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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 256 "
" 0 "" {TEXT -1 43 "Lesson 14:  Solving Trigonometric Equations" }}
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In this lesson, we show how to solve equations for x that contain trig
 functions.  Our general method will be to move all terms over to the \+
left-hand side of the equation and find the roots of the resulting equ
ation.  We'll find these roots both analytically (by solving) and grap
hically by inspecting the plot and seeing where the curve crosses the \+
x-axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 256 9 "Example 1" }{TEXT -1 3 "\n  " }{XPPEDIT 18 0 "sin(x) = sq
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"" 0 "" {TEXT -1 15 "Solutions are: " }{XPPEDIT 18 0 "0, (1/2)*Pi ,Pi,
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$\"31-Spq6\\SFF*$\"3S@8gx_'3y\"F*7$$\"3q)>Y8t\\l%GF*$\"3it3!f$=c\"e\"F
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$zF27$$\"35V?o%e2nM$F*$\"3J(HKyL1k#fF27$$\"3OV5)pR4*QMF*$\"3R)=f7M&)39
%F27$$\"3jV+G476JNF*$\"3+'G/DUQ^S#F27$$\"3m\"pX$yIrKPF*$!3+`VA+O$e9\"F
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$$\"3AgP!=ZWXJ%F*$!3kxn;J['zV)F27$$\"3;B(eVz>gT%F*$!3^3ZAtL1G\"*F27$$
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\"3E(*pkzYSygF*$\"364OJ![_H$fF27$$\"31*\\Lr)\\z!='F*$\"3!*HeU[)pd&zF27
$$\"3)****>YH&=$G'F*$\"2/F3)[(*******F*-%'COLOURG6&%$RGBG\"\"!F\\glF\\
gl-F$6$7S7$$!3;))H)fv()fB&F2$!\"\"F\\gl7$Fagl$!3ommm;p0k&*F27$Fagl$!3w
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L3i.9!zF27$Fagl$!3\"ommT!R=0vF27$Fagl$!3u****\\P8#\\4(F27$Fagl$!3+nm;/
siqmF27$Fagl$!3[++](y$pZiF27$Fagl$!33LLL$yaE\"eF27$Fagl$!3hmmm\">s%HaF
27$Fagl$!3Q+++]$*4)*\\F27$Fagl$!39+++]_&\\c%F27$Fagl$!31+++]1aZTF27$Fa
gl$!3umm;/#)[oPF27$Fagl$!3hLLL$=exJ$F27$Fagl$!3*RLLLtIf$HF27$Fagl$!3]+
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***\\74_c7F27$Fagl$!3)3LLL3x%z#)F[s7$Fagl$!3KMLL3s$QM%F[s7$Fagl$!3]^om
m;zr)*!#@7$Fagl$\"3%pJL$ezw5VF[s7$Fagl$\"3s*)***\\PQ#\\\")F[s7$Fagl$\"
3GKLLe\"*[H7F27$Fagl$\"3I*******pvxl\"F27$Fagl$\"3#z****\\_qn2#F27$Fag
l$\"3U)***\\i&p@[#F27$Fagl$\"3B)****\\2'HKHF27$Fagl$\"3ElmmmZvOLF27$Fa
gl$\"3i******\\2goPF27$Fagl$\"3UKL$eR<*fTF27$Fagl$\"3m******\\)Hxe%F27
$Fagl$\"3ckm;H!o-*\\F27$Fagl$\"3y)***\\7k.6aF27$Fagl$\"3#emmmT9C#eF27$
Fagl$\"33****\\i!*3`iF27$Fagl$\"3%QLLL$*zym'F27$Fagl$\"3wKLL3N1#4(F27$
Fagl$\"3Nmm;HYt7vF27$Fagl$\"3Y*******p(G**yF27$Fagl$\"3]mmmT6KU$)F27$F
agl$\"3fKLLLbdQ()F27$Fagl$\"3[++]i`1h\"*F27$Fagl$\"3W++]P?Wl&*F27$Fagl
$\"\"\"F\\gl-Fifl6&F[gl$\"*++++\"!\")$F\\glF\\glF[am-F$6$7S7$$\"33(G\"
eJlefdF*Fcgl7$F`amFfgl7$F`amFigl7$F`amF\\hl7$F`amF_hl7$F`amFbhl7$F`amF
ehl7$F`amFhhl7$F`amF[il7$F`amF^il7$F`amFail7$F`amFdil7$F`amFgil7$F`amF
jil7$F`amF]jl7$F`amF`jl7$F`amFcjl7$F`amFfjl7$F`amFijl7$F`amF\\[m7$F`am
F_[m7$F`amFb[m7$F`amFe[m7$F`amFh[m7$F`amF[\\m7$F`amF_\\m7$F`amFb\\m7$F
`amFe\\m7$F`amFh\\m7$F`amF[]m7$F`amF^]m7$F`amFa]m7$F`amFd]m7$F`amFg]m7
$F`amFj]m7$F`amF]^m7$F`amF`^m7$F`amFc^m7$F`amFf^m7$F`amFi^m7$F`amF\\_m
7$F`amF__m7$F`amFb_m7$F`amFe_m7$F`amFh_m7$F`amF[`m7$F`amF^`m7$F`amFa`m
7$F`amFd`mFf`m-F$6$7S7$$\"3E#4)=H9>lOF*Fcgl7$FfdmFfgl7$FfdmFigl7$FfdmF
\\hl7$FfdmF_hl7$FfdmFbhl7$FfdmFehl7$FfdmFhhl7$FfdmF[il7$FfdmF^il7$Ffdm
Fail7$FfdmFdil7$FfdmFgil7$FfdmFjil7$FfdmF]jl7$FfdmF`jl7$FfdmFcjl7$Ffdm
Ffjl7$FfdmFijl7$FfdmF\\[m7$FfdmF_[m7$FfdmFb[m7$FfdmFe[m7$FfdmFh[m7$Ffd
mF[\\m7$FfdmF_\\m7$FfdmFb\\m7$FfdmFe\\m7$FfdmFh\\m7$FfdmF[]m7$FfdmF^]m
7$FfdmFa]m7$FfdmFd]m7$FfdmFg]m7$FfdmFj]m7$FfdmF]^m7$FfdmF`^m7$FfdmFc^m
7$FfdmFf^m7$FfdmFi^m7$FfdmF\\_m7$FfdmF__m7$FfdmFb_m7$FfdmFe_m7$FfdmFh_
m7$FfdmF[`m7$FfdmF^`m7$FfdmFa`m7$FfdmFd`mFf`m-%+AXESLABELSG6%Q\"x6\"Q!
6\"%(DEFAULTG-%%VIEWG6$;$!+aEfTJ!\"*$\"+3`=$G'FfhmF_hm" 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" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Solutions are:" }
{XPPEDIT 18 0 "1*Pi/4,5*Pi/4,11*Pi/6,7*Pi/6;" "6&*&*&\"\"\"F%%#PiGF%F%
\"\"%!\"\"*&*&\"\"&F%F&F%F%F'F(*&*&\"#6F%F&F%F%\"\"'F(*&*&\"\"(F%F&F%F
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{PARA 0 "" 0 "" {TEXT 260 9 "Example 5" }{TEXT -1 3 "\n  " }{XPPEDIT 
18 0 "2*sin(x)^2 - sin(x) - 1 = 0 " "6#/,(*&\"\"#\"\"\"*$-%$sinG6#%\"x
GF&F'F'-F*6#F,!\"\"F'F/\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 43 "f5:= x->  2 * sin(x) * sin(x) - sin(x) - 1;" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f5GR6#%\"xG6\"6$%)operatorG%&arrowG
F(,(*$)-%$sinG6#9$\"\"#\"\"\"F3F/!\"\"F4F5F(F(F(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 40 "plot(f5(x), x = 0..2*Pi, color = black);" }
{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&
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%'COLOURG6&%$RGBGF)F)F)-%%VIEWG6$;F($\"+3`=$G'!\"*%(DEFAULTG" 1 2 0 1 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "factor(2 * sin(x) * sin(x) - sin(x)
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oblem 4 for 2 sin(x) + 1 = 0 we have solutions are:   " }{XPPEDIT 18 
0 "(11/6)* Pi, (7/6)* Pi  and (1/2)* Pi" "6$*(\"#6\"\"\"\"\"'!\"\"%#Pi
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 10 "Example \+
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{MPLTEXT 1 0 29 "f6:= x -> sin (2*x) + sin(x);" }{TEXT -1 0 "" }}
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"" {MPLTEXT 1 0 40 "plot(f6(x), x = 0..2*Pi, color = black);" }{TEXT 
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hDF0$\"32*phc:'fMuF07$$\"3Zs[E^dK,RF0$\"3a9!)4D*yP3\"!#<7$$\"3ab^NehL]
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$$\"3#=uye<)G*4#FE$!3fv;9()3U3t!#?7$$\"3Ua[#o!GC>AFE$!3S![-d*R*pl\"F07
$$\"3-YwJf&y(eBFE$!3#*\\^0cV4ZHF07$$\"31\"R2:&\\`?CFE$!39Q3<!RDbJ$F07$
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