{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 128 0 1 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 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 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 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 54 "High School Modul es > Trigonometry by Gregory A. Moore" }{TEXT 266 1 " " }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 256 23 "Trignometric Identities" }} {PARA 0 "" 0 "" {TEXT -1 112 "\nObserving how trig identities can be e quivalent graphically and numerically, and verifying them algebraicall y.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execute the Code \+ Resource section first. Although there will be no output immediately, \+ these definitions are used later in this worksheet.]" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 " 0. Code" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(plots): " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 272 "IdentityPlot := proc( Ide ntity )\nlocal c1,c2;\nc1 := COLOR(RGB, .5,.5, .7);\nc2 := COLOR(RGB, \+ .3,.3, .4);\nplot( [lhs(Identity ), rhs(Identity )+.06],\n x = 0..2*Pi, y = -3..3, discont = true, \n thickness = 3, color = [c1,c2], numpoints = 100 );\nend proc:\n\n\n" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 26 " 1. Visualizing Identities" }}{PARA 0 "" 0 "" {TEXT -1 235 "\nAlthough it is not a valid method of verifying an identity, \+ we can convince ourselves that an identity is true by graphing the lef t and right sides of the identity as separate graphs. If they are the \+ same, then this will reassure us.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "(tan(x) - sec(x))*(sin(x) + 1) = -cos(x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "IdentityPlot( %);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sin(2*x) + sin(2*x) = 4*sin( x)*cos(x);\nIdentityPlot( %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "tan(x) + cot(x) = 1/(sin(x)*cos(x)) ;\nIdentityPlot( %);" }} }{PARA 0 "" 0 "" {TEXT -1 90 "\nAlthough a graph won't prove any ident ity, it can easily show when an identity is false.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "sin(2*x) * cos(2*x) = 4*sin(x)*cos(x) ;\n IdentityPlot( %);" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "sin(2*x) + sin(2*x) = 4*sin(x);\nIdentityPlot( %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "tan(x) + cot(x) = 2*sin(x)*cos(x) ; \nIdentityPlot( %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "si n(x) + sin(2*x) = sin(3*x);\nIdentityPlot( %);" }}}{EXCHG }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 34 " 2. Test ing Identities Numerically" }}{PARA 0 "" 0 "" {TEXT -1 165 "\nAs befor e, this method does not prove any identity, but it helps to convince u s of the validity - an inspiration that can lead to a proof of the val idity.\n\n " }{TEXT 260 7 "Example" }{TEXT 261 2 " :" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(tan(x) - sec(x))*(sin(x) + 1) = -cos(x)" "6 #/*&,&-%$tanG6#%\"xG\"\"\"-%$secG6#F)!\"\"F*,&-%$sinG6#F)F*F*F*F*,$-%$ cosG6#F)F." }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Id1 := \n (tan(x) - sec(x))*(sin(x) + 1) = -cos(x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "array( [[ k*Pi/12, evalf(s ubs( x =k*Pi/12,lhs(Id1)),3), \n evalf(subs( x =k*P i/12,rhs(Id1)),3)] \n $ k= 1..5] );" }}}{PARA 0 "" 0 "" {TEXT -1 66 "\nThis method is valid for proving that a false identity is fal se.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "Id1 := \n si n(2*x) + sin(2*x) = sin(4*x);\n\narray( [[ k*Pi/12, subs( x =k*Pi/12, lhs(Id1)), subs( x =k*Pi/12,rhs(Id1))] \n $ k= 1..12] );" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 37 " 3. Proving Identities Algebraically" }}{PARA 0 "" 0 "" {TEXT -1 272 "\nThe normal way of verifying an identity is to simplify one side or \+ the other (or both) to show that the two expressions are equivalent. I t's quite similar to simplifying an equation. Basically, each side at \+ each step is always equivalent to the original equation.\n\n " } {TEXT 262 7 "Example" }{TEXT 263 5 " 3.1:" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(tan(x) - sec(x))*(sin(x) + 1) = -cos(x)" "6#/*&,&-%$tanG6#%\"x G\"\"\"-%$secG6#F)!\"\"F*,&-%$sinG6#F)F*F*F*F*,$-%$cosG6#F)F." }{TEXT -1 130 "\n\nOne way of solving an identity, although not always the mo st elegant way, is to simply convert everything to sines and cosines. \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Id1 := \n (tan(x) \+ - sec(x))*(sin(x) + 1) = -cos(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "subs( \{tan(x) = sin(x)/cos(x), sec(x) = 1/cos(x)\}, \+ lhs(Id1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify(%, t rig);" }}}{PARA 0 "" 0 "" {TEXT -1 164 "Thus the left side has been re duced to be the same as the right side. Thus we are done.\n\nWe can al so use this command to do these substitutions more automatically.\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "convert( Id1, 'sincos' );\n simplify(%, trig);" }}}{PARA 0 "" 0 "" {TEXT -1 97 "The left side has \+ been reduced to be the same as the right side. Thus we are done...agai n.\n\n " }{TEXT 264 7 "Example" }{TEXT 265 6 " 3.2 :" }{TEXT -1 2 " " }{XPPEDIT 18 0 "tan(x) + cot(x) = 1/(sin(x)*cos(x))" "6#/,&-%$tan G6#%\"xG\"\"\"-%$cotG6#F(F)*&F)F)*&-%$sinG6#F(F)-%$cosG6#F(F)!\"\"" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Id1 := \n \+ tan(x) + cot(x) = 1/(sin(x)*cos(x)) ;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 66 "subs( \{tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin (x)\}, lhs(Id1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal (%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify(%, trig); " }}}{PARA 0 "" 0 "" {TEXT -1 45 "This is the same as the original rig ht side.\n" }}}{PARA 0 "" 0 "" {TEXT -1 4 "\n " }{TEXT 259 34 " \+ \251 2002 Waterloo Maple Inc" }}}{MARK "0 1" 28 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }