=#***F27$$\"3sA%)\\K8\\QJF[r$\"3_%\\mnj2)****F27$$\"3EpD+Vt! 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3]'*=#eVn4O\"F27$Ffcm$!3/UE[]'efD\"!#D-F[dm6&F]dmF(F(F^dmFadm-F$6&7%7$ $\"3;))H)fv()fB&F2$\"3++++++++]F27$$\"3T%\\\"*z(Q*zh#F[rFj_n7$$\"3n*o% Q!)*)Q7ZF[r$!\"\"F)-F[dm6&F]dmF(F^dmF(-%'SYMBOLG6$%&CROSSG\"#5-%&STYLE G6#%&POINTG-F$6&Ff_nFd`n-Fg`n6$%(DIAMONDGFj`nF[an-F$6&Ff_nFd`n-Fg`n6$% 'CIRCLEGFj`nF[an-F$6&Ff_n-F[dm6&F]dmF)F)F)-Fg`n6$Fhan\"#7F[an-%%TEXTG6 &7$$\"#nFc`n$!#8!\"#Q\"q6\"-%&COLORG6&F]dm$F+FhbnF^cnF^cn-%%FONTG6$Fg` n\"#6F_cn-%+AXESLABELSG6%%!GFfcn-F`cn6#%(DEFAULTG-%*AXESTICKSG6$7(/$\" +^v>Z5!\"*%$p/3G/$\"+.^R%4#Fadn%%2p/3G/$\"+aEfTJFadn%\"pG/$\"+0-z)=%Fa dn%%4p/3G/$\"+dx)fB&Fadn%%5p/3G/$\"+3`=$G'Fadn%#2pGFicn-%%VIEWG6$;F(Fd bn;$!#7Fc`n$F_bnFc`n" 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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "_____________ ______________" }}{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 27 "___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }}{PARA 0 "" 0 "" {TEXT -1 19 "Solve the equation " } {XPPEDIT 18 0 "cos*2*theta = -cos*theta;" "6#/*(%$cosG\"\"\"\"\"#F&%&t hetaGF&,$*&F%F&F(F&!\"\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0 <= the ta;" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``<2*Pi" "6#2%!G*&\"\"#\"\"\"% #PiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 43 "Illustrate the \+ solution graphically as the " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 102 " coordinates of the points of intersection of the graphs of the left and right sides of the equation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*cos^2*theta+cos*theta-1 = 0;" "6#/,(*(\"\"#\"\"\"*$%$cosGF&F'%&thetaGF'F'*&F)F'F*F'F'F'!\"\"\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "cos*theta = 1/2;" "6#/*&%$cosG\"\"\"%& thetaGF&*&F&F&\"\"#!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "-1" "6#,$ \"\"\"!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta = Pi/3,Pi;" "6$/%& thetaG*&%#PiG\"\"\"\"\"$!\"\"F&" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "5* Pi/3;" "6#*(\"\"&\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 588 "f := theta -> cos(2*theta): 'f(theta)'=f(theta);\ng := theta -> -cos(theta ): 'g(theta)'=g(theta);\npi := evalf(Pi):\np1 := plot([f(theta),g(thet a)],theta=0..2*Pi,color=[red,blue],thickness=2):\np2 := plot([[[Pi/3,- 1/2],[Pi,1],[5*Pi/3,-1/2]]$4],style=point,\n symbol=[cross,diamond,c ircle$2],symbolsize=[10$3,12],color=[green$3,black]):\nt1 := plots[tex tplot]([6.7,-.13,`q`],font=[SYMBOL,11],color=COLOR(RGB,.01,.01,.01)): \nplots[display]([p1,p2,t1],labels=[``,``],view=[0..6.7,-1.2..1.2],\nx tickmarks=[pi/3=`p/3`,2*pi/3=`2p/3`,pi=`p`,4*pi/3=`4p/3`,5*pi/3=`5p/3` ,2*pi=`2p`],\n font=[SYMBOL,11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"fG6#%&thetaG-%$cosG6#,$*&\"\"#\"\"\"F'F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%&thetaG,$-%$cosGF&!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 659 249 249 {PLOTDATA 2 "6--%'CURVESG6%7]s7$$\"\"!F)$\"\"\"F )7$$\"3Hjqb\"[W>r\"!#>$\"3k))Hyk!RT***!#=7$$\"3eET6j*))QU$F/$\"3!eFW#H 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"" }}}{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 27 "___________________________" }}{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 "C ode for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 258 "" 0 "" {TEXT -1 47 "Code for circle picture in 1st \+ equation example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 824 "darkgreen := COLOR(RGB,0,.7,0):\nmaroon := CO LOR(RGB,.7,0,.9):\ncs := evalf(cos(Pi/3)):\nsn := evalf(sin(Pi/3)):\np 1 := plot([[cos(t),sin(t),t=0..2*Pi],[0.987*cos(t),0.987*sin(t),t=0..P i/3],\n [1.013*cos(t),1.013*sin(t),t=0..5*Pi/3]],color=[red,blue,mar oon],thickness=2):\np2 := plot([[[0,0],[cs,sn]],[[0,0],[cs,-sn]],\n \+ [[1,0],[-1,0]],[[cs,-sn],[cs,sn]]],\n color=[brown$3,black],thic kness=[2$3,1],linestyle=[1$3,2]):\np3 := plot([[[cs,sn],[cs,-sn],[-1,0 ],[1,0]]$4],style=point,\n symbol=[cross,diamond,circle$2],symbolsiz e=[10$3,12],color=[green$3,black]):\nt1 := plots[textplot]([[1.1,.15,` P`],[-1.1,.15,`Q`],[.6,1,`R`],[.6,-1,`S`],\n [1.25,-.06,`x`],[-.06,1.2 5,`y`]],color=COLOR(RGB,.01,.01,.01)):\nplots[display]([p||(1..4),t1], view=[-1.2..1.3,-1.2..1.25],\n tickmarks=[0,0 ],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 47 "Code for circle picture in 2nd equation example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 813 "d arkgreen := COLOR(RGB,0,.7,0):\nmaroon := COLOR(RGB,.7,0,.9):\ncs := e valf(cos(2*Pi/3)):\nsn := evalf(sin(2*Pi/3)):\np1 := plot([[cos(t),sin (t),t=0..2*Pi],[.987*cos(t),.987*sin(t),t=0..2*Pi/3],\n [1.013*cos(t ),1.013*sin(t),t=0..4*Pi/3]],color=[red,blue,maroon],thickness=2):\np2 := plot([[[0,0],[cs,sn]],[[0,0],[cs,-sn]],\n [[0,0],[1,0]],[[cs ,-sn],[cs,sn]]],\n color=[brown$3,black$3],thickness=[2$3,1],linesty le=[1$3,2]):\np3 := plot([[[cs,sn],[cs,-sn],[1,0]]$4],style=point,\n \+ symbol=[cross,diamond,circle$2],symbolsize=[10$3,12],color=[green$3,b lack]):\nt1 := plots[textplot]([[1.1,.15,`P`],[-.6,1,`Q`],\n [-.6, -1,`R`],[1.25,-.06,`x`],[-.06,1.25,`y`]],color=COLOR(RGB,.01,.01,.01)) :\nplots[display]([p||(1..3),t1],view=[-1.2..1.3,-1.2..1.25],\n \+ tickmarks=[0,0],scaling=constrained);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 258 "" 0 "" {TEXT -1 47 "Code for circle picture in 3rd \+ equation example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 786 "darkgreen := COLOR(RGB,0,.7,0):\nmaroon := CO LOR(RGB,.7,0,.9):\ncs := evalf(cos(Pi/6)):\nsn := evalf(sin(Pi/6)):\np 1 := plot([cos(t),sin(t),t=0..2*Pi],color=red,thickness=2):\np2 := plo t([[[0,0],[cs,sn]],[[0,0],[cs,-sn]],[[0,0],[-cs,sn]],\n [[0,0],[-c s,-sn]]],color=brown,thickness=2):\np3 := plot([[[cs,-sn],[cs,sn]],[[- cs,-sn],[-cs,sn]]],linestyle=2,color=black):\np4 := plot([[[cs,sn],[cs ,-sn],[-cs,sn],[-cs,-sn]]$4],style=point,\n symbol=[cross,diamond,ci rcle$2],symbolsize=[10$3,12],color=[green$3,black]):\nt1 := plots[text plot]([[1.,.54,`P`],[-1.,.54,`Q`],\n [-1.,-.54,`R`],[1.,-.54,`S`], \n [1.25,-.06,`x`],[-.06,1.25,`y`]],color=COLOR(RGB,.01,.01,.01)): \nplots[display]([p||(1..4),t1],view=[-1.2..1.3,-1.2..1.25],\n \+ tickmarks=[0,0],scaling=constrained);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 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