{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 265 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 266 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 267 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 269 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " 261 271 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 272 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 273 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 275 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 257 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 257 280 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 281 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "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 72 "Triple angle formulas, converting products to sums and sums to products " }{TEXT 262 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 23.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Triple angle formulas " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "alpha = theta;" "6#/%&alphaG%&thetaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b eta=2*theta" "6#/%%betaG*&\"\"#\"\"\"%&thetaGF'" }{TEXT -1 16 " in the formula " }{XPPEDIT 18 0 "sin(alpha+beta) = sin*alpha*cos*beta+cos*al pha*sin*beta" "6#/-%$sinG6#,&%&alphaG\"\"\"%%betaGF),&**F%F)F(F)%$cosG F)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 (theta+2*theta) = sin*theta*cos*2*theta+cos*theta*sin*2*theta" "6#/-%$ sinG6#,&%&thetaG\"\"\"*&\"\"#F)F(F)F),&*,F%F)F(F)%$cosGF)F+F)F(F)F)*,F .F)F(F)F%F)F+F)F(F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3*t heta=sin*theta*cos*2*theta+cos*theta*sin*2*theta" "6#/*(%$sinG\"\"\"\" \"$F&%&thetaGF&,&*,F%F&F(F&%$cosGF&\"\"#F&F(F&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 -1 32 "Using the double angle formulas " }{XPPEDIT 18 0 "cos*2 *theta = 1-2*sin^2*theta;" "6#/*(%$cosG\"\"\"\"\"#F&%&thetaGF&,&F&F&*( F'F&*$%$sinGF'F&F(F&!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin*2*t heta = 2*sin*theta*cos*theta;" "6#/*(%$sinG\"\"\"\"\"#F&%&thetaGF&*,F' F&F%F&F(F&%$cosGF&F(F&" }{TEXT -1 11 ", we have: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3 *theta = sin*theta*(1-2*sin^2*theta)+cos*theta*``(2*sin*theta*cos*thet a);" "6#/*(%$sinG\"\"\"\"\"$F&%&thetaGF&,&*(F%F&F(F&,&F&F&*(\"\"#F&*$F %F-F&F(F&!\"\"F&F&*(%$cosGF&F(F&-%!G6#*,F-F&F%F&F(F&F1F&F(F&F&F&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= sin*theta-2*sin^3*theta+2*sin*theta *cos^2*theta" "6#/%!G,(*&%$sinG\"\"\"%&thetaGF(F(*(\"\"#F(*$F'\"\"$F(F )F(!\"\"*,F+F(F'F(F)F(%$cosGF+F)F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "cos^2*theta = 1-sin^2*t heta;" "6#/*&%$cosG\"\"#%&thetaG\"\"\",&F(F(*&%$sinGF&F'F(!\"\"" } {TEXT -1 12 " now gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "sin*3*theta = sin*theta-2*sin^3*theta+2*sin*theta*(1-si n^2*theta);" "6#/*(%$sinG\"\"\"\"\"$F&%&thetaGF&,(*&F%F&F(F&F&*(\"\"#F &*$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 2 "or" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "sin*3*theta = sin*theta-2*sin^3*theta+2*sin*theta-2*sin^3*theta;" "6# /*(%$sinG\"\"\"\"\"$F&%&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 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "sin*3*th eta = 3*sin*theta-4*sin^3*theta;" "6#/*(%$sinG\"\"\"\"\"$F&%&thetaGF&, &*(F'F&F%F&F(F&F&*(\"\"%F&*$F%F'F&F(F&!\"\"" }{TEXT -1 14 " ------- (i ). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 273 13 "_____________" }{TEXT -1 19 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "alpha = th eta;" "6#/%&alphaG%&thetaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta=2 *theta" "6#/%%betaG*&\"\"#\"\"\"%&thetaGF'" }{TEXT -1 16 " in the form ula " }{XPPEDIT 18 0 "cos(alpha+beta) = cos*alpha*cos*beta-sin*alpha*s in*beta;" "6#/-%$cosG6#,&%&alphaG\"\"\"%%betaGF),&**F%F)F(F)F%F)F*F)F) **%$sinGF)F(F)F.F)F*F)!\"\"" }{TEXT -1 8 " gives: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(t heta+2*theta) = cos*theta*cos*2*theta-sin*theta*sin*2*theta;" "6#/-%$c osG6#,&%&thetaG\"\"\"*&\"\"#F)F(F)F),&*,F%F)F(F)F%F)F+F)F(F)F)*,%$sinG F)F(F)F/F)F+F)F(F)!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos *3*theta = cos*theta*cos*2*theta-sin*theta*sin*2*theta;" "6#/*(%$cosG \"\"\"\"\"$F&%&thetaGF&,&*,F%F&F(F&F%F&\"\"#F&F(F&F&*,%$sinGF&F(F&F-F& F+F&F(F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "Using the double angle formulas " } {XPPEDIT 18 0 "cos*2*theta = 1-2*sin^2*theta;" "6#/*(%$cosG\"\"\"\"\"# F&%&thetaGF&,&F&F&*(F'F&*$%$sinGF'F&F(F&!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "sin*2*theta = 2*sin*theta*cos*theta;" "6#/*(%$sinG\"\" \"\"\"#F&%&thetaGF&*,F'F&F%F&F(F&%$cosGF&F(F&" }{TEXT -1 11 ", we have : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "cos*3*theta = cos*theta*(2*cos^2*theta-1)-sin*theta* ``(2*sin*theta*cos*theta);" "6#/*(%$cosG\"\"\"\"\"$F&%&thetaGF&,&*(F%F &F(F&,&*(\"\"#F&*$F%F-F&F(F&F&F&!\"\"F&F&*(%$sinGF&F(F&-%!G6#*,F-F&F1F &F(F&F%F&F(F&F&F/" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "`` = 2*cos^3*theta-cos*theta-2*sin^2*theta*cos*theta ;" "6#/%!G,(*(\"\"#\"\"\"*$%$cosG\"\"$F(%&thetaGF(F(*&F*F(F,F(!\"\"*,F 'F(*$%$sinGF'F(F,F(F*F(F,F(F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "sin^2*theta = 1-cos^2*thet a;" "6#/*&%$sinG\"\"#%&thetaG\"\"\",&F(F(*&%$cosGF&F'F(!\"\"" }{TEXT -1 11 " now gives " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*3*theta = 2*cos^3*theta-cos*theta-2*(1-cos^2*theta)*cos*theta;" " 6#/*(%$cosG\"\"\"\"\"$F&%&thetaGF&,(*(\"\"#F&*$F%F'F&F(F&F&*&F%F&F(F&! \"\"**F+F&,&F&F&*&F%F+F(F&F.F&F%F&F(F&F." }{TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 2 "or" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "cos*3*theta \+ = 2*cos^3*theta-cos*theta-2*cos*theta+2*cos^3*theta;" "6#/*(%$cosG\"\" \"\"\"$F&%&thetaGF&,**(\"\"#F&*$F%F'F&F(F&F&*&F%F&F(F&!\"\"*(F+F&F%F&F (F&F.*(F+F&*$F%F'F&F(F&F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "cos*3*theta = 4*cos ^3*theta-3*cos*theta;" "6#/*(%$cosG\"\"\"\"\"$F&%&thetaGF&,&*(\"\"%F&* $F%F'F&F(F&F&*(F'F&F%F&F(F&!\"\"" }{TEXT -1 15 " ------- (ii). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The formu las (i) and (ii) are the " }{TEXT 259 21 "triple angle formulas" } {TEXT -1 22 " for sine and cosine. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([sin*3*the ta = 3*sin*theta-4*sin^3*theta, ``],[cos*3*theta = 4*cos^3*theta-3*cos *theta, ``]);" "6#-%*PIECEWISEG6$7$/*(%$sinG\"\"\"\"\"$F*%&thetaGF*,&* (F+F*F)F*F,F*F**(\"\"%F**$F)F+F*F,F*!\"\"%!G7$/*(%$cosGF*F+F*F,F*,&*(F 0F**$F7F+F*F,F*F**(F+F*F7F*F,F*F2F3" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 272 14 "______________" }{TEXT -1 9 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 65 "Converting sums and differences of sines and cosines to products " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 31 "C onsider the addition formuas: " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {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 14 " ------- (i) " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(alpha-beta) = sin*alpha*cos*beta-co s*alpha*sin*beta;" "6#/-%$sinG6#,&%&alphaG\"\"\"%%betaG!\"\",&**F%F)F( F)%$cosGF)F*F)F)**F.F)F(F)F%F)F*F)F+" }{TEXT -1 14 " ------- (ii) " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(alpha+beta) = cos *alpha*cos*beta-sin*alpha*sin*beta;" "6#/-%$cosG6#,&%&alphaG\"\"\"%%be taGF),&**F%F)F(F)F%F)F*F)F)**%$sinGF)F(F)F.F)F*F)!\"\"" }{TEXT -1 15 " ------- (iii) " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "co s(alpha-beta) = cos*alpha*cos*beta+sin*alpha*sin*beta;" "6#/-%$cosG6#, &%&alphaG\"\"\"%%betaG!\"\",&**F%F)F(F)F%F)F*F)F)**%$sinGF)F(F)F/F)F*F )F)" }{TEXT -1 14 " ------- (iv) " }}{PARA 0 "" 0 "" {TEXT -1 36 "Addi ng equations (i) and (ii) gives " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "sin(alpha+beta)+sin(alpha-beta) = 2*sin*alpha*cos*beta; " "6#/,&-%$sinG6#,&%&alphaG\"\"\"%%betaGF*F*-F&6#,&F)F*F+!\"\"F**,\"\" #F*F&F*F)F*%$cosGF*F+F*" }{TEXT -1 13 " ------- (v)." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Subsituting " } {XPPEDIT 18 0 "theta = alpha+beta;" "6#/%&thetaG,&%&alphaG\"\"\"%%beta GF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi = alpha-beta;" "6#/%$phiG ,&%&alphaG\"\"\"%%betaG!\"\"" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "alpha = (theta+phi)/2;" "6#/%&alphaG*&,&%&thetaG\"\"\"%$phiGF(F(\" \"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta = (theta-phi)/2;" " 6#/%%betaG*&,&%&thetaG\"\"\"%$phiG!\"\"F(\"\"#F*" }{TEXT -1 7 " gives \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "sin*theta+sin*phi = 2*sin((theta+phi)/2)*cos((theta-phi )/2);" "6#/,&*&%$sinG\"\"\"%&thetaGF'F'*&F&F'%$phiGF'F'*(\"\"#F'-F&6#* &,&F(F'F*F'F'F,!\"\"F'-%$cosG6#*&,&F(F'F*F1F'F,F1F'" }{TEXT -1 14 " -- ----- (vi)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "In words \"" }{TEXT 260 14 "sine plus sine" }{TEXT 258 57 " equ als two times sine half sum times cos half difference" }{TEXT -1 2 "\" ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Subt racting equation (ii) from equation (i) gives " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(alpha+beta)-sin(alpha-beta) = 2*cos *alpha*sin*beta;" "6#/,&-%$sinG6#,&%&alphaG\"\"\"%%betaGF*F*-F&6#,&F)F *F+!\"\"F/*,\"\"#F*%$cosGF*F)F*F&F*F+F*" }{TEXT -1 15 " ------- (vii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Subsi tuting " }{XPPEDIT 18 0 "theta = alpha+beta;" "6#/%&thetaG,&%&alphaG\" \"\"%%betaGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi = alpha-beta;" "6#/%$phiG,&%&alphaG\"\"\"%%betaG!\"\"" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "alpha = (theta+phi)/2;" "6#/%&alphaG*&,&%&thetaG\"\"\"% $phiGF(F(\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta = (theta -phi)/2;" "6#/%%betaG*&,&%&thetaG\"\"\"%$phiG!\"\"F(\"\"#F*" }{TEXT -1 7 " gives " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*theta-sin*phi = 2*cos((theta+phi)/2 )*sin((theta-phi)/2);" "6#/,&*&%$sinG\"\"\"%&thetaGF'F'*&F&F'%$phiGF'! \"\"*(\"\"#F'-%$cosG6#*&,&F(F'F*F'F'F-F+F'-F&6#*&,&F(F'F*F+F'F-F+F'" } {TEXT -1 16 " ------- (viii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 10 "In words \"" }{TEXT 260 15 "sine minus si ne" }{TEXT 258 57 " equals two times cos half sum times sine half diff erence" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 38 "Adding equations (iii) and (iv) gives " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(alpha+beta)+cos(alpha-b eta) = 2*cos*alpha*cos*beta;" "6#/,&-%$cosG6#,&%&alphaG\"\"\"%%betaGF* F*-F&6#,&F)F*F+!\"\"F**,\"\"#F*F&F*F)F*F&F*F+F*" }{TEXT -1 14 " ------ - (ix)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Subsituting " }{XPPEDIT 18 0 "theta = alpha+beta;" "6#/%&thetaG,&% &alphaG\"\"\"%%betaGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi = alph a-beta;" "6#/%$phiG,&%&alphaG\"\"\"%%betaG!\"\"" }{TEXT -1 10 ", so th at " }{XPPEDIT 18 0 "alpha = (theta+phi)/2;" "6#/%&alphaG*&,&%&thetaG \"\"\"%$phiGF(F(\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta = (theta-phi)/2;" "6#/%%betaG*&,&%&thetaG\"\"\"%$phiG!\"\"F(\"\"#F*" } {TEXT -1 7 " gives " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta+cos*phi = 2*cos((theta+p hi)/2)*cos((theta-phi)/2);" "6#/,&*&%$cosG\"\"\"%&thetaGF'F'*&F&F'%$ph iGF'F'*(\"\"#F'-F&6#*&,&F(F'F*F'F'F,!\"\"F'-F&6#*&,&F(F'F*F1F'F,F1F'" }{TEXT -1 13 " ------- (x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "In words \"" }{TEXT 260 12 "cos plus cos" } {TEXT 258 56 " equals two times cos half sum times cos half difference " }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Subtracting equation (iv) from equation (iii) gives " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(alpha+beta)-cos(a lpha-beta) = -2*sin*alpha*sin*beta;" "6#/,&-%$cosG6#,&%&alphaG\"\"\"%% betaGF*F*-F&6#,&F)F*F+!\"\"F/,$*,\"\"#F*%$sinGF*F)F*F3F*F+F*F/" } {TEXT -1 15 " ------- (xi). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Subsituting " }{XPPEDIT 18 0 "theta = alpha+bet a;" "6#/%&thetaG,&%&alphaG\"\"\"%%betaGF'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "phi = alpha-beta;" "6#/%$phiG,&%&alphaG\"\"\"%%betaG!\" \"" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "alpha = (theta+phi)/2;" "6#/%&alphaG*&,&%&thetaG\"\"\"%$phiGF(F(\"\"#!\"\"" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "beta = (theta-phi)/2;" "6#/%%betaG*&,&%&thetaG\"\"\" %$phiG!\"\"F(\"\"#F*" }{TEXT -1 7 " gives " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta-c os*phi = -2*sin((theta+phi)/2)*sin((theta-phi)/2);" "6#/,&*&%$cosG\"\" \"%&thetaGF'F'*&F&F'%$phiGF'!\"\",$*(\"\"#F'-%$sinG6#*&,&F(F'F*F'F'F.F +F'-F06#*&,&F(F'F*F+F'F.F+F'F+" }{TEXT -1 15 " ------- (xii)." }} {PARA 0 "" 0 "" {TEXT -1 1 "\004" }}{PARA 0 "" 0 "" {TEXT -1 10 "In wo rds \"" }{TEXT 260 13 "cos minus cos" }{TEXT 258 18 " equals two times " }{TEXT 260 5 "minus" }{TEXT 258 41 " sine half sum times sine half \+ difference" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "Grouping the formulas (vi), (viii), (x) and (x ii) together, gives the following formulas for converting sums to prod ucts: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([sin*theta+sin*phi = 2*sin((theta+phi) /2)*cos((theta-phi)/2),``],[sin*theta-sin*phi = 2*cos((theta+phi)/2)*s in((theta-phi)/2),``],[cos*theta+cos*phi = 2*cos((theta+phi)/2)*cos((t heta-phi)/2),``],[cos*theta-cos*phi = -2*sin((theta+phi)/2)*sin((theta -phi)/2),``])" "6#-%*PIECEWISEG6&7$/,&*&%$sinG\"\"\"%&thetaGF+F+*&F*F+ %$phiGF+F+*(\"\"#F+-F*6#*&,&F,F+F.F+F+F0!\"\"F+-%$cosG6#*&,&F,F+F.F5F+ F0F5F+%!G7$/,&*&F*F+F,F+F+*&F*F+F.F+F5*(F0F+-F76#*&,&F,F+F.F+F+F0F5F+- F*6#*&,&F,F+F.F5F+F0F5F+F;7$/,&*&F7F+F,F+F+*&F7F+F.F+F+*(F0F+-F76#*&,& F,F+F.F+F+F0F5F+-F76#*&,&F,F+F.F5F+F0F5F+F;7$/,&*&F7F+F,F+F+*&F7F+F.F+ F5,$*(F0F+-F*6#*&,&F,F+F.F+F+F0F5F+-F*6#*&,&F,F+F.F5F+F0F5F+F5F;" } {TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 263 22 "____ __________________" }{TEXT -1 11 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The formulas (v), (vii), (ix) and (xi) give rise to the follow ing formulas for converting products to sums: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIEC EWISE([sin*alpha*cos*beta=(sin(alpha+beta)+sin(alpha-beta))/2,``],[cos *alpha*sin*beta=(sin(alpha+beta)-sin(alpha-beta))/2,``],[cos*alpha*cos *beta=(cos(alpha+beta)+cos(alpha-beta))/2,``],[sin*alpha*sin*beta=(cos (alpha-beta)-cos(alpha+beta))/2,``])" "6#-%*PIECEWISEG6&7$/**%$sinG\" \"\"%&alphaGF*%$cosGF*%%betaGF**&,&-F)6#,&F+F*F-F*F*-F)6#,&F+F*F-!\"\" F*F*\"\"#F6%!G7$/**F,F*F+F*F)F*F-F**&,&-F)6#,&F+F*F-F*F*-F)6#,&F+F*F-F 6F6F*F7F6F87$/**F,F*F+F*F,F*F-F**&,&-F,6#,&F+F*F-F*F*-F,6#,&F+F*F-F6F* F*F7F6F87$/**F)F*F+F*F)F*F-F**&,&-F,6#,&F+F*F-F6F*-F,6#,&F+F*F-F*F6F*F 7F6F8" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 281 22 "______________________" }{TEXT -1 11 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Maple can use these fo rmulas to " }{TEXT 259 24 "convert products to sums" }{TEXT -1 5 " via " }{TEXT 0 7 "combine" }{TEXT -1 54 ", but not to convert sums and di fferences to products." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "2*sin((theta+phi)/2)*cos((theta-phi )/2);\n``=combine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\" \"-%$sinG6#,&*&F%!\"\"%&thetaGF&F&*&F%F,%$phiGF&F&F&-%$cosG6#,&*&F%F,F -F&F&*&F%F,F/F&F,F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%$sinG 6#%&thetaG\"\"\"-F'6#%$phiGF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "2*cos((theta+phi)/2)*sin((th eta-phi)/2);\n``=combine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\" \"#\"\"\"-%$cosG6#,&*&F%!\"\"%&thetaGF&F&*&F%F,%$phiGF&F&F&-%$sinG6#,& *&F%F,F-F&F&*&F%F,F/F&F,F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,& -%$sinG6#%&thetaG\"\"\"-F'6#%$phiG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "2*cos((theta+phi)/2)* cos((theta-phi)/2);\n``=combine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*(\"\"#\"\"\"-%$cosG6#,&*&F%!\"\"%&thetaGF&F&*&F%F,%$phiGF&F&F&-F(6 #,&*&F%F,F-F&F&*&F%F,F/F&F,F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G,&-%$cosG6#%$phiG\"\"\"-F'6#%&thetaGF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "-2*sin((theta+phi)/2) *sin((theta-phi)/2);\n``=combine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$*(\"\"#\"\"\"-%$sinG6#,&*&F%!\"\"%&thetaGF&F&*&F%F,%$phiGF&F&F&-F( 6#,&*&F%F,F-F&F&*&F%F,F/F&F,F&F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% !G,&-%$cosG6#%$phiG!\"\"-F'6#%&thetaG\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Some trigonometric equations " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT 270 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 19 "Solve the equation " }{XPPEDIT 18 0 "sin*3*theta = 2*sin*theta;" "6#/*(%$sinG\"\"\"\"\"$F&%&thetaGF&* (\"\"#F&F%F&F(F&" }{TEXT -1 15 " for values of " }{XPPEDIT 18 0 "theta " "6#%&thetaG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "0 <= theta; " "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``< 2*Pi" "6#2%!G*&\"\"#\"\"\"%#P iGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 271 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 37 "The given equation is equivalent to: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3*theta-sin*theta = sin*theta" "6#/,&*(%$ sinG\"\"\"\"\"$F'%&thetaGF'F'*&F&F'F)F'!\"\"*&F&F'F)F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "Using the formula " }{XPPEDIT 18 0 " sin*theta-sin*phi = 2*cos((theta+phi)/2)*sin((theta-phi)/2);" "6#/,&*& %$sinG\"\"\"%&thetaGF'F'*&F&F'%$phiGF'!\"\"*(\"\"#F'-%$cosG6#*&,&F(F'F *F'F'F-F+F'-F&6#*&,&F(F'F*F+F'F-F+F'" }{TEXT -1 48 ", the last equatio n can be written in the form: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2*cos*2*theta*sin*theta=sin*theta" "6#/*.\"\"#\"\"\"%$c osGF&F%F&%&thetaGF&%$sinGF&F(F&*&F)F&F(F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 25 "which can be written as: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*cos*2*theta*sin*theta-sin*theta=0" "6 #/,&*.\"\"#\"\"\"%$cosGF'F&F'%&thetaGF'%$sinGF'F)F'F'*&F*F'F)F'!\"\"\" \"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "Factoring the lef t side gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin *theta*(2*cos*2*theta-1)=0" "6#/*(%$sinG\"\"\"%&thetaGF&,&**\"\"#F&%$c osGF&F*F&F'F&F&F&!\"\"F&\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "sin*theta=0" "6#/*&%$sinG\"\"\"%&the taGF&\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "theta=0" "6#/%&thetaG \"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 5 " \+ for " }{XPPEDIT 18 0 "theta in ``" "6#-%#inG6$%&thetaG%!G" }{TEXT -1 1 "[" }{XPPEDIT 18 0 "0,2*Pi" "6$\"\"!*&\"\"#\"\"\"%#PiGF&" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 5 "Also " }{XPPEDIT 18 0 "2*cos*2*t heta-1=0" "6#/,&**\"\"#\"\"\"%$cosGF'F&F'%&thetaGF'F'F'!\"\"\"\"!" } {TEXT -1 6 " when " }{XPPEDIT 18 0 "cos*2*theta=1/2" "6#/*(%$cosG\"\" \"\"\"#F&%&thetaGF&*&F&F&F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The last equation is satisfied when " }{XPPEDIT 18 0 "2*t heta=Pi/3,5*Pi/3,7*Pi/3" "6%/*&\"\"#\"\"\"%&thetaGF&*&%#PiGF&\"\"$!\" \"*(\"\"&F&F)F&F*F+*(\"\"(F&F)F&F*F+" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "11*Pi/3" "6#*(\"#6\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "2*theta in``" "6#-%#inG6$*&\"\"#\"\"\"%&thetaGF(%!G" } {TEXT -1 1 "[" }{XPPEDIT 18 0 "0,4*Pi" "6$\"\"!*&\"\"%\"\"\"%#PiGF&" } {TEXT -1 16 "), that is, for " }{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 5 " for " }{XPPEDIT 18 0 "theta in``" "6#-%#inG6 $%&thetaG%!G" }{TEXT -1 1 "[" }{XPPEDIT 18 0 "0,2*Pi" "6$\"\"!*&\"\"# \"\"\"%#PiGF&" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 21 "The so lution set is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "\{ 0,Pi/6, 5*Pi/6, Pi,7*Pi/6,11*Pi/6\}" "6#<(\"\"!*&%#PiG\"\"\"\"\"'!\"\" *(\"\"&F'F&F'F(F)F&*(\"\"(F'F&F'F(F)*(\"#6F'F&F'F(F)" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 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*3*theta;" "6#/-%\"fG6# %&thetaG*(%$sinG\"\"\"\"\"$F*F'F*" }{TEXT -1 11 " (drawn in " }{TEXT 260 3 "red" }{TEXT -1 6 ") and " }{XPPEDIT 18 0 "g(theta) = 2*sin*thet a;" "6#/-%\"gG6#%&thetaG*(\"\"#\"\"\"%$sinGF*F'F*" }{TEXT -1 11 " (dra wn in " }{TEXT 256 4 "blue" }{TEXT -1 55 "), which give the left and r ight sides of the equation " }{XPPEDIT 18 0 "sin*3*theta = 2*sin*theta ;" "6#/*(%$sinG\"\"\"\"\"$F&%&thetaGF&*(\"\"#F&F%F&F(F&" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 630 "theta:='theta':\nf := theta -> sin(3*theta): 'f(thet a)'=f(theta);\ng := theta -> 2*sin(theta): 'g(theta)'=g(theta);\npi := evalf(Pi):\np1 := plot([f(theta),g(theta)],theta=0..2*Pi,color=[red,b lue],thickness=2):\np2 := plot([[[0,0],[Pi/6,1],[5*Pi/6,1],[Pi,0],[7*P i/6,-1],[11*Pi/6,-1]]$4],style=point,\n symbol=[cross,diamond,circle $2],symbolsize=[10$3,12],color=[green$3,black]):\nt1 := plots[textplot ]([6.7,-.13,`q`],font=[SYMBOL,11],color=COLOR(RGB,.01,.01,.01)):\nplot s[display]([p1,p2,t1],labels=[``,``],view=[0..6.7,-2.2..2.2],\nxtickma rks=[pi/3=`p/3`,2*pi/3=`2p/3`,pi=`p`,4*pi/3=`4p/3`,5*pi/3=`5p/3`,2*pi= `2p`],\n font=[SYMBOL,11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"f G6#%&thetaG-%$sinG6#,$*&\"\"$\"\"\"F'F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%&thetaG,$*&\"\"#\"\"\"-%$sinGF&F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 585 220 220 {PLOTDATA 2 "6--%'CURVESG6%7gt7$$\"\" !F)F(7$$\"3eET6j*))QU$!#>$\"3Bq(pyTh`-\"!#=7$$\"3;`#Gi#zxZoF-$\"3%HL') *[S\"*R?F07$$\"3\"zBM*)omr-\"F0$\"3?xGyYI'H.$F07$$\"3i]cC&eb&p8F0$\"3O \\;(RCSS*RF07$$\"3q5)>63x`'>F0$\"3'fMxb5&RgbF07$$\"3YqR*pd)>hDF0$\"3G5 7IDic\\pF07$$\"3Y@%HT;i7B$F0$\"3?RZK[IMX#)F07$$\"3Zs[E^dK,RF0$\"37#[S8 )p04#*F07$$\"3CVu.`$y&QUF0$\"3W\"G/ssecb*F07$$\"3+9+\"[&4$ed%F0$\"3;?T p1KY_p])**F07$$\"3]9*[\">z&z)*F07$$\"3Ls#3kbN; #fF0$\"3?2UwOS>*y*F07$$\"3;I[Vb_GdiF0$\"3q$z#))*\\)GM&*F07$$\"35*QhW& \\$Hf'F0$\"3[.1RO?z#=*F07$$\"3]:5wGaJ:sF0$\"31\\ZkIY@)G)F07$$\"3zS11.f pPyF0$\"3%oj+G`Ib5(F07$$\"3#e!R^M[8#[)F0$\"3T\\Vu&*RC@cF07$$\"3upr'fwt l7*F0$\"3?OZ!*f]]FRF07$$\"3!Rbb2V`Iz*F0$\"3>S8tlJrA?F07$$\"3!QRa&4L&f/ \"!#<$\"3&>!=YeirKP!#?7$$\"3_y%f^`(Q76Ffs$!3$z(>KMlDV>F07$$\"3YjXwg<#) y6Ffs$!3]QS=Ci!p%QF07$$\"3!or.w_drC\"Ffs$!31m7N1PVXcF07$$\"36qGW%H$\\: 8Ffs$!3KG\"33wCv?(F07$$\"3SH22vMov8Ffs$!3_Sf`\")3DN$)F07$$\"3$*)e)pbO( eV\"Ffs$!3weA\\&zP>>*F07$$\"3Q>$*Q*f`(p9Ffs$!3#*=)yMvtSa*F07$$\"3/]+3V Nj.:Ffs$!3-]**F07$$\"3/YhheMXa:Ffs$!3a(Q2rQ$)z)**F07$$\"396 :YIMRr:Ffs$!2'*zGg&R)*****Ffs7$$\"3[K)e%fHS)e\"Ffs$!37_$=[``g)**F07$$ \"3\"Q:c%)[7ag\"Ffs$!3e>f3/f7Y**F07$$\"39vMXoZG6<*F07$$\"33u?N%*p.t#)o9_@)F0 7$$\"3LmSEEVgQ=Ffs$!3S`wCSSYUpF07$$\"3')4G5Xd9)*=Ffs$!3#\\9g<4.Kb&F07$ $\"3S`:%R;(od>Ffs$!3;:$*o&yGs)RF07$$\"3M]e#pT(3$*>Ffs$!3m()p(*=wl#*HF0 7$$\"3^Z,\"*pw[G?Ffs$!37?\"*Rs^Ok>F07$$\"3oWW*G#z)Q1#Ffs$!3')yj!**\\#Q R\"*F-7$$\"3#=uye<)G*4#Ffs$\"35(>x$G$pyY\"F-7$$\"3!zz^8\\l#f@Ffs$\"3OT !Q\\c]Q$>F07$$\"3Ua[#o!GC>AFfs$\"391g5VNZeOF07$$\"3W]72$o5!*G#Ffs$\"3g *faAIvB^&F07$$\"3-YwJf&y(eBFfs$\"3aF?%R%Q@*p#=*F07$$\"3X@Su='ph^#Ffs$ \"34(=A/u[q`*F07$$\"3C24z$*y/]DFfs$\"3u-M:\"yoHz*F07$$\"3O]VJJq)pc#Ffs $\"3we*y:%=:$))*F07$$\"3Y$zP)oh#Re#Ffs$\"3sF>a\\!=y%**F07$$\"3eO7O1`'3 g#Ffs$\"3im>NY/!o)**F07$$\"3EzY)QW/yh#Ffs$\"2kSi^Q)******Ffs7$$\"3&)yS c3X$Rj#Ffs$\"3C9**Q7vc))**F07$$\"3+yMCtX1]EFfs$\"3'Rxo\\'=v`**F07$$\"3 exG#zj%>mEFfs$\"3UI2p^Hj&*)*F07$$\"3=xAg-ZK#o#Ffs$\"3:y7)y$oM9)*F07$$ \"3Ow5'>$[e9FFfs$\"3)f8'\\%ou#Ffs$\"3Ur:%pxR@E*F0 7$$\"3Vz?n#3lT\"GFfs$\"3%\\>94GnwJ)F07$$\"3w$GCS?&[\")GFfs$\"3'HX.rW&3 NqF07$$\"3*Q3$\\!41L%HFfs$\"36+)z(3b#Rg&F07$$\"3Z%)='p(p70IFfs$\"3=m(p 5fi0)RF07$$\"3g`,ta\"4=2$Ffs$\"3/V$e\\@Y#y?F07$$\"3sA%)\\K8\\QJFfs$\"3 N?osV!HQI*Fis7$$\"3#er1NNBJ<$Ffs$!3m`i7x*4^W*F-7$$\"3#*3]^u`v2KFfs$!3% )\\O!=.u=(>F07$$\"3--L_&R(QUKFfs$!3S>=iC5(z(HF07$$\"3c&fJlT>qF$Ffs$!3k 3+_O:&>&RF07$$\"3e!=@(oRJPLFfs$!3CP:qa>-SbF07$$\"39l2\"4_3wR$Ffs$!3-ct ^`EKZpF07$$\"3_]()4*GGFY$Ffs$!3[`ME$R29@)F07$$\"3MOnGd![y_$Ffs$!3W%e&> /r4j\"*F07$$\"3?#yUNg&[hNFfs$!3g7)3Nz?*>&*F07$$\"3iF))z\\J7&f$Ffs$!3)o j?m$>))z(*F07$$\"3N]o#H#>%>h$Ffs$!3e!pSp#>d@6w$Ffs$!3E&[!*znL()e*F07$$\"3%G ,f%[$HSz$Ffs$!3#>t5Xl)Gi#*F07$$\"3eS3B!G4x&QFfs$!3'>,w'o&3!y$)F07$$\"3 KoE+7#*Q@RFfs$!3QjF^(y\"*))=(F07$$\"31LT5>\\4#*RFfs$!3%)*e[(HOKkbF07$$ \"3y(f0ii+G1%Ffs$!3c\"eN@UL.p$F07$$\"3ZG'*z[GLETFfs$!3M!4BU0wF'=F07$$ \"3;fORr]')*=%Ffs$\"3;nJbLUfCKFis7$$\"3))=Rav@yBUFfs$\"3S-rKP@$y/\"F07 $$\"3sxTpz#*pdUFfs$\"3a7o]P\"zD0#F07$$\"3cOW%QQ;;H%Ffs$\"3V?w&zV$4OIF0 7$$\"3R&p%*z[LbK%Ffs$\"3kkDIt6?))RF07$$\"3'epgGO,qQ%Ffs$\"3dww$3z^?g&F 07$$\"3M'pExBp%[WFfs$\"3\"G'R*y`Xf-(F07$$\"3,z&[2')pc^%Ffs$\"3.i^\\/7` 3$)F07$$\"3oh/x$[qGe%Ffs$\"3[7xJ2Gda#*F07$$\"3>](o$ed[9YFfs$\"3ERU(>( \\wr&*F07$$\"3qQq'H.,hk%Ffs$\"3;7RI\\m\"H!)*F07$$\"3Q$=m-n3>m%Ffs$\"3% )4V'e@Zb))*F07$$\"33G`c2jrxYFfs$\"39'[([>+&f%**F07$$\"3)=Zk[%R_$p%Ffs$ \"3_NszV#*)R)**F07$$\"3e;O;#eJ$4ZFfs$\"3mRV:%[Ffs$\"3/CW[T>(*e#*F07$$\"3; @9vt*Qh!\\Ffs$\"3#G)f7w(pxN)F07$$\"3Ou\"4%z!e2(\\Ffs$\"3c%)\\cC&fM9(F0 7$$\"3t:eW%H3%Q]Ffs$\"3mI\"GBpVje&F07$$\"3*zX#[4&eg5&Ffs$\"3)>1?!\\3#* *z$F07$$\"3#Q_\")pq87<&Ffs$\"3JqL&R%[,J>F07$$\"3n*e![/*ojB&Ffs$!3BnSjV _WV6Fis7$$\"3Dtq\\/&**HI&Ffs$!3Y()Q4P$Qo*>F07$$\"3#ob8X5I'p`Ffs$!3AH%[ QS?F!RF07$$\"3fR3_p*3dV&Ffs$!3\\#eT!))*>&RcF07$$\"3PA\"GX$yy,bFfs$!3SG cTaRUbrF07$$\"3wxVy[u]ibFfs$!3w&G/]'o^-$)F07$$\"39L1/jqABcFfs$!3_!y$3% *y([<*F07$$\"3CW>6GG-ecFfs$!3D*=!)\\k'RR&*F07$$\"3WaK=$f=Gp&Ffs$!3]8+ \\Q<1+)*F07$$\"3W5*=dZ;-r&Ffs$!3+e^8^s^!*)*F07$$\"3blXDeVhFdFfs$!3G%Ql ;ONS&**F07$$\"3m?-zSA,XdFfs$!39SAzuIW!***F07$$\"3mweKB,TidFfs$!3ec*z>B T'****F07$$\"3AIu'o*4(zx&Ffs$!3W'f4(pVz%)**F07$$\"3w$)*3/(=`$z&Ffs$!3] $e]_#>>[**F07$$\"3@Q0&Ru#44eFfs$!3#\\v>1l8**))*F07$$\"3w\"4#\\vXOvd&eFfs$!3/'4'=O!Gle*F07$$\"3'oIe;6(*o)eF fs$!3u*zuJ\"QYz#*F07$$\"3yp>qe;E`fFfs$!31oj^nF+f$)F07$$\"3nKcu0ii>gFfs $!3yP<2p#)H3rF07$$\"3G:=>Xb9$3'Ffs$!3+Z\\J.#4uk&F07$$\"3w)*zj%)[mYhFfs $!3aB)p.>p?)RF07$$\"31*\\Lr)\\z!='Ffs$!3RL*QkHKO-$F07$$\"3P***G'*3D\\@ 'Ffs$!3_6ky0S_L?F07$$\"3o*\\C@>b!\\iFfs$!38jKmD`6A5F07$$\"3)****>YH&=$ G'Ffs$!35ndESg(yw$!#D-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG 6#\"\"#-F$6%7enF'7$F<$\"3,GqLfjbIFF07$FF$\"3kgvj3zdm]F07$FP$\"3y+bMM1A 1wF07$Fho$\"3E!ybf=%[-5Ffs7$Ffq$\"3PQ^yJr6D7Ffs7$F`r$\"3&RE9y`3>T\"Ffs 7$Fjr$\"3kzn_2QE#e\"Ffs7$Fds$\"3p*o@bA03t\"Ffs7$F`t$\"3[sSH+JJ[=Ffs7$F jt$\"3;(Q\"[DOFfs7$F_u$\"3U!**HGf^?'>Ffs7$Fdu$\"3=>h#>ZB=)>Ffs7$F^v $\"3+j\"oe&3\\&*>Ffs7$Fbw$\"3/AsYV'*****>Ffs7$Ffx$\"3kv-jb5H&*>Ffs7$F` y$\"3M.Z^e%\\8)>Ffs7$Fey$\"3yRx1KzBf>Ffs7$Fjy$\"3S,&3>^1(G>Ffs7$Fdz$\" 3q%[5.Jt@&=Ffs7$Fh[l$\"3V\")R7>q8F>Wb59Ffs7$Ff]l$\"3d,(=_[H^A\"Ffs7$Fd_l$\"3W(3`:4G.+\"Ffs7$Fb al$\"3C$)z)y%f]\"p(F07$F\\bl$\"3ypL#=x(oV^F07$Ffbl$\"3jA@!p8]3s#F07$F` cl$\"3-(R6(RAj-iFis7$Fddl$!3x*G3Tdj-q#F07$F^el$!3_$yjl\"fck]F07$Fhel$! 3FyfXLpWMvF07$Ffgl$!3i,igP;e^**F07$Fdil$!3=@h'4%)[U@\"Ffs7$F^jl$!3O'=Q 0bpiS\"Ffs7$Fhjl$!3))3a42dm#f\"Ffs7$Fb[m$!3#y,_=Ffs7$F`]m$!3z=?DM(\\2$>Ffs7$Fe]m$!3USnNCiUh>Ffs7$Fj]m$!3=\\F4' R[K)>Ffs7$Fd^m$!3m\">&z;vg&*>Ffs7$Fh_m$!3C*RBBl!****>Ffs7$F\\am$!3A\") )\\WoFg*>Ffs7$Ffam$!3AIAnr$[L)>Ffs7$F[bm$!3Umz(\\AyD'>Ffs7$F`bm$!3]=GI JfhL>Ffs7$Fjbm$!3]5dxYa,Z=Ffs7$Fdcm$!3My$pK`p;t\"Ffs7$F^dm$!3Eu8Ms9O$e \"Ffs7$Fhdm$!3:AGvGWa39Ffs7$Fbem$!3+$>m#Hx;E7Ffs7$F`gm$!3C\\a/eT0^**F0 7$F^im$!3Wxbq%QP*>xF07$Fhim$!39\\A(o#zO5_F07$Fbjm$!3+$zV;([$>s#F07$Ff[ n$!33%Gl4I<>^#Fj[n-F\\\\n6&F^\\nF(F(F_\\nFb\\n-F$6&7(F'7$$\"3;))H)fv() fB&F0$\"\"\"F)7$$\"3T%\\\"*z(Q*zh#FfsFign7$$\"37$z*e`EfTJFfsF(7$$\"3E# 4)=H9>lOFfs$!\"\"F)7$$\"33(G\"eJlefdFfsFdhn-F\\\\n6&F^\\nF(F_\\nF(-%'S YMBOLG6$%&CROSSG\"#5-%&STYLEG6#%&POINTG-F$6&FegnFihn-F\\in6$%(DIAMONDG F_inF`in-F$6&FegnFihn-F\\in6$%'CIRCLEGF_inF`in-F$6&Fegn-F\\\\n6&F^\\nF )F)F)-F\\in6$F]jn\"#7F`in-%%TEXTG6&7$$\"#nFehn$!#8!\"#Q\"q6\"-%&COLORG 6&F^\\n$FjgnF][oFc[oFc[o-%%FONTG6$F\\in\"#6-%+AXESLABELSG6%%!GF[\\o-Fe [o6#%(DEFAULTGFd[o-%*AXESTICKSG6$7(/$\"+^v>Z5!\"*%$p/3G/$\"+.^R%4#Ff\\ o%%2p/3G/$\"+aEfTJFf\\o%\"pG/$\"+0-z)=%Ff\\o%%4p/3G/$\"+dx)fB&Ff\\o%%5 p/3G/$\"+3`=$G'Ff\\o%#2pGF^\\o-%%VIEWG6$;F(Fijn;$!#AFehn$\"#AFehn" 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*3*theta = 2*sin*theta;" "6#/*(%$sinG\"\"\"\"\"$F&%& thetaGF&*(\"\"#F&F%F&F(F&" }{TEXT -1 9 " are the " }{XPPEDIT 18 0 "the ta" "6#%&thetaG" }{TEXT -1 62 " coordinates of the points of intersect ion 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 36 " finds 6 solutions in the interval (" }{XPPEDIT 18 0 "-Pi ,Pi" "6$,$%#PiG!\"\"F$" }{TEXT -1 2 "]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve(sin(3*theta)=2*si n(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%#PiG\"\"!,$*&\"\"'!\"\" F#\"\"\"F(,$*(\"\"&F)F'F(F#F)F(,$*&F'F(F#F)F),$*(F,F)F'F(F#F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 " " 0 "" {TEXT 268 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 19 "Solve the equation " }{XPPEDIT 18 0 "cos*3*theta+2*cos*th eta = 0;" "6#/,&*(%$cosG\"\"\"\"\"$F'%&thetaGF'F'*(\"\"#F'F&F'F)F'F'\" \"!" }{TEXT -1 15 " for values of " }{XPPEDIT 18 0 "theta" "6#%&thetaG " }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "0 <= theta;" "6#1\"\"!%&t hetaG" }{XPPEDIT 18 0 "``< 2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 37 "The given equation is equivalent to: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "cos*3*theta+cos*theta = -cos*theta;" "6#/,&*(%$cosG\"\" \"\"\"$F'%&thetaGF'F'*&F&F'F)F'F',$*&F&F'F)F'!\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 18 "Using the formula " }{XPPEDIT 18 0 "cos* theta+cos*phi = 2*cos((theta+phi)/2)*cos((theta-phi)/2);" "6#/,&*&%$co sG\"\"\"%&thetaGF'F'*&F&F'%$phiGF'F'*(\"\"#F'-F&6#*&,&F(F'F*F'F'F,!\" \"F'-F&6#*&,&F(F'F*F1F'F,F1F'" }{TEXT -1 48 ", the last equation can b e written in the form: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*cos*2*theta*cos*theta = -cos*theta;" "6#/*.\"\"#\"\"\"%$cosGF& F%F&%&thetaGF&F'F&F(F&,$*&F'F&F(F&!\"\"" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 25 "which can be written as: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*cos*2*theta*cos*theta+cos*theta = 0; " "6#/,&*.\"\"#\"\"\"%$cosGF'F&F'%&thetaGF'F(F'F)F'F'*&F(F'F)F'F'\"\"! " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "Factoring the left s ide gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*th eta*(2*cos*2*theta+1) = 0;" "6#/*(%$cosG\"\"\"%&thetaGF&,&**\"\"#F&F%F &F*F&F'F&F&F&F&F&\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "cos*theta = 0;" "6#/*&%$cosG\"\"\"%&thetaGF&\" \"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "theta = Pi/2;" "6#/%&thetaG* &%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "3*Pi/2;" "6 #*(\"\"$\"\"\"%#PiGF%\"\"#!\"\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "t heta in ``" "6#-%#inG6$%&thetaG%!G" }{TEXT -1 1 "[" }{XPPEDIT 18 0 "0, 2*Pi" "6$\"\"!*&\"\"#\"\"\"%#PiGF&" }{TEXT -1 3 "). " }}{PARA 0 "" 0 " " {TEXT -1 5 "Also " }{XPPEDIT 18 0 "2*cos*2*theta+1 = 0;" "6#/,&**\" \"#\"\"\"%$cosGF'F&F'%&thetaGF'F'F'F'\"\"!" }{TEXT -1 6 " when " } {XPPEDIT 18 0 "cos*2*theta = -1/2;" "6#/*(%$cosG\"\"\"\"\"#F&%&thetaGF &,$*&F&F&F'!\"\"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "Th e last equation is satisfied when " }{XPPEDIT 18 0 "2*theta = 2*Pi/3,4 *Pi/3,8*Pi/3;" "6%/*&\"\"#\"\"\"%&thetaGF&*(F%F&%#PiGF&\"\"$!\"\"*(\" \"%F&F)F&F*F+*(\"\")F&F)F&F*F+" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "10* Pi/3;" "6#*(\"#5\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "2*theta in``" "6#-%#inG6$*&\"\"#\"\"\"%&thetaGF(%!G" } {TEXT -1 1 "[" }{XPPEDIT 18 0 "0,4*Pi" "6$\"\"!*&\"\"%\"\"\"%#PiGF&" } {TEXT -1 16 "), that is, for " }{XPPEDIT 18 0 "theta = Pi/3,2*Pi/3,4*P i/3;" "6%/%&thetaG*&%#PiG\"\"\"\"\"$!\"\"*(\"\"#F'F&F'F(F)*(\"\"%F'F&F 'F(F)" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "5*Pi/3;" "6#*(\"\"&\"\"\"%#P iGF%\"\"$!\"\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "theta in``" "6#-%# inG6$%&thetaG%!G" }{TEXT -1 1 "[" }{XPPEDIT 18 0 "0,2*Pi" "6$\"\"!*&\" \"#\"\"\"%#PiGF&" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 21 "The solution set is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "\{Pi/3, Pi/2, 2*Pi/3, 4*Pi/3, 3*Pi/2, 5*Pi/3\};" "6#<(*&%#PiG\"\"\"\" \"$!\"\"*&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 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The follo wing picture shows the graphs of the functions " }{XPPEDIT 18 0 "f(the ta) = cos*3*theta;" "6#/-%\"fG6#%&thetaG*(%$cosG\"\"\"\"\"$F*F'F*" } {TEXT -1 11 " (drawn in " }{TEXT 260 3 "red" }{TEXT -1 6 ") and " } {XPPEDIT 18 0 "g(theta) = -2*cos*theta;" "6#/-%\"gG6#%&thetaG,$*(\"\"# \"\"\"%$cosGF+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*3*theta = -2*cos*theta;" "6#/*(%$cosG\"\"\"\"\" $F&%&thetaGF&,$*(\"\"#F&F%F&F(F&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 635 "theta:=' theta':\nf := theta -> cos(3*theta): 'f(theta)'=f(theta);\ng := theta \+ -> -2*cos(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/3,-1],[Pi/2,0],[2*Pi/3,1],[4*Pi/3,1],[3*Pi/2,0],[5*Pi/3,-1 ]]$4],\n style=point,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,p2,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=[SYMBO L,11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%&thetaG-%$cosG6#, $*&\"\"$\"\"\"F'F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%&the taG,$*&\"\"#\"\"\"-%$cosGF&F+!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 585 220 220 {PLOTDATA 2 "6--%'CURVESG6%7[u7$$\"\"!F)$\"\"\"F)7$$\"3Hjqb\"[ W>r\"!#>$\"3c:W01X\"o)**!#=7$$\"3eET6j*))QU$F/$\"3Czxm&z#HZ**F27$$\"3_ *=rYWLe8&F/$\"3M%p%\\\"4R:))*F27$$\"3;`#Gi#zxZoF/$\"3i>)[DzE(*y*F27$$ \"3\"zBM*)omr-\"F2$\"3KgvMjH'*G&*F27$$\"3i]cC&eb&p8F2$\"31P'\\X8]x;*F2 7$$\"3q5)>63x`'>F2$\"3n\"*Q]%fe:J)F27$$\"3YqR*pd)>hDF2$\"3[gxA@m^!>(F2 7$$\"3Y@%HT;i7B$F2$\"3#oL(o`'>\"ecF27$$\"3Zs[E^dK,RF2$\"3Af%)yvW&y*QF2 7$$\"3CVu.`$y&QUF2$\"3u9Z>*\\wx%HF27$$\"3+9+\"[&4$ed%F2$\"3q>QlE#\\v'> F27$$\"3y%e#ecN38\\F2$\"3)\\f!)zN#)>n*F/7$$\"3ab^NehL]_F2$!3Xfxl=u]/V! #?7$$\"3]9O3@\"pC/#F27 $$\"3;I[Vb_GdiF2$!3J*o)>n,=;IF27$$\"35*QhW&\\$Hf'F2$!3%yfIEfM$fRF27$$ \"3]:5wGaJ:sF2$!3agF'\\lJ^f&F27$$\"3zS11.fpPyF2$!32Z'*3.jVOqF27$$\"3#e !R^M[8#[)F2$!3WKn`Wp_q#)F27$$\"3upr'fwtl7*F2$!3m%*)3*>1X'>*F27$$\"3#=O h$)f8)f%*F2$!31*RVR0>Da*F27$$\"3!Rbb2V`Iz*F2$!3sZ$[$H_H$z*F27$$\"3%*\\ E&pMt'f**F2$!3g0L`9#G@))*F27$$\"3gu\\JE$HE,\"!#<$!3u6K!y([FY**F27$$\"3 4%oMzJ\"HH5Fis$!3_UL#e(\\d&)**F27$$\"3!QRa&4L&f/\"Fis$!3K,S!RLI*****F2 7$$\"3/lc&f'=ci5Fis$!3uul.$3z$*)**F27$$\"31OpNA/%zqcL.a**F2 7$$\"3G2#e(y*yd4\"Fis$!3]N4(f)3)R*)*F27$$\"3_y%f^`(Q76Fis$!3\"*>r3E2P4 )*F27$$\"3+@?'zk/c9\"Fis$!3?n&p)4yPn&*F27$$\"3YjXwg<#)y6Fis$!3;8l$Rqb/ B*F27$$\"3!or.w_drC\"Fis$!3!HA:-1MSD)F27$$\"36qGW%H$\\:8Fis$!3x\"eV()= D>$pF27$$\"3SH22vMov8Fis$!3dHX<#3;[_&F27$$\"3$*)e)pbO(eV\"Fis$!3sJ\"yA Ib!QRF27$$\"3Q>$*Q*f`(p9Fis$!3_@\"[\\As])HF27$$\"3/]+3VNj.:Fis$!3#*GMC MzF,?F27$$\"3q!yqn[8v`\"Fis$!3eOPIM#p#o**F/7$$\"396:YIMRr:Fis$\"3[U=&= /58z\"F^p7$$\"3\"Q:c%)[7ag\"Fis$\"3s`(fZE>m.\"F27$$\"3Z'z]kaJ%R;Fis$\" 35JQf;x`W?F27$$\"39RaW/1Xt;Fis$\"3Q5p8A%z6.$F27$$\"3!=3SCmpuq\"Fis$\"3 oI%=owwi)RF27$$\"33u?N%*p.t(F27$$\"3')4G5Xd9)*=Fis$\"3lu-cFis$\"3 /xYJ=Yrq\"*F27$$\"3M]e#pT(3$*>Fis$\"30-Ry\")zpT&*F27$$\"3^Z,\"*pw[G?Fi s$\"3!\\QZ$[a;0)*F27$$\"3'eH-kz(=Y?Fis$\"3;N))Q^Yg&*)*F27$$\"3oWW*G#z) Q1#Fis$\"3C\\cPl#[\"e**F27$$\"3[$f'Q\\!)e\"3#Fis$\"3%yK@1)*>E***F27$$ \"3#=uye<)G*4#Fis$\"319-y@E#*)***F27$$\"3>1qu/DG9@Fis$\"3]ZS.%Q2A)**F2 7$$\"34q_hLoFH@Fis$\"3U]$zR'oHX**F27$$\"3+MN[i6FW@Fis$\"3!RA>jtl#)))*F 27$$\"3!zz^8\\l#f@Fis$\"3Hm%>BQH7\")*F27$$\"3;E$)3\\TD*=#Fis$\"3]<\"42 d/yf*F27$$\"3Ua[#o!GC>AFis$\"3Xr#e$)p[nI*F27$$\"3W]72$o5!*G#Fis$\"37)e @mf$[V$)F27$$\"3-YwJf&y(eBFis$\"3pc.%R]Rg,(F27$$\"31\"R2:&\\`?CFis$\"3 3J7ABTM$e&F27$$\"3oNrpV8H#[#Fis$\"3%QNao')\\&fRF27$$\"3X@Su='ph^#Fis$ \"3#)fsS#*4W2IF27$$\"3C24z$*y/]DFis$\"36&)=ZuIHC?F27$$\"3Y$zP)oh#Re#Fi s$\"3z#o(*e$HD?5F27$$\"3EzY)QW/yh#Fis$\"3'eU_&)*G-$o&!#@7$$\"3=xAg-ZK# o#Fis$!3,!4>'pm'z\">F27$$\"35v)>8'\\%ou#Fis$!3?eE'yK#)*pPF27$$\"3Vz?n# 3lT\"GFis$!39V\"R@5`7b&F27$$\"3w$GCS?&[\")GFis$!3^]3x)*z'o5(F27$$\"3*Q 3$\\!41L%HFis$!33bkog8F#G)F27$$\"3Z%)='p(p70IFis$!3-?s%f05O<*F27$$\"3D >g%e1o%QIFis$!3%)GYo?UCD&*F27$$\"3g`,ta\"4=2$Fis$!3MzFRQ5m\"y*F27$$\"3 )4As\"*pz%)3$Fis$!3kYXH()RKt)*F27$$\"3$zG9OC]^5$Fis$!3HMmk&G(HS**F27$$ \"3Kbj0)y?=7$Fis$!3gnL>dMT#)**F27$$\"3sA%)\\K8\\QJFis$!3gh\"fW=n&****F 27$$\"3EpD+Vt!e:$Fis$!3%Qfa]z34***F27$$\"3#er1NNBJ<$Fis$!3!z72?-&Hb**F 27$$\"3Oi3,k$R/>$Fis$!3WN<'4&>#G*)*F27$$\"3#*3]^u`v2KFis$!3v[?rL\"eO!) *F27$$\"3--L_&R(QUKFis$!3+RRsv=HY&*F27$$\"3c&fJlT>qF$Fis$!33hF>i=(f=*F 27$$\"3e!=@(oRJPLFis$!3=/M:b?:D$)F27$$\"39l2\"4_3wR$Fis$!3m,bMNVo#>(F2 7$$\"3_]()4*GGFY$Fis$!3m^=ZRvD2dF27$$\"3MOnGd![y_$Fis$!3O7^#om.Z+%F27$ $\"3?#yUNg&[hNFis$!3=$G*\\llAhIF27$$\"3iF))z\\J7&f$Fis$!3cX'>`L-m3#F27 $$\"31t[0'pg(GOFis$!3g&4Yn]Z24\"F27$$\"3#*=4JU#)RiOFis$!3y.?@nDYz$)F^p 7$$\"36m\\Q&z8#GPFis$\"3g&3?z*pUz=F27$$\"3%G,f%[$HSz$Fis$\"3%)y@!Ha;'p PF27$$\"3eS3B!G4x&QFis$\"3y]36u$f(faF27$$\"3KoE+7#*Q@RFis$\"3'ya$*3+Z7 &pF27$$\"31LT5>\\4#*RFis$\"3oKCB&3H*3$)F27$$\"3y(f0ii+G1%Fis$\"3QU$)*p chTH*F27$$\"378E]Pnc%4%Fis$\"3Sy0*)=.9.'*F27$$\"3ZG'*z[GLETFis$\"3)4x_ ,_r\\#)*F27$$\"39O\"[W!f@UTFis$\"3W+J^aia-**F27$$\"3#Qk'4g*)4eTFis$\"3 skm_u?kd**F27$$\"3[^^u:?)R<%Fis$\"3EUX$>\"R8!***F27$$\"3;fORr]')*=%Fis $\"3)f*\\#)4![*****F27$$\"3Y*yoMiBo?%Fis$\"3'=\\a/Qp`)**F27$$\"3))=Rav @yBUFis$\"3!e*\\Eq3&\\%**F27$$\"3I[!>wsS2C%Fis$\"3#pfKK2(zy)*F27$$\"3s xTpz#*pdUFis$\"3ufNW!=zqy*F27$$\"3cOW%QQ;;H%Fis$\"3)*zkFGg'z_*F27$$\"3 R&p%*z[LbK%Fis$\"3k9v'Qy\"Hq\"*F27$$\"3'epgGO,qQ%Fis$\"3ewe9T)QNG)F27$ $\"3M'pExBp%[WFis$\"3OCpx\"\\c& F27$$\"3oh/x$[qGe%Fis$\"3S%3RSy>&)y$F27$$\"3qQq'H.,hk%Fis$\"3!45Id#pbv >F27$$\"3e;O;#eJ$4ZFis$\"3'pP(fbg1s\"*F^p7$$\"3sU'G^sDax%Fis$!3c'=5em] )z=F27$$\"3&)oO4o)>:%[Fis$!3'Rj21wbxx$F27$$\"3;@9vt*Qh!\\Fis$!3Kp`c+2p !\\&F27$$\"3Ou\"4%z!e2(\\Fis$!3Up#[gEFz*pF27$$\"3t:eW%H3%Q]Fis$!3!z'GM ].9%H)F27$$\"3*zX#[4&eg5&Fis$!3WK_.!Q(*)\\#*F27$$\"3Y!*>B3hjQ^Fis$!3'* p8*4PUld*F27$$\"3#Q_\")pq87<&Fis$!3&o5vP*yy6)*F27$$\"3]!Hcj]-v=&Fis$!3 38)z%3&*R%*)*F27$$\"3>d5t08z._Fis$!3'4!\\$=u)Q`**F27$$\"3(Q#e50,3?_Fis $!31P;,gZh))**F27$$\"3n*e![/*ojB&Fis$!2utiEY$******Fis7$$\"3!3r%[al-`_ Fis$!3?I.'\\TQp)**F27$$\"3-J))[/Uop_Fis$!3AS0U=#[*[**F27$$\"39_H\\a=M' G&Fis$!3]Xp0Ex6'))*F27$$\"3Dtq\\/&**HI&Fis$!3S!GR:\"Qg)z*F27$$\"3[:`]/ [JO`Fis$!3/.s053Y]&*F27$$\"3#ob8X5I'p`Fis$!3lt\"4r*e*p?*F27$$\"3fR3_p* 3dV&Fis$!3mpMF-d2e#)F27$$\"3PA\"GX$yy,bFis$!3+'zwg4$p&)pF27$$\"3wxVy[u ]ibFis$!3]6GG-ecF is$!3-\\1#\\F&)***HF27$$\"3WaK=$f=Gp&Fis$!33M5R23n*)>F27$$\"3blXDeVhFd Fis$!3KKx'ff^pd*F/7$$\"3mweKB,TidFis$\"3Q?63KRnq%)F^p7$$\"3w\"4#\\F27$$\"3'oIe;6(*o)eFis$\"3#*\\+o]r8FPF27$$\"3yp>qe;E `fFis$\"3U')yagFis$\"3`eQ?*GSO.(F27$$\"3G:=>Xb 9$3'Fis$\"3wGc;^Do_#)F27$$\"3w)*zj%)[mYhFis$\"3E+vhth&H<*F27$$\"31*\\L r)\\z!='Fis$\"3\"3&z%z!y#>`*F27$$\"3P***G'*3D\\@'Fis$\"3#R`H)Gh0\"z*F2 7$$\"3'*\\n(39!*>B'Fis$\"3q@@2J\"*G#))*F27$$\"3o*\\C@>b!\\iFis$\"39!)G zloiZ**F27$$\"3Q\\APV-7miFis$\"3E#HiQ7)*o)**F27$$\"3)****>YH&=$G'Fis$ \"2M***************Fis-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESS G6#\"\"#-F$6%7en7$F($!\"#F)7$F>$!3;b)[ei7`*>Fis7$FH$!3)R&=([Zs7)>Fis7$ FM$!3'\\fco5(\\h>Fis7$FR$!3]c4/?/wM>Fis7$Ffn$!3??TS'H<(\\=Fis7$Fjo$!3- 0-)[=91t\"Fis7$F_q$!3mF]H3/&3e\"Fis7$Fiq$!3W!R%3O[^;9Fis7$Fcr$!3x\"**Q B:\"HB7Fis7$Fbt$!3q&f;^Ja@+\"Fis7$F`v$!3H?(RdLz-k(F27$Fjv$!3G1Ol1Ex]]F 27$Fdw$!3iI\"Geqt-p#F27$Fhx$\"3!GqmAE2U>\"F^p7$F\\z$\"3DUO#H&F27$F`[l$\"3)y%Q\"\\yF is7$F^cl$\"3-lFis7$Fccl$\"3I:A9-gS\")>Fis7$F]dl$\"3_(yg\"RA8&* >Fis7$Fael$\"3S(>:=Q!****>Fis7$Fefl$\"3Y9k+\"4Cc*>Fis7$F_gl$\"3if*Gcg( o\")>Fis7$Fdgl$\"3qg9A&H:='>Fis7$Figl$\"3.<]6(48[$>Fis7$Fchl$\"370pE[C l_=Fis7$Fgil$\"3#[%4=Ms$[t\"Fis7$Fajl$\"3iPj1kN@*e\"Fis7$F[[m$\"3K>9;* >8@U\"Fis7$Fe[m$\"3$H%G!)*H<(47Fis7$Fc]m$\"3iHG9\"*pP\")**F27$Fa_m$\"3 sF40MJcXvF27$F[`m$\"3nsX[X8L<_F27$Fe`m$\"3Q=9.WP8$e#F27$F_am$\"3@;.i:p y9hF^p7$Fiam$!3Qz:y-jWvDF27$Fcbm$!33[3t'[#35^F27$F]cm$!3%R$RGqlfrwF27$ F[em$!3Q,+\"f4g1+\"Fis7$Fifm$!3TgDzh+(=A\"Fis7$Fcgm$!3D()4zg+')>9Fis7$ F]hm$!3mv91kc.!e\"Fis7$Faim$!3&o2J5\\n[t\"Fis7$F[jm$!35r`'4=+]%=Fis7$F ejm$!3*=`lrwP4$>Fis7$Fjjm$!3!Q)z:bt6g>Fis7$F_[n$!39*HIO5\"R\")>Fis7$Fi [n$!3iK3j^BM&*>Fis7$F]]n$!3y**************>Fis-Fb]n6&Fd]nF(F(Fe]nFh]n- F$6&7(7$$\"3k(f'>^v>Z5Fis$!\"\"F)7$$\"3c'*[zEjzq:FisF(7$$\"3E&>$R-^R%4 #FisF*7$$\"3_!R'y/-z)=%FisF*7$$\"3n*o%Q!)*)Q7ZFisF(7$$\"3$)))H)fv()fB& FisFbin-Fb]n6&Fd]nF(Fe]nF(-%'SYMBOLG6$%&CROSSG\"#5-%&STYLEG6#%&POINTG- F$6&F^inFcjn-Ffjn6$%(DIAMONDGFijnFjjn-F$6&F^inFcjn-Ffjn6$%'CIRCLEGFijn Fjjn-F$6&F^in-Fb]n6&Fd]nF)F)F)-Ffjn6$Fg[o\"#7Fjjn-%%TEXTG6&7$$\"#nFcin $!#8Fa^nQ\"q6\"-%&COLORG6&Fd]n$F+Fa^nF\\]oF\\]o-%%FONTG6$Ffjn\"#6-%+AX ESLABELSG6%%!GFd]o-F^]o6#%(DEFAULTGF]]o-%*AXESTICKSG6$7(/$\"+^v>Z5!\"* %$p/3G/$\"+.^R%4#F_^o%%2p/3G/$\"+aEfTJF_^o%\"pG/$\"+0-z)=%F_^o%%4p/3G/ $\"+dx)fB&F_^o%%5p/3G/$\"+3`=$G'F_^o%#2pGFg]o-%%VIEWG6$;F(Fc\\o;$!#AFc in$\"#AFcin" 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 equa tion " }{XPPEDIT 18 0 "cos*3*theta = -2*cos*theta;" "6#/*(%$cosG\"\"\" \"\"$F&%&thetaGF&,$*(\"\"#F&F%F&F(F&!\"\"" }{TEXT -1 9 " are the " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 62 " coordinates of the p oints of intersection of the two graphs. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 10 ": Maple's " } {TEXT 0 5 "solve" }{TEXT -1 19 " finds 3 solutions." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "solve(cos( 3*theta)=-2*cos(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,$*&\"\"#! \"\"%#PiG\"\"\"F(,$*&\"\"$F&F'F(F(,$*(F%F(F+F&F'F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT 264 8 "Question" }{TEXT -1 22 ": Solve the equation " } {XPPEDIT 18 0 "cos*3*theta+cos*theta = cos*2*theta;" "6#/,&*(%$cosG\" \"\"\"\"$F'%&thetaGF'F'*&F&F'F)F'F'*(F&F'\"\"#F'F)F'" }{TEXT -1 15 " f or values of " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "0<= x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``< 2*P i" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 8 "Solution" }{TEXT -1 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT 279 8 "Method I" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 18 "Using the formula " }{XPPEDIT 18 0 "cos*theta+cos* phi = 2*cos((theta+phi)/2)*cos((theta-phi)/2);" "6#/,&*&%$cosG\"\"\"%& thetaGF'F'*&F&F'%$phiGF'F'*(\"\"#F'-F&6#*&,&F(F'F*F'F'F,!\"\"F'-F&6#*& ,&F(F'F*F1F'F,F1F'" }{TEXT -1 15 ", the equation " }{XPPEDIT 18 0 "cos *3*theta+cos*theta = cos*2*theta;" "6#/,&*(%$cosG\"\"\"\"\"$F'%&thetaG F'F'*&F&F'F)F'F'*(F&F'\"\"#F'F)F'" }{TEXT -1 28 " can be written in th e form " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*cos*2*th eta*cos*theta = cos*2*theta;" "6#/*.\"\"#\"\"\"%$cosGF&F%F&%&thetaGF&F 'F&F(F&*(F'F&F%F&F(F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 " Collecting all the terms on the left of the equation, we have " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*cos*2*theta*cos*the ta-cos*2*theta = 0;" "6#/,&*.\"\"#\"\"\"%$cosGF'F&F'%&thetaGF'F(F'F)F' F'*(F(F'F&F'F)F'!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "co s*2*theta*(2*cos*theta-1) = 0;" "6#/**%$cosG\"\"\"\"\"#F&%&thetaGF&,&* (F'F&F%F&F(F&F&F&!\"\"F&\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "co s*2*theta = 0;" "6#/*(%$cosG\"\"\"\"\"#F&%&thetaGF&\"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "cos*theta = 1/2;" "6#/*&%$cosG\"\"\"%&thetaGF&* &F&F&\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*2*theta = 0;" "6#/* (%$cosG\"\"\"\"\"#F&%&thetaGF&\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "2*theta = Pi/2,3*Pi/2,5*Pi/2,7*Pi/2;" "6&/*&\"\"#\"\"\"%&thetaGF &*&%#PiGF&F%!\"\"*(\"\"$F&F)F&F%F**(\"\"&F&F)F&F%F**(\"\"(F&F)F&F%F*" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0 <= 2*theta;" "6#1\"\"!*&\"\"#\" \"\"%&thetaGF'" }{XPPEDIT 18 0 "``<4*Pi" "6#2%!G*&\"\"%\"\"\"%#PiGF'" }{TEXT -1 17 ", that is, when " }{XPPEDIT 18 0 "x = Pi/4, 3*Pi/4, 5*P i/4, 7*Pi/4" "6&/%\"xG*&%#PiG\"\"\"\"\"%!\"\"*(\"\"$F'F&F'F(F)*(\"\"&F 'F&F'F(F)*(\"\"(F'F&F'F(F)" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0<= x " "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``< 2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF '" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta = 1/2;" "6#/*&%$cosG\"\"\"%&t hetaGF&*&F&F&\"\"#!\"\"" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x=Pi/3" "6#/%\"xG*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x = 5*Pi/3;" "6#/%\"xG*(\"\"&\"\"\"%#PiGF'\"\"$!\"\"" }{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 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 " for the equation " }{XPPEDIT 18 0 " cos*3*theta+cos*theta = cos*2*theta;" "6#/,&*(%$cosG\"\"\"\"\"$F'%&the taGF'F'*&F&F'F)F'F'*(F&F'\"\"#F'F)F'" }{TEXT -1 4 " is " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "\{Pi/4, Pi/3, 3*Pi/4, 5*Pi/4, 5* Pi/3, 7*Pi/4\};" "6#<(*&%#PiG\"\"\"\"\"%!\"\"*&F%F&\"\"$F(*(F*F&F%F&F' F(*(\"\"&F&F%F&F'F(*(F-F&F%F&F*F(*(\"\"(F&F%F&F'F(" }{TEXT -1 2 ". " } {TEXT 266 0 "" }}{PARA 0 "" 0 "" {TEXT 280 9 "Method II" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "Using the double and triple angle fo rmulas: " }{XPPEDIT 18 0 "cos*2*theta = 2*cos^2*theta-1;" "6#/*(%$cosG \"\"\"\"\"#F&%&thetaGF&,&*(F'F&*$F%F'F&F(F&F&F&!\"\"" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "cos*3*theta = 4*cos^3*theta-3*cos*theta;" "6#/*(%$ cosG\"\"\"\"\"$F&%&thetaGF&,&*(\"\"%F&*$F%F'F&F(F&F&*(F'F&F%F&F(F&!\" \"" }{TEXT -1 15 ", the equation " }{XPPEDIT 18 0 "cos*3*theta+cos*the ta = cos*2*theta;" "6#/,&*(%$cosG\"\"\"\"\"$F'%&thetaGF'F'*&F&F'F)F'F' *(F&F'\"\"#F'F)F'" }{TEXT -1 28 " can be written in the form " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4*cos^3*theta-3*cos*t heta+cos*theta = 2*cos^2*theta-1;" "6#/,(*(\"\"%\"\"\"*$%$cosG\"\"$F'% &thetaGF'F'*(F*F'F)F'F+F'!\"\"*&F)F'F+F'F',&*(\"\"#F'*$F)F1F'F+F'F'F'F -" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4*cos^3*theta-2*cos*theta = 2*cos^2*theta-1;" "6#/,&*(\"\"%\"\"\"*$%$cosG\"\"$F'%&thetaGF'F'*(\" \"#F'F)F'F+F'!\"\",&*(F-F'*$F)F-F'F+F'F'F'F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 "Collecting all the terms on the left of the equ ation, we have " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4* cos^3*theta-2*cos^2*theta-2*cos*theta+1 = 0;" "6#/,**(\"\"%\"\"\"*$%$c osG\"\"$F'%&thetaGF'F'*(\"\"#F'*$F)F-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 -1 20 "Temporarily writing " }{XPPEDIT 18 0 "cos*x=c" "6#/*&%$ cosG\"\"\"%\"xGF&%\"cG" }{TEXT -1 40 ", the left side of this equation becomes" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4*c^3-2*c ^2-2*c+1" "6#,**&\"\"%\"\"\"*$%\"cG\"\"$F&F&*&\"\"#F&*$F(F+F&!\"\"*&F+ F&F(F&F-F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 19 "Taking a factor of " }{XPPEDIT 18 0 "2*c^ 2;" "6#*&\"\"#\"\"\"*$%\"cGF$F%" }{TEXT -1 43 " from the first two ter ms, and a factor of " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 34 " from the second two terms gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*c^2*(2*c-1)-1*(2*c-1)" "6#,&*(\"\"#\"\"\"*$%\" cGF%F&,&*&F%F&F(F&F&F&!\"\"F&F&*&F&F&,&*&F%F&F(F&F&F&F+F&F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 44 "so the left side of the equat ion factors as " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(2 *c^2-1)*(2*c-1)" "6#*&,&*&\"\"#\"\"\"*$%\"cGF&F'F'F'!\"\"F',&*&F&F'F)F 'F'F'F*F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "The equatio n therefore becomes: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(2*cos^2*theta-1)*(2*cos*theta-1) = 0;" "6#/*&,&*(\"\"#\"\"\"*$% $cosGF'F(%&thetaGF(F(F(!\"\"F(,&*(F'F(F*F(F+F(F(F(F,F(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }{XPPEDIT 18 0 "cos*theta = 1/2;" "6#/*&%$cosG\"\"\"% &thetaGF&*&F&F&\"\"#!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "1/sqrt(2 )" "6#*&\"\"\"F$-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "-1/sqrt(2)" "6#,$*&\"\"\"F%-%%sqrtG6#\"\"#!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Given t hat " }{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 257 "" 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 "theta = 5*Pi/3;" "6#/%&thetaG*(\"\"&\"\"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta = 1/sqrt(2);" "6# /*&%$cosG\"\"\"%&thetaGF&*&F&F&-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 6 " whe n " }{XPPEDIT 18 0 "theta = Pi/4;" "6#/%&thetaG*&%#PiG\"\"\"\"\"%!\"\" " }{TEXT -1 4 " or " }{XPPEDIT 18 0 "theta = 7*Pi/4;" "6#/%&thetaG*(\" \"(\"\"\"%#PiGF'\"\"%!\"\"" }{TEXT -1 3 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta = -1/sqrt(2);" "6#/*&%$cosG\" \"\"%&thetaGF&,$*&F&F&-%%sqrtG6#\"\"#!\"\"F." }{TEXT -1 6 " when " } {XPPEDIT 18 0 "theta = 3*Pi/4;" "6#/%&thetaG*(\"\"$\"\"\"%#PiGF'\"\"%! \"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "theta = 5*Pi/4;" "6#/%&thetaG *(\"\"&\"\"\"%#PiGF'\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The set of solutions " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "0 <= theta;" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``< 2*Pi" "6#2%!G*& \"\"#\"\"\"%#PiGF'" }{TEXT -1 18 " for the equation " }{XPPEDIT 18 0 " cos*3*theta+cos*theta = cos*2*theta;" "6#/,&*(%$cosG\"\"\"\"\"$F'%&the taGF'F'*&F&F'F)F'F'*(F&F'\"\"#F'F)F'" }{TEXT -1 4 " is " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "\{Pi/4, Pi/3, 3*Pi/4, 5*Pi/4, 5* Pi/3, 7*Pi/4\};" "6#<(*&%#PiG\"\"\"\"\"%!\"\"*&F%F&\"\"$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 2 ". " } {TEXT 267 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(x) = cos*3*theta+cos*theta;" "6#/-%\"fG6#%\"xG,&*( %$cosG\"\"\"\"\"$F+%&thetaGF+F+*&F*F+F-F+F+" }{TEXT -1 11 " (drawn in \+ " }{TEXT 260 3 "red" }{TEXT -1 6 ") and " }{XPPEDIT 18 0 "g(x) = cos*t heta;" "6#/-%\"gG6#%\"xG*&%$cosG\"\"\"%&thetaGF*" }{TEXT -1 11 " (draw n in " }{TEXT 256 4 "blue" }{TEXT -1 55 "), which give the left and ri ght sides of the equation " }{XPPEDIT 18 0 "cos*3*theta+cos*theta = co s*2*theta;" "6#/,&*(%$cosG\"\"\"\"\"$F'%&thetaGF'F'*&F&F'F)F'F'*(F&F' \"\"#F'F)F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 674 "f := x -> cos(3*theta)+cos( theta): 'f(theta)'=f(theta);\ng := x -> cos(2*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/4,0],[Pi/3,-1/2],[3*Pi/ 4,0],[5*Pi/4,0],[5*Pi/3,-1/2],[7*Pi/4,0]]$4],\n style=point,symbol=[cr oss,diamond,circle$2],symbolsize=[10$3,12],color=[green$3,black]):\nt1 := plots[textplot]([6.7,-.2,`q`],font=[SYMBOL,11],color=COLOR(RGB,.01 ,.01,.01)):\nplots[display]([p1,p2,t1],labels=[``,``],view=[0..6.7,-2. 2..2.2],\nxtickmarks=[pi/4=`p/4`,pi/3=`p/3`,pi/2=`p/2`,3*pi/4=`3p/4`,p i=`p`,5*pi/4=`5p/4`,\n 3*pi/2=`3p/2`,5*pi/3=`5p/3`,7*pi/4=`7p/4`,2*p i=`2p`],font=[SYMBOL,11]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6 #%&thetaG,&-%$cosG6#,$*&\"\"$\"\"\"F'F/F/F/-F*F&F/" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"gG6#%&thetaG-%$cosG6#,$*&\"\"#\"\"\"F'F.F." }} {PARA 13 "" 1 "" {GLPLOT2D 585 266 266 {PLOTDATA 2 "6--%'CURVESG6%7jr7 $$\"\"!F)$\"\"#F)7$$\"3Hjqb\"[W>r\"!#>$\"3e#*))[;\\`)*>!#<7$$\"3eET6j* ))QU$F/$\"3!p2Xg=VT*>F27$$\"3_*=rYWLe8&F/$\"3)3\"\\vf`$o)>F27$$\"3;`#G i#zxZoF/$\"3m4$z@**Gm(>F27$$\"3\"zBM*)omr-\"!#=$\"3'R`,fdDw%>F27$$\"3i ]cC&eb&p8FE$\"3q!*3*3D6u!>F27$$\"3q5)>63x`'>FE$\"3`'oyGT/>\"=F27$$\"3Y qR*pd)>hDFE$\"3GaK9s=V'o\"F27$$\"3Y@%HT;i7B$FE$\"3PTWOV$fS^\"F27$$\"3Z s[E^dK,RFE$\"3-14y&4WYJ\"F27$$\"3+9+\"[&4$ed%FE$\"37b1A#yxQ4\"F27$$\"3 ab^NehL]_FE$\"3xZW@]e-5')FE7$$\"3Ls#3kbN;#fFE$\"3!o\\n40\")[D'FE7$$\"3 5*QhW&\\$Hf'FE$\"3VSX%)[u\"\\%RFE7$$\"3]:5wGaJ:sFE$\"3du*fT1=G\">FE7$$ \"3zS11.fpPyFE$\"3<70BLxy8Y!#?7$$\"3upr'fwtl7*FE$!3#e$R@e[**zIFE7$$\"3 !Rbb2V`Iz*FE$!3S,*4')*>I&e[Q2&FE7$$\" 3W$HB5TD8G\"F2$!3u<4*ey#fyZFE7$$\"36qGW%H$\\:8F2$!3jynT&))QlS%FE7$$\"3 SH22vMov8F2$!3S^7o:O/'e$FE7$$\"3$*)e)pbO(eV\"F2$!3omSO\\%=Hf#FE7$$\"39 6:YIMRr:F2$\"33\"\\=2T1U>\"Fip7$$\"3!=3SCmpuq\"F2$\"3g4$)fVWzBEFE7$$\" 33u?N%*p.t>^XFE7$$\"3 ')4G5Xd9)*=F2$\"3UFlN7A-,^FE7$$\"3S`:%R;(od>F2$\"3a_x&es0wR&FE7$$\"3)> qL/H(Qv>F2$\"3B4#\\'=Y)HV&FE7$$\"3M]e#pT(3$*>F2$\"3)GZ^2[%=VaFE7$$\"3: **zTVvy5?F2$\"3sPm^?/xFaFE7$$\"3^Z,\"*pw[G?F2$\"3e%=t*oHQ'Q&FE7$$\"3oW W*G#z)Q1#F2$\"3n70**\\PjC_FE7$$\"3#=uye<)G*4#F2$\"3S_Gw'Q2m&\\FE7$$\"3 !zz^8\\l#f@F2$\"3+$)**o,gMgUFE7$$\"3Ua[#o!GC>AF2$\"37>%o'f#fsE$FE7$$\" 3-YwJf&y(eBF2$!3Y'Hz(R2oFtFip7$$\"31\"R2:&\\`?CF2$!3:B+tQYtF>FE7$$\"3o NrpV8H#[#F2$!3d>zJ1VlWRFE7$$\"3C24z$*y/]DF2$!3))>QBk$4lF'FE7$$\"3EzY)Q W/yh#F2$!3A*fH!*oBOl)FE7$$\"3=xAg-ZK#o#F2$!3i]j8;R<)3\"F27$$\"35v)>8' \\%ou#F2$!31\"Q]y&=4+8F27$$\"3Vz?n#3lT\"GF2$!3$\\go.5)*>]\"F27$$\"3w$G CS?&[\")GF2$!3)fOR#\\\"\\qn\"F27$$\"3*Q3$\\!41L%HF2$!3-GXV#oK'3=F27$$ \"3Z%)='p(p70IF2$!3lHem1S13>F27$$\"3D>g%e1o%QIF2$!3m[4X#z6s%>F27$$\"3g `,ta\"4=2$F2$!3qr'>MAKd(>F27$$\"3)4As\"*pz%)3$F2$!3iGlJPA#f)>F27$$\"3$ zG9OC]^5$F2$!3S:rUxdO$*>F27$$\"3Kbj0)y?=7$F2$!39qNv))e/)*>F27$$\"3sA%) \\K8\\QJF2$!3'\\^`$4>&***>F27$$\"3EpD+Vt!e:$F2$!3%*H!4D&)*)*)*>F27$$\" 3#er1NNBJ<$F2$!3)y@k3XK]*>F27$$\"3Oi3,k$R/>$F2$!3mF27$$\"3#* 3]^u`v2KF2$!377W()eyF27$$\"3--L_&R(QUKF2$!3wkl#op`&\\>F27$$\"3c&fJl T>qF$F2$!3\"evL!*)4W4>F27$$\"3e!=@(oRJPLF2$!3xqg7`GU8=F27$$\"39l2\"4_3 wR$F2$!3ye?4#)\\n'o\"F27$$\"3_]()4*GGFY$F2$!3'\\.^qK.'>:F27$$\"3MOnGd! [y_$F2$!3!Q'f\"3f'zE8F27$$\"3iF))z\\J7&f$F2$!37xL9$3mv5\"F27$$\"3#*=4J U#)RiOF2$!39Wod'z!)zv)FE7$$\"36m\\Q&z8#GPF2$!3cyu\\u4t[kFE7$$\"3%G,f%[ $HSz$F2$!3G4&HuF^k<%FE7$$\"3eS3B!G4x&QF2$!3#G=HL`ZQ3#FE7$$\"3KoE+7#*Q@ RF2$!3G([N\"\\**=$f\"F/7$$\"31LT5>\\4#*RF2$\"34Fo3\")=(Gr\"FE7$$\"3y(f 0ii+G1%F2$\"3wFT)z1vbC$FE7$$\"3ZG'*z[GLETF2$\"3mqRZ\"ovTH%FE7$$\"3;fOR r]')*=%F2$\"3=\"e`U^f#4]FE7$$\"3))=Rav@yBUF2$\"3ohwlq\"*)4D&FE7$$\"3sx Tpz#*pdUF2$\"3#3tx;bV_R&FE7$$\"392$p<$yluUF2$\"3=!*pY-$y6V&FE7$$\"3cOW %QQ;;H%F2$\"3>^Bz@#3LW&FE7$$\"3(fc>f$\\d3VF2$\"37,Ci0m&>V&FE7$$\"3R&p% *z[LbK%F2$\"3y]?%o@5vR&FE7$$\"3'epgGO,qQ%F2$\"3C%\\iA'zx'3&FE7$$\"3M'p ExBp%[WF2$\"3-QYV'HQs]%FE7$$\"3,z&[2')pc^%F2$\"31^\"Q#e)*Q5OFE7$$\"3oh /x$[qGe%F2$\"3AvL-7H&p\\#FE7$$\"3e;O;#eJ$4ZF2$\"3&)=sy(fsY6'Fip7$$\"3& )oO4o)>:%[F2$!3GWo@4E.!\\#FE7$$\"3;@9vt*Qh!\\F2$!3S1%4D-!HlNFE7$$\"3Ou \"4%z!e2(\\F2$!3QXGoAg)GW%FE7$$\"3/&\\Fp=$e/]F2$!3#fr-wuG]![FE7$$\"3t: eW%H3%Q]F2$!3K0<::YR\"4&FE7$$\"3IPT'>SLA2&F2$!3CQ?*f,\"4'H&FE7$$\"3*zX #[4&eg5&F2$!3YlK*[4*49aFE7$$\"3yBs&)3tMA^F2$!3/dBk,!f(QaFE7$$\"3Y!*>B3 hjQ^F2$!33yNBXf(>W&FE7$$\"39dng2\\#\\:&F2$!3:5(H\\NEMU&FE7$$\"3#Q_\")p q87<&F2$!3KGEVa4%GQ&FE7$$\"3>d5t08z._F2$!3\"*p)HVJhZB&FE7$$\"3n*e![/*o jB&F2$!3YIF6$)Hp'*\\FE7$$\"3Dtq\\/&**HI&F2$!3mR-TE:#*HUFE7$$\"3#ob8X5I 'p`F2$!3erj9)eXw4$FE7$$\"3PA\"GX$yy,bF2$\"3r-9yy?2O6F/7$$\"3wxVy[u]ibF 2$\"3M8\"3=5*pR>FE7$$\"39L1/jqABcF2$\"3z1h(f')QD#RFE7$$\"3WaK=$f=Gp&F2 $\"3XwT>r^pwW>W!f()FE7$$\"3w\"4#\\qe;E`fF2 $\"3M%=/PU[\\\\\"F27$$\"3nKcu0ii>gF2$\"3!=:.D\"H$)o;F27$$\"3G:=>Xb9$3' F2$\"3yabpKpK0=F27$$\"3w)*zj%)[mYhF2$\"3f**oF27$$\"31*\\Lr)\\z! ='F2$\"3A_:%*pa&z%>F27$$\"3P***G'*3D\\@'F2$\"3#)p$)p)ywn(>F27$$\"3'*\\ n(39!*>B'F2$\"3Sg)yns=p)>F27$$\"3o*\\C@>b!\\iF2$\"39o\"Q2J!=%*>F27$$\" 3Q\\APV-7miF2$\"3UX%=x?W&)*>F27$$\"3)****>YH&=$G'F2$\"37************** >F2-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#F+-F$6%7]s7$F($ \"\"\"F)7$F-$\"3k))Hyk!RT***FE7$F4$\"3!eFW#HJcw**FE7$F9$\"3Czxm&z#HZ** FE7$F>$\"3#*p#fVPij!**FE7$FC$\"3i>)[DzE(*y*FE7$FI$\"3>0UU)4.si*FE7$FN$ \"3;kR.,XNP#*FE7$FS$\"3F#H&y9%*[;()FE7$FX$\"3k#p9vzhM)zFE7$Fgn$\"3aOG3 $Qqs5(FE7$F\\o$\"3)\\gKF])e'4'FE7$Fao$\"3M#)Ga%GF^(\\FE7$Ffo$\"3KMt_vR ?pPFE7$F[p$\"3[zF.n+W&\\#FE7$F`p$\"3EJ](pzhQF\"FE7$Fep$\"3mDNfB/9dKFip 7$$\"3#e!R^M[8#[)FE$!3'f`c>G/ID\"FE7$F[q$!3L5d/$y$z$*Q*f`(p9F2$!3#R_h,J,lz*FE7$$\"3/]+3VNj.:F2$!3rssEz( =*4**FE7$$\"3P:a#\\^t0_\"F2$!3!ez[My&f\\**FE7$$\"3q!yqn[8v`\"F2$!3-Jjo IK&y(**FE7$$\"3/YhheMXa:F2$!3G4]v)oeY***FE7$Ffu$!2M2X$pG******F27$$\"3 [K)e%fHS)e\"F2$!3uB\"Q.x+Q***FE7$$\"3\"Q:c%)[7ag\"F2$!3k*o&)z*R/w**FE7 $$\"39vMX$f!**FE7$ $\"39RaW/1Xt;F2$!31Ow(=E\")**y*FE7$F[v$!3#*RDy$QG(G'*FE7$F`v$!3;n`Esj1 $>*FE7$Fev$!3!3H\"RXSa*f)FE7$Fjv$!3r)GOf[8B$zFE7$F_w$!3)f\\Kr&)HF:(FE7 $Fcx$!3E=^**zA([4'FE7$F]y$!3whB999,:\\FE7$Fby$!3_z-$*Q(Qv$QFE7$Fgy$!3G Y2*RY8\\q#FE7$$\"3W]72$o5!*G#F2$!3PDuZAljR8FE7$F\\z$\"3'H..TC:\"o^Fip7 $Faz$\"3iOqkL3E$G\"FE7$Ffz$\"3m&zQAs)G&\\#FE7$F[[l$\"3%H^%GDUm!y$FE7$F `[l$\"3'*)y[ia=n*\\FE7$Fe[l$\"3%3y]nXV)pgFE7$Fj[l$\"3+)=!o7o.UqFE7$F_ \\l$\"3kulq9;OJzFE7$Fd\\l$\"3#ot*H0Q7x')FE7$Fi\\l$\"3b#p**oq+RA*FE7$F^ ]l$\"3o%)fiE([)H'*FE7$Fc]l$\"3o)[ofHf!)y*FE7$Fh]l$\"3z*o$peKw-**FE7$F] ^l$\"3Qhre9LjV**FE7$Fb^l$\"3GT.7r2Xt**FE7$Fg^l$\"3gTyV&[#=#***FE7$F\\_ l$\"3_%\\mnj2)****FE7$Fa_l$\"3#f!>FC\"ff***FE7$Ff_l$\"3MC3u()G7!)**FE7 $F[`l$\"3M)*3*)=zJ_**FE7$F``l$\"3W*)e-jvd7**FE7$Fe`l$\"3GaIH#e&\\(z*FE 7$Fj`l$\"3`'><:$)Gaj*FE7$F_al$\"36NV#GE'fV#*FE7$Fdal$\"3gA!RQg3vr)FE7$ Fial$\"3VN!*yuHP2!)FE7$F^bl$\"3=!zY.Z0;;(FE7$Fcbl$\"3E8BJHXjghFE7$Fhbl $\"3i`H]X6I[]FE7$F]cl$\"3]Z%3F/V;(QFE7$Fbcl$\"3Nc@1h()*zi#FE7$Fgcl$\"3 -*R&zM5?\"Q\"FE7$F\\dl$\"3!HNc9c(H?6F/7$Fadl$!3;sx#>(eS)H\"FE7$Ffdl$!3 ![f)ys-#Ho#FE7$F[el$!37a1'f NUhFE7$F^gl$!3%od-&\\QA`rFE7$Fcgl$!3)y9[m(R9czFE7$Fhgl$!3Y17d\"os*Q')F E7$F]hl$!3<%prz;kfB*FE7$Fbhl$!3c6B-.5Pm'*FE7$$\"3>](o$ed[9YF2$!3i#)o(o )3\"*3)*FE7$$\"3qQq'H.,hk%F2$!3qctW0oC7**FE7$$\"3Q$=m-n3>m%F2$!3gf)=?/ y!\\**FE7$$\"33G`c2jrxYF2$!3#p'oy[d'f(**FE7$$\"3)=Zk[%R_$p%F2$!3M-*QI0 $)G***FE7$Fghl$!3#H)[!p/8)****FE7$$\"3ns[!z6bes%F2$!3?F3-#[tj***FE7$$ \"3lHhk`'yBu%F2$!3W2a(=yB_Zgx;NDFE7$$\"3fR3_p*3dV&F2$ !3u!4#)3?w$Q7FE7$Fg]m$\"3(4*Hp9hH,!)Fip7$F\\^m$\"3M\\'Hc_%z!H\"FE7$Fa^ m$\"3+S4`)\\jD[#FE7$Ff^m$\"3I,5V+fY-QFE7$F[_m$\"3[=B=dg#)[]FE7$F`_m$\" 3Um+'>@R:3'FE7$Fe_m$\"33g7\")Q$G,-(FE7$Fj_m$\"3-\"*['4Ze3!zFE7$F_`m$\" 3)*[5L.LgU')FE7$Fd`m$\"3>Ww&*H-I5#*FE7$Fi`m$\"3LuksFNbH'*FE7$F^am$\"3u _(pV:c5z*FE7$Fcam$\"3p2$e5]bp!**FE7$Fham$\"3+G-jyoiZ**FE7$F]bm$\"3'yh( e\\^v>Z5F2$!3+++++ +++]FE7$$\"3%[M#>!\\%>cBF2F(7$$\"3RTs)p\"3*p#RF2F(7$$\"3$)))H)fv()fB&F 2Fd_o7$$\"3'z8#yVry(\\&F2F(-F\\cm6&F^cmF(F_cmF(-%'SYMBOLG6$%&CROSSG\"# 5-%&STYLEG6#%&POINTG-F$6&F]_oFb`o-Fe`o6$%(DIAMONDGFh`oFi`o-F$6&F]_oFb` o-Fe`o6$%'CIRCLEGFh`oFi`o-F$6&F]_o-F\\cm6&F^cmF)F)F)-Fe`o6$Ffao\"#7Fi` o-%%TEXTG6&7$$\"#n!\"\"$!\"#FdboQ\"q6\"-%&COLORG6&F^cm$FjcmFfboF\\coF \\co-%%FONTG6$Fe`o\"#6-%+AXESLABELSG6%%!GFdco-F^co6#%(DEFAULTGF]co-%*A XESTICKSG6$7,/$\"+N;)R&y!#5%$p/4G/$\"+^v>Z5!\"*%$p/3G/$\"+Fjzq:Fddo%$p /2G/$\"+!\\%>cBFddo%%3p/4G/$\"+aEfTJFddo%\"pG/$\"+=3*p#RFddo%%5p/4G/$ \"+\")*)Q7ZFddo%%3p/2G/$\"+dx)fB&Fddo%%5p/3G/$\"+Wry(\\&Fddo%%7p/4G/$ \"+3`=$G'Fddo%#2pGFgco-%%VIEWG6$;F(Fbbo;$!#AFdbo$\"#AFdbo" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 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* 3*theta+cos*theta = cos*2*theta;" "6#/,&*(%$cosG\"\"\"\"\"$F'%&thetaGF 'F'*&F&F'F)F'F'*(F&F'\"\"#F'F)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 259 4 "Note" }{TEXT -1 10 ": Maple's " }{TEXT 0 5 "solve" }{TEXT -1 36 " finds 3 solutions for the equation " } {XPPEDIT 18 0 "cos*3*theta+cos*theta = cos*2*theta;" "6#/,&*(%$cosG\" \"\"\"\"$F'%&thetaGF'F'*&F&F'F)F'F'*(F&F'\"\"#F'F)F'" }{TEXT -1 15 " b etween 0 and " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "co s(3*theta)+cos(theta)-cos(2*theta)=0;\nexpand(%);\nfactor(%);\nsolve(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%$cosG6#,$*&\"\"$\"\"\"%&the taGF+F+F+-F&6#F,F+-F&6#,$*&\"\"#F+F,F+F+!\"\"\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,**&\"\"%\"\"\")-%$cosG6#%&thetaG\"\"$F'F'*&\"\"#F'F )F'!\"\"*&F/F')F)F/F'F0F'F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/* &,&*&\"\"#\"\"\"-%$cosG6#%&thetaGF(F(F(!\"\"F(,&*&F'F()F)F'F(F(F(F-F( \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,$*&\"\"$!\"\"%#PiG\"\"\"F(,$ *&\"\"%F&F'F(F(,$*(F%F(F+F&F'F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 4 " }}{PARA 0 "" 0 "" {TEXT 274 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 19 "Solve the equation " } {XPPEDIT 18 0 "cos*3*theta = sin*theta;" "6#/*(%$cosG\"\"\"\"\"$F&%&th etaGF&*&%$sinGF&F(F&" }{TEXT -1 15 " 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 275 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 95 "This equation can be solved by a different approach than that u sed for the previous equations. " }}{PARA 0 "" 0 "" {TEXT -1 16 "First note that " }{XPPEDIT 18 0 "cos*alpha=cos*beta" "6#/*&%$cosG\"\"\"%&a lphaGF&*&F%F&%%betaGF&" }{TEXT -1 22 " excactly when either " } {XPPEDIT 18 0 "alpha=beta+2*k*Pi" "6#/%&alphaG,&%%betaG\"\"\"*(\"\"#F' %\"kGF'%#PiGF'F'" }{TEXT -1 8 ", where " }{TEXT 276 1 "k" }{TEXT -1 19 " is an integer, or " }{XPPEDIT 18 0 "alpha=-beta+2*k*Pi" "6#/%&alp haG,&%%betaG!\"\"*(\"\"#\"\"\"%\"kGF*%#PiGF*F*" }{TEXT -1 8 ", where \+ " }{TEXT 277 1 "k" }{TEXT -1 16 " is an integer. " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "cos*3*theta = sin*theta" " 6#/*(%$cosG\"\"\"\"\"$F&%&thetaGF&*&%$sinGF&F(F&" }{TEXT -1 18 " is eq uivalent to " }{XPPEDIT 18 0 "cos*3*theta=cos(Pi/2-theta)" "6#/*(%$cos G\"\"\"\"\"$F&%&thetaGF&-F%6#,&*&%#PiGF&\"\"#!\"\"F&F(F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*theta=Pi/2-theta+2*k*Pi" "6#/*&\"\"$\"\"\"% &thetaGF&,(*&%#PiGF&\"\"#!\"\"F&F'F,*(F+F&%\"kGF&F*F&F&" }{TEXT -1 14 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "3*theta=theta-Pi/2+2*k*Pi" "6#/*&\"\" $\"\"\"%&thetaGF&,(F'F&*&%#PiGF&\"\"#!\"\"F,*(F+F&%\"kGF&F*F&F&" } {TEXT -1 15 " ------- (ii), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {TEXT 278 1 "k" }{TEXT -1 16 " is an integer. " }}{PARA 0 "" 0 "" {TEXT -1 20 "Equation (i) gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4*theta=Pi/2+2*k*Pi" "6#/*&\"\"%\"\"\"%&thetaGF&,&*&%# PiGF&\"\"#!\"\"F&*(F+F&%\"kGF&F*F&F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "theta=Pi/8+k*Pi/2" "6#/%&thetaG,&*&%#PiG\"\"\"\"\")!\" \"F(*(%\"kGF(F'F(\"\"#F*F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "With the restriction " }{XPPEDIT 18 0 "0 <= theta;" "6#1\"\"!%& thetaG" }{XPPEDIT 18 0 "``< 2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 42 ", we obtain from this equation the values " }{XPPEDIT 18 0 "the ta=Pi/8,5*Pi/8,9*Pi/8,13*Pi/8" "6&/%&thetaG*&%#PiG\"\"\"\"\")!\"\"*(\" \"&F'F&F'F(F)*(\"\"*F'F&F'F(F)*(\"#8F'F&F'F(F)" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 21 "Equation (ii) gives: " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "2*theta = -Pi/2+2*k*Pi;" "6#/*&\"\"# \"\"\"%&thetaGF&,&*&%#PiGF&F%!\"\"F+*(F%F&%\"kGF&F*F&F&" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta = -Pi/4+k*Pi;" "6#/%&thetaG,&*&%# PiG\"\"\"\"\"%!\"\"F**&%\"kGF(F'F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "With the restriction " }{XPPEDIT 18 0 "0 <= theta;" "6 #1\"\"!%&thetaG" }{XPPEDIT 18 0 "``< 2*Pi" "6#2%!G*&\"\"#\"\"\"%#PiGF' " }{TEXT -1 42 ", we obtain from this equation the values " }{XPPEDIT 18 0 "theta = 3*Pi/4,7*Pi/4;" "6$/%&thetaG*(\"\"$\"\"\"%#PiGF'\"\"%!\" \"*(\"\"(F'F(F'F)F*" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "The solution set is: " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "\{Pi/8, 5/8*Pi, 3/4*Pi, 9/8*Pi, \+ 13/8*Pi, 7/4*Pi\};" "6#<(*&%#PiG\"\"\"\"\")!\"\"*(\"\"&F&F'F(F%F&*(\" \"$F&\"\"%F(F%F&*(\"\"*F&F'F(F%F&*(\"#8F&F'F(F%F&*(\"\"(F&F-F(F%F&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 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*3*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) = 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 equation " }{XPPEDIT 18 0 "cos*3*theta = sin*t heta;" "6#/*(%$cosG\"\"\"\"\"$F&%&thetaGF&*&%$sinGF&F(F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 722 "theta:='theta':\nf := theta -> cos(3*theta): 'f(thet a)'=f(theta);\ng := theta -> sin(theta): 'g(theta)'=g(theta);\npi := e valf(Pi):\np1 := plot([f(theta),g(theta)],theta=0..2*Pi,color=[red,blu e],thickness=2):\np2 := plot([[[Pi/8,sin(Pi/8)],[5*Pi/8,sin(5*Pi/8)],[ 3*Pi/4,sqrt(2)/2],\n [9*Pi/8,sin(9*Pi/8)],[13*Pi/8,sin(13*Pi/8)],[7 *Pi/4,-sqrt(2)/2]]$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[d isplay]([p1,p2,t1],labels=[``,``],view=[0..6.7,-1.2..1.2],\nxtickmarks =[pi/4=`p/4`,pi/2=`p/2`,3*pi/4=`3p/4`,pi=`p`,5*pi/4=`5p/4`,3*pi/2=`3p/ 2`,\n 7*pi/4=`7p/4`,2*pi=`2p`],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 585 220 220 {PLOTDATA 2 "6--%'CURVESG6%7[u7$$\"\" !F)$\"\"\"F)7$$\"3Hjqb\"[W>r\"!#>$\"3c:W01X\"o)**!#=7$$\"3eET6j*))QU$F /$\"3Czxm&z#HZ**F27$$\"3_*=rYWLe8&F/$\"3M%p%\\\"4R:))*F27$$\"3;`#Gi#zx ZoF/$\"3i>)[DzE(*y*F27$$\"3\"zBM*)omr-\"F2$\"3KgvMjH'*G&*F27$$\"3i]cC& eb&p8F2$\"31P'\\X8]x;*F27$$\"3q5)>63x`'>F2$\"3n\"*Q]%fe:J)F27$$\"3YqR* pd)>hDF2$\"3[gxA@m^!>(F27$$\"3Y@%HT;i7B$F2$\"3#oL(o`'>\"ecF27$$\"3Zs[E ^dK,RF2$\"3Af%)yvW&y*QF27$$\"3CVu.`$y&QUF2$\"3u9Z>*\\wx%HF27$$\"3+9+\" [&4$ed%F2$\"3q>QlE#\\v'>F27$$\"3y%e#ecN38\\F2$\"3)\\f!)zN#)>n*F/7$$\"3 ab^NehL]_F2$!3Xfxl=u]/V!#?7$$\"3]9O3@\"pC/#F27$$\"3;I[Vb_GdiF2$!3J*o)>n,=;IF27$$\"35*QhW&\\$ Hf'F2$!3%yfIEfM$fRF27$$\"3]:5wGaJ:sF2$!3agF'\\lJ^f&F27$$\"3zS11.fpPyF2 $!32Z'*3.jVOqF27$$\"3#e!R^M[8#[)F2$!3WKn`Wp_q#)F27$$\"3upr'fwtl7*F2$!3 m%*)3*>1X'>*F27$$\"3#=Oh$)f8)f%*F2$!31*RVR0>Da*F27$$\"3!Rbb2V`Iz*F2$!3 sZ$[$H_H$z*F27$$\"3%*\\E&pMt'f**F2$!3g0L`9#G@))*F27$$\"3gu\\JE$HE,\"!# <$!3u6K!y([FY**F27$$\"34%oMzJ\"HH5Fis$!3_UL#e(\\d&)**F27$$\"3!QRa&4L&f /\"Fis$!3K,S!RLI*****F27$$\"3/lc&f'=ci5Fis$!3uul.$3z$*)**F27$$\"31OpNA /%zqcL.a**F27$$\"3G2#e(y*yd4\"Fis$!3]N4(f)3)R*)*F27$$\"3_y% f^`(Q76Fis$!3\"*>r3E2P4)*F27$$\"3+@?'zk/c9\"Fis$!3?n&p)4yPn&*F27$$\"3Y jXwg<#)y6Fis$!3;8l$Rqb/B*F27$$\"3!or.w_drC\"Fis$!3!HA:-1MSD)F27$$\"36q GW%H$\\:8Fis$!3x\"eV()=D>$pF27$$\"3SH22vMov8Fis$!3dHX<#3;[_&F27$$\"3$* )e)pbO(eV\"Fis$!3sJ\"yAIb!QRF27$$\"3Q>$*Q*f`(p9Fis$!3_@\"[\\As])HF27$$ \"3/]+3VNj.:Fis$!3#*GMCMzF,?F27$$\"3q!yqn[8v`\"Fis$!3eOPIM#p#o**F/7$$ \"396:YIMRr:Fis$\"3[U=&=/58z\"F^p7$$\"3\"Q:c%)[7ag\"Fis$\"3s`(fZE>m.\" F27$$\"3Z'z]kaJ%R;Fis$\"35JQf;x`W?F27$$\"39RaW/1Xt;Fis$\"3Q5p8A%z6.$F2 7$$\"3!=3SCmpuq\"Fis$\"3oI%=owwi)RF27$$\"33u?N%*p.t(F27$$\"3')4G5Xd9)*=Fis$\"3lu-cFis$\"3/xYJ=Yrq\"*F27$$\"3M]e#pT(3$*>Fis$\"30-Ry\")zp T&*F27$$\"3^Z,\"*pw[G?Fis$\"3!\\QZ$[a;0)*F27$$\"3'eH-kz(=Y?Fis$\"3;N)) Q^Yg&*)*F27$$\"3oWW*G#z)Q1#Fis$\"3C\\cPl#[\"e**F27$$\"3[$f'Q\\!)e\"3#F is$\"3%yK@1)*>E***F27$$\"3#=uye<)G*4#Fis$\"319-y@E#*)***F27$$\"3>1qu/D G9@Fis$\"3]ZS.%Q2A)**F27$$\"34q_hLoFH@Fis$\"3U]$zR'oHX**F27$$\"3+MN[i6 FW@Fis$\"3!RA>jtl#)))*F27$$\"3!zz^8\\l#f@Fis$\"3Hm%>BQH7\")*F27$$\"3;E $)3\\TD*=#Fis$\"3]<\"42d/yf*F27$$\"3Ua[#o!GC>AFis$\"3Xr#e$)p[nI*F27$$ \"3W]72$o5!*G#Fis$\"37)e@mf$[V$)F27$$\"3-YwJf&y(eBFis$\"3pc.%R]Rg,(F27 $$\"31\"R2:&\\`?CFis$\"33J7ABTM$e&F27$$\"3oNrpV8H#[#Fis$\"3%QNao')\\&f RF27$$\"3X@Su='ph^#Fis$\"3#)fsS#*4W2IF27$$\"3C24z$*y/]DFis$\"36&)=ZuIH C?F27$$\"3Y$zP)oh#Re#Fis$\"3z#o(*e$HD?5F27$$\"3EzY)QW/yh#Fis$\"3'eU_&) *G-$o&!#@7$$\"3=xAg-ZK#o#Fis$!3,!4>'pm'z\">F27$$\"35v)>8'\\%ou#Fis$!3? eE'yK#)*pPF27$$\"3Vz?n#3lT\"GFis$!39V\"R@5`7b&F27$$\"3w$GCS?&[\")GFis$ !3^]3x)*z'o5(F27$$\"3*Q3$\\!41L%HFis$!33bkog8F#G)F27$$\"3Z%)='p(p70IFi s$!3-?s%f05O<*F27$$\"3D>g%e1o%QIFis$!3%)GYo?UCD&*F27$$\"3g`,ta\"4=2$Fi s$!3MzFRQ5m\"y*F27$$\"3)4As\"*pz%)3$Fis$!3kYXH()RKt)*F27$$\"3$zG9OC]^5 $Fis$!3HMmk&G(HS**F27$$\"3Kbj0)y?=7$Fis$!3gnL>dMT#)**F27$$\"3sA%)\\K8 \\QJFis$!3gh\"fW=n&****F27$$\"3EpD+Vt!e:$Fis$!3%Qfa]z34***F27$$\"3#er1 NNBJ<$Fis$!3!z72?-&Hb**F27$$\"3Oi3,k$R/>$Fis$!3WN<'4&>#G*)*F27$$\"3#*3 ]^u`v2KFis$!3v[?rL\"eO!)*F27$$\"3--L_&R(QUKFis$!3+RRsv=HY&*F27$$\"3c&f JlT>qF$Fis$!33hF>i=(f=*F27$$\"3e!=@(oRJPLFis$!3=/M:b?:D$)F27$$\"39l2\" 4_3wR$Fis$!3m,bMNVo#>(F27$$\"3_]()4*GGFY$Fis$!3m^=ZRvD2dF27$$\"3MOnGd! [y_$Fis$!3O7^#om.Z+%F27$$\"3?#yUNg&[hNFis$!3=$G*\\llAhIF27$$\"3iF))z\\ J7&f$Fis$!3cX'>`L-m3#F27$$\"31t[0'pg(GOFis$!3g&4Yn]Z24\"F27$$\"3#*=4JU #)RiOFis$!3y.?@nDYz$)F^p7$$\"36m\\Q&z8#GPFis$\"3g&3?z*pUz=F27$$\"3%G,f %[$HSz$Fis$\"3%)y@!Ha;'pPF27$$\"3eS3B!G4x&QFis$\"3y]36u$f(faF27$$\"3Ko E+7#*Q@RFis$\"3'ya$*3+Z7&pF27$$\"31LT5>\\4#*RFis$\"3oKCB&3H*3$)F27$$\" 3y(f0ii+G1%Fis$\"3QU$)*pchTH*F27$$\"378E]Pnc%4%Fis$\"3Sy0*)=.9.'*F27$$ \"3ZG'*z[GLETFis$\"3)4x_,_r\\#)*F27$$\"39O\"[W!f@UTFis$\"3W+J^aia-**F2 7$$\"3#Qk'4g*)4eTFis$\"3skm_u?kd**F27$$\"3[^^u:?)R<%Fis$\"3EUX$>\"R8!* **F27$$\"3;fORr]')*=%Fis$\"3)f*\\#)4![*****F27$$\"3Y*yoMiBo?%Fis$\"3'= \\a/Qp`)**F27$$\"3))=Rav@yBUFis$\"3!e*\\Eq3&\\%**F27$$\"3I[!>wsS2C%Fis $\"3#pfKK2(zy)*F27$$\"3sxTpz#*pdUFis$\"3ufNW!=zqy*F27$$\"3cOW%QQ;;H%Fi s$\"3)*zkFGg'z_*F27$$\"3R&p%*z[LbK%Fis$\"3k9v'Qy\"Hq\"*F27$$\"3'epgGO, qQ%Fis$\"3ewe9T)QNG)F27$$\"3M'pExBp%[WFis$\"3OCpx\"\\c&F27$$\"3oh/x$[qGe%Fis$\"3S%3RSy>&)y$F27$$\"3qQ q'H.,hk%Fis$\"3!45Id#pbv>F27$$\"3e;O;#eJ$4ZFis$\"3'pP(fbg1s\"*F^p7$$\" 3sU'G^sDax%Fis$!3c'=5em])z=F27$$\"3&)oO4o)>:%[Fis$!3'Rj21wbxx$F27$$\"3 ;@9vt*Qh!\\Fis$!3Kp`c+2p!\\&F27$$\"3Ou\"4%z!e2(\\Fis$!3Up#[gEFz*pF27$$ \"3t:eW%H3%Q]Fis$!3!z'GM].9%H)F27$$\"3*zX#[4&eg5&Fis$!3WK_.!Q(*)\\#*F2 7$$\"3Y!*>B3hjQ^Fis$!3'*p8*4PUld*F27$$\"3#Q_\")pq87<&Fis$!3&o5vP*yy6)* F27$$\"3]!Hcj]-v=&Fis$!338)z%3&*R%*)*F27$$\"3>d5t08z._Fis$!3'4!\\$=u)Q `**F27$$\"3(Q#e50,3?_Fis$!31P;,gZh))**F27$$\"3n*e![/*ojB&Fis$!2utiEY$* *****Fis7$$\"3!3r%[al-`_Fis$!3?I.'\\TQp)**F27$$\"3-J))[/Uop_Fis$!3AS0U =#[*[**F27$$\"39_H\\a=M'G&Fis$!3]Xp0Ex6'))*F27$$\"3Dtq\\/&**HI&Fis$!3S !GR:\"Qg)z*F27$$\"3[:`]/[JO`Fis$!3/.s053Y]&*F27$$\"3#ob8X5I'p`Fis$!3lt \"4r*e*p?*F27$$\"3fR3_p*3dV&Fis$!3mpMF-d2e#)F27$$\"3PA\"GX$yy,bFis$!3+ 'zwg4$p&)pF27$$\"3wxVy[u]ibFis$!3]6GG-ecFis$!3-\\1#\\F&)***HF27$$\"3WaK=$f=Gp&Fis$!33M5 R23n*)>F27$$\"3blXDeVhFdFis$!3KKx'ff^pd*F/7$$\"3mweKB,TidFis$\"3Q?63KR nq%)F^p7$$\"3w\"4#\\F27$$\"3'oIe;6(*o)eFis$\"3#* \\+o]r8FPF27$$\"3yp>qe;E`fFis$\"3U')yagFis$\"3 `eQ?*GSO.(F27$$\"3G:=>Xb9$3'Fis$\"3wGc;^Do_#)F27$$\"3w)*zj%)[mYhFis$\" 3E+vhth&H<*F27$$\"31*\\Lr)\\z!='Fis$\"3\"3&z%z!y#>`*F27$$\"3P***G'*3D \\@'Fis$\"3#R`H)Gh0\"z*F27$$\"3'*\\n(39!*>B'Fis$\"3q@@2J\"*G#))*F27$$ \"3o*\\C@>b!\\iFis$\"39!)GzloiZ**F27$$\"3Q\\APV-7miFis$\"3E#HiQ7)*o)** F27$$\"3)****>YH&=$G'Fis$\"2M***************Fis-%'COLOURG6&%$RGBG$\"*+ +++\"!\")F(F(-%*THICKNESSG6#\"\"#-F$6%7en7$F(F(7$FH$\"3+9&o'z\"y_O\"F2 7$FR$\"3K!y=V&*)GLDF27$Ffn$\"3R]F<<.6.QF27$Fjo$\"3I,*y(H4U7]F27$F_q$\" 3%=pD*eceDhF27$Fiq$\"3w>82*oU&fqF27$Fcr$\"3@)*QjP!>8\"zF27$Fbt$\"3[[%3 w7ESl)F27$F`v$\"3Pi.Z,bcT#*F27$Fjv$\"3!e$pSF\"oen*F27$F_w$\"35_*\\T'zD 5)*F27$Fdw$\"3#ffI'ft64**F27$F^x$\"3)\\\"3MzUXx**F27$Fhx$\"2-6Ot@)**** **Fis7$Fby$\"3Ay8:y_Xw**F27$F\\z$\"3s;Nd#HZn!**F27$Faz$\"3'))pQ.m*='z* F27$Ffz$\"3*p]U&fD`V'*F27$F`[l$\"3cBCb^l'3E*F27$F^]l$\"3;2*>c4&oN')F27 $F\\_l$\"3Se*Rp1I-(zF27$Ff_l$\"3c:s(f4sF0(F27$F``l$\"3(y]$4EukDhF27$Fd al$\"3?Pawd/k,]F27$F_bl$\"3i\"*R%R(HvXQF27$Fibl$\"3!\\o6f)Q%=d#F27$Fcc l$\"3Kh5Xo]Ug8F27$Fael$\"3^)pb)>hJ,JF^p7$F_gl$!3)[9aqyJ,N\"F27$Figl$!3 w\"*=GeHGKDF27$Fchl$!39*)zsmMAnPF27$Fgil$!3#35.)=3zv\\F27$Fajl$!3!fgI[ ?W72'F27$F[[m$!3#=$4p_xMJqF27$Fe[m$!3WWqZN&GL'zF27$Fc]m$!33*3&3nLil')F 27$Fa_m$!3]$[$Q0***4E*F27$F[`m$!3'R4g7n[Pl*F27$F``m$!30-Py@682)*F27$Fe `m$!3$euj/)>C;**F27$Fj`m$!3Ief(ReP!y**F27$F_am$!3;'*phhK&*****F27$Fdam $!311%\\AUQ,)**F27$Fiam$!32^6Oe=u;**F27$F^bm$!31K)*)[7\"*G\")*F27$Fcbm $!3_#49llz!o'*F27$F]cm$!3]_&yQBx]B*F27$F[em$!3o\"*oMmwMe')F27$Fifm$!3F roqht!o\"zF27$Fcgm$!3x5TwV@sUqF27$F]hm$!3)\\'4LY'Q38'F27$Faim$!3iCF-zq _v\\F27$F[jm$!3s)y_Bpo*fQF27$Fejm$!3dChVjR=0EF27$F_[n$!3]'*=#eVn4O\"F2 7$F]]n$!3/UE[]'efD\"!#D-Fb]n6&Fd]nF(F(Fe]nFh]n-F$6&7(7$$\"3STs)p\"3*p# RF2$\"3#y*3lBV$o#QF27$$\"3q?O\\3a\\j>Fis$\"3QnG6D`zQ#*F27$$\"3%[M#>!\\ %>cBFis$\"3sva'=\"y1rqF27$$\"3E<&)GNcBF__o%%3p/4G/$\"+aEfTJF__o%\"pG/$\"+=3*p#RF__o%% 5p/4G/$\"+\")*)Q7ZF__o%%3p/2G/$\"+Wry(\\&F__o%%7p/4G/$\"+3`=$G'F__o%#2 pGFb^o-%%VIEWG6$;F(F\\]o;$!#7F^]o$Fg\\oF^]o" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The solutions of the equation " }{XPPEDIT 18 0 "cos*3*the ta = sin*theta;" "6#/*(%$cosG\"\"\"\"\"$F&%&thetaGF&*&%$sinGF&F(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 259 4 "Note" } {TEXT -1 10 ": Maple's " }{TEXT 0 5 "solve" }{TEXT -1 36 " finds 6 sol utions in the interval (" }{XPPEDIT 18 0 "-Pi,Pi" "6$,$%#PiG!\"\"F$" } {TEXT -1 2 "]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(cos(3*theta)=sin(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(,$*&\"\"%!\"\"%#PiG\"\"\"F&,$*(\"\"$F(F%F&F'F(F(,$ *&\"\")F&F'F(F(,$*(\"\"(F(F.F&F'F(F&,$*(F+F(F.F&F'F(F&,$*(\"\"&F(F.F&F 'F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ta sks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 35 "Solve the follo wing equations for " }{XPPEDIT 18 0 "0 <= theta;" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "`` < 2*Pi;" "6#2%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 58 ". In each case illustrate the solution graphically as the " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 100 " coordinates of the \+ points of intersection of the graphs of the left and right sides of th e equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " (a) " }{XPPEDIT 18 0 "sin*3*theta = sin*theta" "6#/*(%$sinG\"\" \"\"\"$F&%&thetaGF&*&F%F&F(F&" }{TEXT -1 29 " \+ (b) " }{XPPEDIT 18 0 "sin*3*theta = 2*sin*theta;" "6#/*(%$sinG\"\"\"\" \"$F&%&thetaGF&*(\"\"#F&F%F&F(F&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "\{0, Pi/4, 3/4*Pi, Pi, 5/4*Pi , 7/4*Pi\};" "6#<(\"\"!*&%#PiG\"\"\"\"\"%!\"\"*(\"\"$F'F(F)F&F'F&*(\" \"&F'F(F)F&F'*(\"\"(F'F(F)F&F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "\{0,Pi/6,5*Pi/6,Pi,7*Pi/6,11*Pi/6\} " "6#<(\"\"!*&%#PiG\"\"\"\"\"'!\"\"*(\"\"&F'F&F'F(F)F&*(\"\"(F'F&F'F(F )*(\"#6F'F&F'F(F)" }{TEXT -1 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 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 35 "Solve t he following equations 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 6 " (a) " }{XPPEDIT 18 0 "cos*3*theta-cos*theta = sin*2*thet a;" "6#/,&*(%$cosG\"\"\"\"\"$F'%&thetaGF'F'*&F&F'F)F'!\"\"*(%$sinGF'\" \"#F'F)F'" }{TEXT -1 11 " (b) " }{XPPEDIT 18 0 "cos*3*theta-cos* theta=sin*theta" "6#/,&*(%$cosG\"\"\"\"\"$F'%&thetaGF'F'*&F&F'F)F'!\" \"*&%$sinGF'F)F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "\{0, Pi/2, Pi, 7*Pi/6, 3*Pi/2, 11*Pi/6\};" "6 #<(\"\"!*&%#PiG\"\"\"\"\"#!\"\"F&*(\"\"(F'F&F'\"\"'F)*(\"\"$F'F&F'F(F) *(\"#6F'F&F'F,F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "\{0,7*Pi/12,Pi,11*Pi/12,19*Pi/12,23*Pi/12\}" "6#<(\"\" !*(\"\"(\"\"\"%#PiGF'\"#7!\"\"F(*(\"#6F'F(F'F)F**(\"#>F'F(F'F)F**(\"#B F'F(F'F)F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{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 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 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }