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-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 }{PSTYLE "Section hea ding" -1 259 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 3 0 3 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 57 "Trigonometric identities includin g the addition formulas " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Ston e, 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 33 "Proving trigonometric ident ities " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 3 "An " }{TEXT 259 8 "identity" }{TEXT -1 75 " is an equat ion which is true for all values of the variables which occur. " }} {PARA 0 "" 0 "" {TEXT -1 72 "For example, the difference of two square s formula of ordinary algebra: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a^2-b^2=(a+b)*(a-b)" "6#/,&*$%\"aG\"\"#\"\"\"*$%\"bGF'! \"\"*&,&F&F(F*F(F(,&F&F(F*F+F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "is an identity.\nThe formulas " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos^2*theta+sin^2*theta=1" "6#/,&*&%$co sG\"\"#%&thetaG\"\"\"F)*&%$sinGF'F(F)F)F)" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 4 "and " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+tan^2*theta=sec^2*theta" "6#/,&\"\"\"F%*&%$tanG\"\"#%&thetaGF% F%*&%$secGF(F)F%" }}{PARA 0 "" 0 "" {TEXT -1 42 "are examples of trigo nometric identities. " }}{PARA 0 "" 0 "" {TEXT -1 149 "It is possible \+ to establish (prove) many other trigonometric identities. Suitable str ategies for doing this are suggested by the following examples. " }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT 305 8 "Question" }{TEXT -1 23 ": Verify the identity: " } {XPPEDIT 18 0 "sin^2*theta/(cos*theta)+cos*theta=sec*theta" "6#/,&*(%$ sinG\"\"#%&thetaG\"\"\"*&%$cosGF)F(F)!\"\"F)*&F+F)F(F)F)*&%$secGF)F(F) " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 306 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 58 "This identity can be proved by simplifying the left side. " }}{PARA 0 "" 0 "" {TEXT -1 82 "We start by combining the two terms on the left side using the common denominator " }{XPPEDIT 18 0 "cos*theta;" "6#*& %$cosG\"\"\"%&thetaGF%" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin^2*theta/(cos*theta)+cos*theta/1=(sin^2*theta +cos^2*theta)/(cos*theta)" "6#/,&*(%$sinG\"\"#%&thetaG\"\"\"*&%$cosGF) F(F)!\"\"F)*(F+F)F(F)F)F,F)*&,&*&F&F'F(F)F)*&F+F'F(F)F)F)*&F+F)F(F)F, " }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` `=1/(cos*theta)" "6#/%!G*&\"\"\"F&*&%$cosGF&%&thetaGF&!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "( using the basic identity " } {XPPEDIT 18 0 "cos^2*theta+sin^2*theta = 1" "6#/,&*&%$cosG\"\"#%&theta G\"\"\"F)*&%$sinGF'F(F)F)F)" }{TEXT -1 2 " )" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sec*theta;" "6#/%!G*&%$secG\"\"\"% &thetaGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sin(theta)^2/cos(theta)+cos(theta); \nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%&theta G\"\"#-%$cosGF'!\"\"\"\"\"F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\" \"\"F$-%$cosG6#%&thetaG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "The graphs of the left and right sides co incide. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 631 "pi := evalf(Pi):\nf := theta->sin(theta)^2/cos(theta )+cos(theta):'f(theta)'=f(theta);\ng := theta->sec(theta):'g(theta)'=g (theta);\np1:=plot([f(theta),g(theta)],theta=-Pi/2..10.3,color=[red,gr een],thickness=[1,2],\n thickness=2,discont=true):\np2:=plot([[[-pi/ 2,-8],[-pi/2,8]],[[pi/2,-8],[pi/2,8]],[[3*pi/2,-8],[3*pi/2,8]],\n [[ 5*pi/2,-8],[5*pi/2,8]]],color=black,linestyle=3):\nt1:=plots[textplot] ([10.3,-.8,`q`],font=[SYMBOL,11]):\nplots[display]([p1,p2,t1],xtickmar ks=[-pi/2=`-p/2`,pi/2=`p/2`,\n pi=`p`,3*pi/2=`3p/2`,2*pi=`2p`,5*pi/2 =`5p/2`,3*pi=`3p`],\n ytickmarks=3,font=[SYMBOL,11],\n labels=[``, ``],view=[-1.58..10.3,-8..8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"fG6#%&thetaG,&*&-%$sinGF&\"\"#-%$cosGF&!\"\"\"\"\"F-F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"gG6#%&thetaG-%$secGF&" }}{PARA 13 "" 1 "" {GLPLOT2D 420 282 282 {PLOTDATA 2 "6--%'CURVESG6(7ao7$$!3-+++Cjzq:!#<$ \"3W$GW#**)\\zd$!\"*7$$!3o$e(R$Rc'o:F*$\"37s8w+i/tY!#:7$$!3an^zik^m:F* $\"3RN<0w*HlL#F37$$!3?^F>KlPk:F*$\"3?\"yoS$Hpd:F37$$!3%[L!f,mBi:F*$\"3 AIGDpgFo6F37$$!39-bQSn&zb\"F*$\"3+ksQkoi)y(!#;7$$!3Wp1=zon`:F*$\"3S#yP @I&fTeFH7$$!3//5xcr6X:F*$\"3WV$>l([j%*QFH7$$!3jQ8OMubO:F*$\"3LZ>!)[fA@ HFH7$$!3/3?a*)zV>:F*$\"33d4Wq(fz%>FH7$$!3XxEsW&=B]\"F*$\"3UkB#>fp9Y\"F H7$$!3AEC#*pu_s9F*$\"3f./ndgD>5FH7$$!3Av@7&ROFW\"F*$\"36>G$3+G-$yF*7$$ !3M8RRO+tv8F*$\"3!*=iGQ$=\"f^F*7$$!3f%eCg^z#38F*$\"3-2w/7qQ`QF*7$$!3/H X?w&\\6C\"F*$\"3'yoA=.'>*3$F*7$$!3)o*3wG:\"*y6F*$\"3HR$oK%RF=EF*7$$!3E ;9gNwY96F*$\"3iUVU'yY$pAF*7$$!3yRz5p'>y/\"F*$\"3N\"[x?zd@+#F*7$$!3JPE) [VaQ\")*!#=$\"3CLV2Gs(*)z\"F*7$$!3E:5Mmn\\I\"*Fgq$\"3ym\"p@!RwN;F*7$$! 3VA-$*f\\fG&)Fgq$\"36$y:XdW,_\"F*7$$!3A)ymf3'*4&yFgq$\"3b)eR6`\"z89F*7 $$!37,X#f#\\hqrFgq$\"3L#)*e@GfFgq$\"3\\q+W$=L]?\"F*7$$!3P<4IeB_6_Fgq$\"3Q,96g_2`6F*7 $$!3FC'QM?\\Fgq$\"3kb/gt%*z>5F*7$$!3^@Ur;s`+8Fgq$\"3K]Pa)*p^35F*7$$ !3W*R/*=NGBo!#>$\"3[)RTUQKB+\"F*7$$!3?_#HhSd1b\"!#?$\"37+/F-7++5F*7$$ \"3C*)pci%Q8x'F^v$\"31\\lBRpH-5F*7$$\"3Eu\\Go$z+G\"Fgq$\"3!*zY6H$\\#35 F*7$$\"3zbyI]qFJ>Fgq$\"3\")fDiWM%*=5F*7$$\"35Eldvz-/EFgq$\"3&\\oF#=**) [.\"F*7$$\"3?^:Y1N=iKFgq$\"3a]\"z]Evc0\"F*7$$\"3-!p>V#G)*)*QFgq$\"3]8x )zfFgq$\"3-,&=z]Z]?\"F*7$$\"3s`'=N&HQMlFgq$\" 3O'3%Hg&[%f7F*7$$\"3bt])Q=*Q1sFgq$\"3K\"3*3?p(3L\"F*7$$\"3eB(*)fn%pQyF gq$\"3h_Y83m079F*7$$\"3#)[^y0hj*\\)Fgq$\"3K3_LZu7::F*7$$\"3#Gg/D;Fe9*F gq$\"3KsS,&[=!R;F*7$$\"3w3'>IJHB#)*Fgq$\"3pP;HA2E,=F*7$$\"31P`\")G\")Q Z5F*$\"3KLwcM/m+?F*7$$\"3qak%)G(=S6\"F*$\"33N!)HEOFnAF*7$$\"3u]#oQf(4! =\"F*$\"34$[Ws#>\"ei#F*7$$\"3pTy83s\"3C\"F*$\"3WD([T1)>'3$F*7$$\"3b[dH Q(3/J\"F*$\"3UL+#>^sT)QF*7$$\"31McZKAls8F*$\"3q#Q#eK\"3+3&F*7$$\"3'G(Q `zn,R9F*$\"3))[YDD)Q/h(F*7$$\"39SRE\\kxq9F*$\"3]h;\")4-Z,5FH7$$\"3T2S* *=h`-:F*$\"35/I@W17m9FH7$$\"3X0bCq6g>:F*$\"3oaS/%HjT&>FH7$$\"3r.q\\@im O:F*$\"3Al?6.N`IHFH7$$\"3%GvAru)>X:F*$\"3!=1&e1g/2RFH7$$\"3)>][FFJPb\" F*$\"3aWEyGJ@geFH7$$\"3Ww8cNv*zb\"F*$\"3uH;Xf6X8yFH7$$\"3)3Du$)zjAc\"F *$\"3!)o_Op(**><\"F37$$\"37)o!yHpRk:F*$\"3'p)=\"H*yli:F37$$\"3MDr=h+`m :F*$\"3C3%*RRu(RM#F37$$\"3!Gc$f#>j'o:F*$\"3AF'o>9Tzo%F37$$\"3-+++Cjzq: F*F+7ao7$$\"3!******fL'zq:F*$!3+f#3f\"RN'3\"F-7$$\"3MUUfmi$Hd\"F*$!3[S _[?A.tYF37$$\"3![[)=(>w]d\"F*$!39xJW&[ElL#F37$$\"3-FFyFh@x:F*$!37.]n%Q \"pd:F37$$\"3YppPegNz:F*$!3qo'Q))>v#o6F37$$\"37aac>fj$e\"F*$!3/lM&e+B' )y(FH7$$\"3-RRv!y:ze\"F*$!3O_hLPJfTeFH7$$\"3e348.bZ'f\"F*$!3Yc:S>Rj%*Q FH7$$\"39yy]D_.0;F*$!3aowD8aA@HFH7$$\"3E<=EqY:A;F*$!3;3v&\\`fz%>FH7$$ \"3gcd,:TFR;F*$!3yd5FH7$$\"3 wr')Qki&))p\"F*$!3B)4r2kF-$yF*7$$\"3Q))4'Giiew\"F*$!35C0O!>=\"f^F*7$$ \"3+vE(H98L$=F*$!3k0_]MpQ`QF*7$$\"3A8j`#3V/!>F*$!3'yK7`)f>*3$F*7$$\"3a 8AuH6oi>F*$!3mM#R@\"RF=EF*7$$\"3SOblA]7F?F*$!3a$=[[wY$pAF*7$$\"3_OW*)) )Hx$4#F*$!3hXLvwx:-?F*7$$\"3M_.E9s?g@F*$!3%H1')o@x*)z\"F*7$$\"3M\"\\`3 )HaGAF*$!3<2c\"R*QwN;F*7$$\"3'3mk7;L()G#F*$!3DK?8oX9?:F*7$$\"3)3=-%[I \\cBF*$!3HcFKE:z89F*7$$\"3.Bl9k6`CCF*$!3\"e>;&ysrE8F*7$$\"3AnC#e\\)4! \\#F*$!3Sk-55!zvD\"F*7$$\"3!p&oW9*R'\\DF*$!3i^cF\"=L]?\"F*7$$\"3cj0;?/ W?EF*$!3e(4A&e_2`6F*7$$\"3!fp\"GF*$!3AW_lK(3p0\"F*7$$\"3k!=IKb)pzG F*$!3ea*=\"fCIN5F*7$$\"3P+br6)=U%HF*$!3W[IOt%*z>5F*7$$\"3dY`#G$*Q:,$F* $!3\"\\6C%)*p^35F*7$$\"3]'er!>)fL2$F*$!3BZa>%QKB+\"F*7$$\"3)Qz*f'*>/SJ F*$!3'[vpA?,++\"F*7$$\"3*R-n$QgI4KF*$!3/^oCRpH-5F*7$$\"35V&R.f+'pKF*$! 3_gW5H$\\#35F*7$$\"3*>4$He.sMLF*$!3N,tbWM%*=5F*7$$\"3C()Hc]a*>S$F*$!3v ?l1=**)[.\"F*7$$\"3'G4+M+6yY$F*$!3!Hj!yk_nb5F*7$$\"33nE%\\$4\\JNF*$!3y #H3v-OF*$!3YtWw..M;6F*7$$\"3%[*[:k)Gdm$F*$!3WC3T 1\"f]:\"F*7$$\"3K/7@sIcLPF*$!3H#p=k]Z]?\"F*7$$\"37:f&o%4.&z$F*$!3aEGGe &[%f7F*7$$\"3cpejp:BiQF*$!3J28N5(\\/m0 79F*7$$\"37$*=Lhib\"*RF*$!3\"zaVCWF^^\"F*7$$\"3'>,dnOvh0%F*$!3Bs:Xy%=! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 289 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 58 "This identity can be proved by sim plifying the left side. " }}{PARA 0 "" 0 "" {TEXT -1 91 "We start by c ombining the two rational terms on the left side using the common deno minator " }{XPPEDIT 18 0 "(1-sin*theta)*cos*theta" "6#*(,&\"\"\"F%*&%$ sinGF%%&thetaGF%!\"\"F%%$cosGF%F(F%" }{TEXT -1 3 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*theta/(1-sin*theta)+(1-sin*the ta)/(cos*theta)=(cos^2*theta+(1-sin*theta)^2)/((1-sin*theta)*cos*theta )" "6#/,&*(%$cosG\"\"\"%&thetaGF',&F'F'*&%$sinGF'F(F'!\"\"F,F'*&,&F'F' *&F+F'F(F'F,F'*&F&F'F(F'F,F'*&,&*&F&\"\"#F(F'F'*$,&F'F'*&F+F'F(F'F,F4F 'F'*(,&F'F'*&F+F'F(F'F,F'F&F'F(F'F," }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(cos^2*theta+1-2*sin*theta+sin^2* theta)/((1-sin*theta)*cos*theta)" "6#/%!G*&,**&%$cosG\"\"#%&thetaG\"\" \"F+F+F+*(F)F+%$sinGF+F*F+!\"\"*&F-F)F*F+F+F+*(,&F+F+*&F-F+F*F+F.F+F(F +F*F+F." }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= (cos^2*theta+sin^2*theta+1-2*sin*theta)/((1-sin*theta)*cos*t heta)" "6#/%!G*&,**&%$cosG\"\"#%&thetaG\"\"\"F+*&%$sinGF)F*F+F+F+F+*(F )F+F-F+F*F+!\"\"F+*(,&F+F+*&F-F+F*F+F/F+F(F+F*F+F/" }{TEXT -1 1 " " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(2-2*sin*theta)/(( 1-sin*theta)*cos*theta)" "6#/%!G*&,&\"\"#\"\"\"*(F'F(%$sinGF(%&thetaGF (!\"\"F(*(,&F(F(*&F*F(F+F(F,F(%$cosGF(F+F(F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "( using the basic identity " }{XPPEDIT 18 0 "co s^2*theta+sin^2*theta = 1" "6#/,&*&%$cosG\"\"#%&thetaG\"\"\"F)*&%$sinG F'F(F)F)F)" }{TEXT -1 2 " )" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=2*(1-sin*theta)/((1-sin*theta)*cos*theta)" "6#/%!G*( \"\"#\"\"\",&F'F'*&%$sinGF'%&thetaGF'!\"\"F'*(,&F'F'*&F*F'F+F'F,F'%$co sGF'F+F'F," }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=2/(cos*theta)" "6#/%!G*&\"\"#\"\"\"*&%$cosGF'%&theta GF'!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=2*sec*theta" "6#/%!G*(\"\"#\"\"\"%$secGF'%&thetaGF' " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "cos(theta)/(1-sin(theta))+(1-sin(theta))/cos( theta);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$cosG6# %&thetaG\"\"\",&F)F)-%$sinGF'!\"\"F-F)*&F*F)F%F-F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&\"\"#\"\"\"-%$cosG6#%&thetaG!\"\"F&" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The graphs of the \+ left and right sides coincide. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 650 "pi := evalf(Pi):\nf := thet a->cos(theta)/(1-sin(theta))+(1-sin(theta))/cos(theta):'f(theta)'=f(th eta);\ng := theta->2*sec(theta):'g(theta)'=g(theta);\np1:=plot([f(thet a),g(theta)],theta=-Pi/2..10.3,color=[red,green],thickness=[1,2],\n \+ thickness=2,discont=true):\np2:=plot([[[-pi/2,-8],[-pi/2,8]],[[pi/2,-8 ],[pi/2,8]],[[3*pi/2,-8],[3*pi/2,8]],\n [[5*pi/2,-8],[5*pi/2,8]]],co lor=black,linestyle=3):\nt1:=plots[textplot]([10.3,-.8,`q`],font=[SYMB OL,11]):\nplots[display]([p1,p2,t1],xtickmarks=[-pi/2=`-p/2`,pi/2=`p/2 `,\n pi=`p`,3*pi/2=`3p/2`,2*pi=`2p`,5*pi/2=`5p/2`,3*pi=`3p`],\n yt ickmarks=3,font=[SYMBOL,11],\n labels=[``,``],view=[-1.58..10.3,-8.. 8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%&thetaG,&*&-%$cosGF& \"\"\",&F,F,-%$sinGF&!\"\"F0F,*&F-F,F*F0F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%&thetaG,$*&\"\"#\"\"\"-%$secGF&F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 420 282 282 {PLOTDATA 2 "6--%'CURVESG6(7ao7$$!3-+ ++Cjzq:!#<$\"3)oc)[)z**e:(!\"*7$$!3++++%Rc'o:F*$\"3-]SbK]4Y$*!#:7$$!3# ******HY;lc\"F*$\"3irEvv,1tYF37$$!3/+++KlPk:F*$\"3]M4gudQ:JF37$$!3/+++ -mBi:F*$\"3OI%H.D_lL#F37$$!31+++Sn&zb\"F*$\"3;4,6E`sd:F37$$!3'*******y on`:F*$\"3mZ*)4[!>$o6F37$$!35+++cr6X:F*$\"3kl7A>&p#*y(!#;7$$!36+++MubO :F*$\"3L`*pf$=XUeFR7$$!3!********)zV>:F*$\"3#)>Nfv&>f*QFR7$$!31+++X&=B ]\"F*$\"3U&fjc>RH#HFR7$$!3'*******pu_s9F*$\"37J[%p67&Q?FR7$$!31+++&ROF W\"F*$\"3Gf1o)fXgc\"FR7$$!3)******f.IdP\"F*$\"3D$4+cmB=.\"FR7$$!3.+++; &z#38F*$\"3SN,RBSx1xF*7$$!3)******fd\\6C\"F*$\"3-o=&*f?RyhF*7$$!31+++H :\"*y6F*$\"3.Jkc*)yaO_F*7$$!34+++OwY96F*$\"3)>(R`wNpQXF*7$$!3!*******o '>y/\"F*$\"3i[_S$e:V+%F*7$$!35+++NW&Q\")*!#=$\"3I8=@cW&zf$F*7$$!3N+++n n\\I\"*Fgq$\"3c7uh/y_rKF*7$$!3&*******f\\fG&)Fgq$\"3zae0\\\"*GSIF*7$$! 3a+++'3'*4&yFgq$\"3/i&)GiIeFGF*7$$!3i*****f#\\hqrFgq$\"37JaLkXV`EF*7$$ !3\"******pgT\\^'Fgq$\"377`#e-e^^#F*7$$!3C+++>u_>fFgq$\"3\"47>qOm+T#F* 7$$!3Z+++gB_6_Fgq$\"3:_xW?0:1BF*7$$!3))*******>\\l]'H$Fgq$\"3dGagmu\"Q6 #F*7$$!3<+++?5%*=EFgq$\"3Uo$[\">\\gq?F*7$$!3%*******H%QP(>Fgq$\"3PV_=Z *)fR?F*7$$!35+++?s`+8Fgq$\"3I$=ur*R.$\"3+**oXoZm/ ?F*7$$!3+++++ql]:!#?$\"3\"3aRXS-++#F*7$$\"35++++&Q8x'F^v$\"3u')R_yQf/? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 287 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 34 "When both sides are multiplied by " }{XPPEDIT 18 0 "1+sin*theta" "6#,&\"\"\"F$* &%$sinGF$%&thetaGF$F$" }{TEXT -1 29 " the given equation becomes: " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos^2*theta = (1-sin* theta)*(1+sin*theta)" "6#/*&%$cosG\"\"#%&thetaG\"\"\"*&,&F(F(*&%$sinGF (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 "cos^2*theta = 1-sin^2*theta" "6#/*&%$cosG\"\"#%&thetaG\"\"\",&F( F(*&%$sinGF&F'F(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 91 "This last equation is an identity because it is just a rearrangement \+ of the basic identity " }{XPPEDIT 18 0 "cos^2*theta+sin^2*theta = 1" " 6#/,&*&%$cosG\"\"#%&thetaG\"\"\"F)*&%$sinGF'F(F)F)F)" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 120 "The previous steps may be reversed i n order to prove the required identity. In detail, starting with the b asic identity " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos ^2*theta+sin^2*theta = 1" "6#/,&*&%$cosG\"\"#%&thetaG\"\"\"F)*&%$sinGF 'F(F)F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 16 "it follows \+ that " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos^2*theta = 1-sin^2*theta" "6#/*&%$cosG\"\"#%& thetaG\"\"\",&F(F(*&%$sinGF&F'F(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "On factoring the right side of the last identity we \+ obtain: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos^2*the ta=(1-sin*theta)*(1+sin*theta)" "6#/*&%$cosG\"\"#%&thetaG\"\"\"*&,&F(F (*&%$sinGF(F'F(!\"\"F(,&F(F(*&F,F(F'F(F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "Then dividing both sides by " }{XPPEDIT 18 0 "1 +sin*theta" "6#,&\"\"\"F$*&%$sinGF$%&thetaGF$F$" }{TEXT -1 30 " gives \+ the required identity: " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "cos^2*theta/(1+sin*theta)=1-sin*theta" "6#/*(%$cosG\"\" #%&thetaG\"\"\",&F(F(*&%$sinGF(F'F(F(!\"\",&F(F(*&F+F(F'F(F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 158 "Since this approach needs careful explanation, it is probably bet ter to construct a proof which involves rearranging the left side to p roduce the right side. " }}{PARA 0 "" 0 "" {TEXT -1 29 "This can be do ne as follows. " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "co s^2*theta/(1+sin*theta)=(1-sin^2*theta)/(1+sin*theta)" "6#/*(%$cosG\" \"#%&thetaG\"\"\",&F(F(*&%$sinGF(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 27 "( usi ng the basic identity " }{XPPEDIT 18 0 "cos^2*theta = 1-sin^2*theta;" "6#/*&%$cosG\"\"#%&thetaG\"\"\",&F(F(*&%$sinGF&F'F(!\"\"" }{TEXT -1 3 " ) " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(1-sin*the ta)*(1+sin*theta)/(1+sin*theta)" "6#/%!G*(,&\"\"\"F'*&%$sinGF'%&thetaG F'!\"\"F',&F'F'*&F)F'F*F'F'F',&F'F'*&F)F'F*F'F'F+" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1-sin*theta;" "6#/%!G,&\"\"\"F&*&%$sinGF&%&thetaGF &!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "cos(theta)^2/(1+sin(theta));\nsimpl ify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$cosG6#%&thetaG\"\"#,& \"\"\"F*-%$sinGF&F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$sinG6 #%&thetaG!\"\"\"\"\"F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The graphs of the left and right sides coincide, exc ept where " }{XPPEDIT 18 0 "theta = ` . . . `,-Pi/2,3*Pi/2,7*Pi/2,11*P i/2,` . . . `;" "6(/%&thetaG%(~.~.~.~G,$*&%#PiG\"\"\"\"\"#!\"\"F+*(\" \"$F)F(F)F*F+*(\"\"(F)F(F)F*F+*(\"#6F)F(F)F*F+F%" }{TEXT -1 21 ". At t hese values of " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 82 " th e left side of the identity is not defined but the right side has the \+ value 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 457 "pi := evalf(Pi):\nf := theta->cos(theta)^2/(1+sin(th eta)):'f(theta)'=f(theta);\ng := theta->1-sin(theta):'g(theta)'=g(thet a);\np1:=plot([f(theta),g(theta)],theta=-Pi/2..10.2,color=[red,green], thickness=[1,2]):\nt1:=plots[textplot]([10.3,-.13,`q`],font=[SYMBOL,11 ]):\nplots[display]([p1,t1],xtickmarks=[-pi/2=`-p/2`,pi/2=`p/2`,\n p i=`p`,3*pi/2=`3p/2`,2*pi=`2p`,5*pi/2=`5p/2`,3*pi=`3p`],font=[SYMBOL,11 ],thickness=2,\n labels=[``,``],view=[-Pi/2..10.3,-.13..2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%&thetaG*&-%$cosGF&\"\"#,&\" \"\"F--%$sinGF&F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%&t hetaG,&\"\"\"F)-%$sinGF&!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 619 163 163 {PLOTDATA 2 "6*-%'CURVESG6%7jq7$$!31+++;&yFc\"!#<$\"3c)=Ev&y'***>F *7$$!35+++32wa:F*$\"3m>(=AVr)**>F*7$$!3#*********GuY:F*$\"3%fWx-t5(**> F*7$$!35+++\"4D(Q:F*$\"3c\\r*>w&[**>F*7$$!35+++u%*oA:F*$\"3W\\+])3V))* >F*7$$!3'******z&Ql1:F*$\"3y:y/xN%z*>F*7$$!3!******fi#eu9F*$\"3#H#*[8. v`*>F*7$$!3(******HR6DW\"F*$\"3e*GFkw#y\"*>F*7$$!3)******z#*o$y8F*$\"3 )\\G+m!Ha\")>F*7$$!3$******>YEUJ\"F*$\"3k_NF7hEn>F*7$$!3*)******Hkg-7F *$\"3+^%G\"z.)H$>F*7$$!3++++(R')44\"F*$\"3Cr#>sn#3()=F*7$$!3y*****R'fH *R)!#=$\"3O='Rj7'fW6pNrFao$\"3M.%R?HOXl\"F*7$$!3a***** *zi3seFao$\"3I!)e'QRRSb\"F*7$$!35+++Ir[9YFao$\"3uI5ajdCX9F*7$$!35+++!) z)oN$Fao$\"3[eQw*f>%H8F*7$$!3)******\\eK4>#Fao$\"3KZIxYYM<7F*7$$!3,+++ !>x\\-\"Fao$\"3fF7YV$=B5\"F*7$$\"35+++]3-B=!#>$\"3'p%*fD,3x\")*Fao7$$ \"35+++g8e*Q\"Fao$\"3:LY.8j)[h)Fao7$$\"3A+++:8:QEFao$\"3m5tNJUM#R(Fao7 $$\"3!)******p7s')QFao$\"3F)))oY\"3S5iFao7$$\"3K+++XqGJ^Fao$\"3'[gUB(e %44&Fao7$$\"3w******>G&eP'Fao$\"3#))>7!H8VZSFao7$$\"3')*****\\pTgl(Fao $\"33y5S1HFqIFao7$$\"3********p0BO*)Fao$\"3o(e-%>'Hl?#Fao7$$\"3++++p79 >6F*$\"3T'=enBSF+\"Fao7$$\"33+++:83Y7F*$\"3=GS>$GJeA&Feq7$$\"3%******4 O@IP\"F*$\"3mY>;^-Q\\>Feq7$$\"33+++)*>vO9F*$\"3+lf1Hs\\q*)!#?7$$\"34++ +OE[+:F*$\"3])4.UD*)4Z#F_u7$$\"3,+++azMK:F*$\"3C#e;WNw/R(!#@7$$\"3-+++ tK@k:F*$\"3[sO3Q?#o;#!#A7$$\"3-+++#fygf\"F*$\"3C)=ImV%z&>$Fju7$$\"3%** *****4R%zi\"F*$\"3.4;9v(yCj\"F_u7$$\"3)******z=g$*o\"F*$\"32^+>\"pk/-( F_u7$$\"3/+++mkx]7=F*$\"33v$[AT!\\**GF eq7$$\"3!******>-4O(=F*$\"3S+p\"**p\\)\\XFeq7$$\"3#*******\\F:&)>F*$\" 3arbmN&QCY)Feq7$$\"39+++ykp'4#F*$\"3;6C!Qwv7N\"Fao7$$\"37+++U!p>O#F*$ \"3_a\"eA5#))pHFao7$$\"3)*******p$HVZ#F*$\"33V`tLVh6QFao7$$\"3#)*****z p*o'e#F*$\"3CY![O!=RJZFao7$$\"33+++s9R#HJ#)Fao7$$\"3 9+++x'z%zIF*$\"3%*p\"pdap#z$*Fao7$$\"3$******>Z8k?$F*$\"3#44\"fMaxk5F* 7$$\"33+++msMLLF*$\"3]rO>S;e!>\"F*7$$\"3!)*****\\D=UX$F*$\"3x<7VI!evI \"F*7$$\"3%******RC*3vNF*$\"3dVw^Gl/?9F*7$$\"3,+++Ua?,PF*$\"3pk%464e3` \"F*7$$\"3++++R;KFQF*$\"3&)[ca$QPKj\"F*7$$\"3)******znNJ%RF*$\"3k(3Z>F I%=>F*7$$\"3W+++)y\\&QWF*$\"3gM#))GwRF'>F*7$$\"3)******zo2$oXF*$\" 3esgcM#Q'*)>F*7$$\"3n*****\\+&yCYF*$\"3%R;u^Alh*>F*7$$\"3C+++ABE\"o%F* $\"39k1o/c^**>F*7$$\"3h******z4]4ZF*$\"3!p\\n(He****>F*7$$\"3()*****zj Rxt%F*$\"3YbwG*oy'**>F*7$$\"37+++'Hyfw%F*$\"3'yWJiVk&)*>F*7$$\"3c***** \\&p@%z%F*$\"3K6/&*eRl'*>F*7$$\"3p*****R06i\"\\F*$\"3?1TO#4+$z>F*7$$\" 3#)*****H:0#Q]F*$\"3koRs*z*QZ>F*7$$\"3W+++&*pBk^F*$\"3y!)=cU(R'o`&F*$\"3+UT$e^U*y;F* 7$$\"3)*******42;ccF*$\"3%*p'3j%zt'e\"F*7$$\"3'******pWdax&F*$\"3!)pT! 4*H>'[\"F*7$$\"3z*****\\8;z!fF*$\"3IU)yD)H_m8F*7$$\"3M+++@[PSgF*$\"30! 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 53 "Since the expression on the right side only involves " }{XPPEDIT 18 0 "cos*theta" "6#*&%$c osG\"\"\"%&thetaGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin*theta" "6 #*&%$sinG\"\"\"%&thetaGF%" }{TEXT -1 50 ", this suggests that we shoul d start by replacing " }{XPPEDIT 18 0 "cot*theta" "6#*&%$cotG\"\"\"%&t hetaGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "tan*theta" "6#*&%$tanG\" \"\"%&thetaGF%" }{TEXT -1 43 " by their defining expressions in terms \+ of " }{XPPEDIT 18 0 "cos*theta" "6#*&%$cosG\"\"\"%&thetaGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin*theta" "6#*&%$sinG\"\"\"%&thetaGF%" } {TEXT -1 27 ". This gives the identity: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cot*theta-tan*theta=cos*theta/(sin*theta)-sin *theta/(cos*theta)" "6#/,&*&%$cotG\"\"\"%&thetaGF'F'*&%$tanGF'F(F'!\" \",&*(%$cosGF'F(F'*&%$sinGF'F(F'F+F'*(F0F'F(F'*&F.F'F(F'F+F+" }{TEXT -1 3 ". " }}{PARA 258 "" 0 "" {TEXT -1 31 "By taking a common denomin ator " }{XPPEDIT 18 0 "sin*theta*cos*theta" "6#**%$sinG\"\"\"%&thetaGF %%$cosGF%F&F%" }{TEXT -1 66 " on the right side of this last identity, we obtain the identity: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "cot*theta-tan*theta=(cos^2*theta-sin^2*theta)/(sin*thet a*cos*theta)" "6#/,&*&%$cotG\"\"\"%&thetaGF'F'*&%$tanGF'F(F'!\"\"*&,&* &%$cosG\"\"#F(F'F'*&%$sinGF0F(F'F+F'**F2F'F(F'F/F'F(F'F+" }{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 25 "Comparing the expres sion " }{XPPEDIT 18 0 "(2*cos^2*theta-1)/(sin*theta*cos*theta)" "6#*&, &*(\"\"#\"\"\"*$%$cosGF&F'%&thetaGF'F'F'!\"\"F'**%$sinGF'F*F'F)F'F*F'F +" }{TEXT -1 82 " on the right of the identity that we are attempting \+ to prove with the expression " }{XPPEDIT 18 0 "(cos^2*theta-sin^2*thet a)/(sin*theta*cos*theta)" "6#*&,&*&%$cosG\"\"#%&thetaG\"\"\"F)*&%$sinG F'F(F)!\"\"F)**F+F)F(F)F&F)F(F)F," }{TEXT -1 61 " on the right side of the identity (i), we need to show that " }{XPPEDIT 18 0 "cos^2*theta- sin^2*theta=2*cos^2*theta-1" "6#/,&*&%$cosG\"\"#%&thetaG\"\"\"F)*&%$si nGF'F(F)!\"\",&*(F'F)*$F&F'F)F(F)F)F)F," }{TEXT -1 34 ". This follows \+ from the fact that " }{XPPEDIT 18 0 "sin^2*theta=1-cos^2*theta" "6#/*& %$sinG\"\"#%&thetaG\"\"\",&F(F(*&%$cosGF&F'F(!\"\"" }{TEXT -1 7 ", sin ce" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos^2*theta-sin ^2*theta=cos^2*theta-(1-cos^2*theta)" "6#/,&*&%$cosG\"\"#%&thetaG\"\" \"F)*&%$sinGF'F(F)!\"\",&*&F&F'F(F)F),&F)F)*&F&F'F(F)F,F," }{XPPEDIT 18 0 "``=cos^2*theta-1+cos^2*theta" "6#/%!G,(*&%$cosG\"\"#%&thetaG\"\" \"F*F*!\"\"*&F'F(F)F*F*" }{XPPEDIT 18 0 "``=2*cos^2*theta-1" "6#/%!G,& *(\"\"#\"\"\"*$%$cosGF'F(%&thetaGF(F(F(!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "The proo f of the identity in this example could be given by following steps wh ich construct the right side from the left side. " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "cot*theta-tan*theta = cos*theta/(sin* theta)-sin*theta/(cos*theta)" "6#/,&*&%$cotG\"\"\"%&thetaGF'F'*&%$tanG F'F(F'!\"\",&*(%$cosGF'F(F'*&%$sinGF'F(F'F+F'*(F0F'F(F'*&F.F'F(F'F+F+ " }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` `=(cos^2*theta-sin^2*theta)/(sin*theta*cos*theta)" "6#/%!G*&,&*&%$cosG \"\"#%&thetaG\"\"\"F+*&%$sinGF)F*F+!\"\"F+**F-F+F*F+F(F+F*F+F." } {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=( cos^2*theta-(1-cos^2*theta))/(sin*theta*cos*theta)" "6#/%!G*&,&*&%$cos G\"\"#%&thetaG\"\"\"F+,&F+F+*&F(F)F*F+!\"\"F.F+**%$sinGF+F*F+F(F+F*F+F ." }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``=(2*cos^2*theta-1)/(sin*theta*cos*theta)" "6#/%!G*&,&*(\"\"#\"\"\"*$ %$cosGF(F)%&thetaGF)F)F)!\"\"F)**%$sinGF)F,F)F+F)F,F)F-" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "cot(theta)-tan(theta);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$cotG6#%&thetaG\"\"\"-%$tanGF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&*&\"\"#\"\"\")-%$cosG6#%&thetaGF&F'F'F'! \"\"F'-%$sinGF+F-F)F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The graphs of the left and right sides coincide. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 599 "pi := evalf(Pi):\nf := theta->cot(theta)-tan(theta):'f(theta)'= f(theta);\ng := theta->(2*cos(theta)^2-1)/sin(theta)/cos(theta):'g(the ta)'=g(theta);\np1:=plot([f(theta),g(theta)],theta=-Pi/2..5.3,color=[r ed,green],thickness=[1,2],\n thickness=2,discont=true):\np2:=plot([[ [-pi/2,-8],[-pi/2,8]],[[pi/2,-8],[pi/2,8]],[[pi,-8],[pi,8]],\n [[3*p i/2,-8],[3*pi/2,8]]],color=black,linestyle=3):\nt1:=plots[textplot]([5 .3,-.8,`q`],font=[SYMBOL,11]):\nplots[display]([p1,p2,t1],xtickmarks=[ -pi/2=`-p/2`,pi/2=`p/2`,\n pi=`p`,3*pi/2=`3p/2`],ytickmarks=3,font=[ SYMBOL,11],\n labels=[``,``],view=[-1.58..5.3,-8..8]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"fG6#%&thetaG,&-%$cotGF&\"\"\"-%$tanGF&!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%&thetaG*(,&*&\"\"#\"\"\" )-%$cosGF&F+F,F,F,!\"\"F,-%$sinGF&F0F.F0" }}{PARA 13 "" 1 "" {GLPLOT2D 512 300 300 {PLOTDATA 2 "6--%'CURVESG6)7ao7$$!3-+++Cjzq:!#<$ \"3W$GW#**)\\zd$!\"*7$$!3!=$4qejsp:F*$\"3Y4eD9)egM*!#:7$$!3Oj=S$Rc'o:F *$\"3e;%p569In%F37$$!39&z-\"Gken:F*$\"3e42*)>.K:JF37$$!3#ps.GY;lc\"F*$ \"3cA2Z#ylkL#F37$$!3[!f0A`wVc\"F*$\"3!e+t#Rmfd:F37$$!3/aug,mBi:F*$\"3k f&fLnZ\"o6F37$$!3;\"=6/uczb\"F*$\"3W&4EE'3q'y(!#;7$$!3E3\\@zon`:F*$\"3 &3?9&3s-ReFR7$$!3]iB#o:<^a\"F*$\"3]^nltBy!*QFR7$$!3_;)HWVdl`\"F*$\"3p7 L&\\h)3;HFR7$$!3=#[fq*=m@:F*$\"3**e_-gsnG?FR7$$!3jZ\"*ofjw1:F*$\"3EnMH Y?A`:FR7$$!3'G-#R!=jKZ\"F*$\"3a-R)e$\\D75FR7$$!344[F?z`R9F*$\"3NFCRp!R FW(F*7$$!3<6>V]H(fS\"F*$\"3k1@!*HtsXeF*7$$!3?LBxER&[P\"F*$\"3Je#>ubp&R [F*7$$!3+PqD!)>jU8F*$\"3#*)R[2vxU2%F*7$$!3cYp2(*zI48F*$\"3Wa(*o8R3pMF* 7$$!3yUNL%)34w7F*$\"3aTX-ha'3*HF*7$$!3Z(zu4+B>C\"F*$\"3=75U6[**)e#F*7$ $!31[Wr5z#=@\"F*$\"3w\"37K+Q)*G#F*7$$!3EOS3nz%z<\"F*$\"3IJQIKh!))*>F*7 $$!3/g*\\\"4*GR9\"F*$\"3_D::)fkPu\"F*7$$!3xIBDV_966F*$\"3cT65l%H^_\"F* 7$$!3shf)Q`u83\"F*$\"3!=xF=.vZM\"F*7$$!3#**ok4Guf/\"F*$\"3e$)*p)>c=[6F *7$$!3t_:GBc)f,\"F*$\"3r@[`w*e#Q**!#=7$$!3R5Y?_o,6)*Fat$\"3+G$\\%*38SD )Fat7$$!3(4j0=*[B-&*Fat$\"3w@R`\"e\"pUoFat7$$!3_H*R>9_M;*Fat$\"3I!4B*y T,h`Fat7$$!3D)GE*[3&3%))Fat$\"3-_QG\"H[&**RFat7$$!3MQXwU-D/&)Fat$\"3I] hVn\"Reh#Fat7$$!3Oc3(4\"e9&>)Fat$\"3Qzf=z%ynO\"Fat7$$!3#[5BF#\\thyFat$ \"3t%>'>&)zJ,J!#?7$$!3[\\nD8ZT:vFat$!3QAbs/8Mc8Fat7$$!3Gdb%G&>%R@(Fat$ !3+Se)[gMUd#Fat7$$!3]9`[7JM))oFat$!3251`uPI6RFat7$$!35IP_]w'>b'Fat$!3) 4>u6zd!H`Fat7$$!3It$Rd))*)GA'Fat$!3#o8%Q\\(*4mnFat7$$!3R.rWF-\\/fFat$! 3fB[E)G?(=#)Fat7$$!3!=>pDph4b&Fat$!3ie'[r'[-C**Fat7$$!3@)ek,e+LB&Fat$! 31%z%yVT8c6F*7$$!3#z%eER&HT*[Fat$!3&p*)eHPE[M\"F*7$$!3?@I[l,z'e%Fat$!3 kFZ2=zGJ:F*7$$!3?==(40(y]UFat$!3;!RE;(yRcF*7$$!31E]J\"fjTg$Fat$!3gYT@Q$)\\wAF*7$$!31\"\\,O1o5G$Fat$!3')[E z2^@(f#F*7$$!3Q\"\\?!*)p\"G%HFat$!3%=M`1!zN'*HF*7$$!3mi^5-5/nilMF*7$$!3W\")eh-!))QG#Fat$!3[czq'Ge'pSF*7$$!3;)on\"yO\\`>Fat$! 3)))p*Hk\"ye&[F*7$$!3)*HpU2c*)\\;Fat$!3$zZ\"[3)*QReF*7$$!335LLdz$>I\"F at$!3c9./vgZ1vF*7$$!3*)zJcq[?2**!#>$!3![\"\\fw6Ah**F*7$$!3!po(GCv(*)e' F]]l$!3]tLEp1*)3:FR7$$!3([Ig4/%*4+&F]]l$!3eUzD>+$H*>FR7$$!3$G#Hjd0,8MF ]]l$!3`?_&=Z7a#HFR7$$!3W3Xw'*zvfDF]]l$!3r\\!y`p0K!RFR7$$!30%4'*eV0lq\" F]]l$!3+A?#p2`w&eFR7$$!3'o)=Yb\"z)z7F]]l$!32Q!yg1J:\"yFR7$$!3e'zw-vGD` )Fev$!3S9&GQtr=<\"F37$$!3ggd5[tR*R'Fev$!3y+iV>=ci:F37$$!3_DZ$f%fEmUFev $!35[fKfK\"RM#F37$$!3_2#\\[C+(*>$Fev$!3Y7mF%4]_7$F37$$!3(**ojPaMJ8#Fev $!3a3\\O#R3zo%F37$$!3+s\"yE%)ol1\"Fev$!35:W3Rc%eP*F37$$!3*)*****Rl#fTJ !#E$!3m!GA9'))4$=$F-7ao7$$\"3*)*****Rl#fTJF_al$\"3m!GA9'))4$=$F-7$$\"3 O))3mS%o*p5Fev$\"3!R#Q*eybgM*F37$$\"3pA\"H(RP$*R@Fev$\"3#o!Q(RN8In%F37 $$\"3#oN(zQ!**)4KFev$\"35z3S$)*>`6$F37$$\"3'4flyLk)zUFev$\"38wI>$flkL# F37$$\"3Af?+O\\z>kFev$\"3w$*p9blfd:F37$$\"3[F&QT`D(f&)Fev$\"3]Sm.Ew9o6 F37$$\"3SY6/t'eRG\"F]]l$\"3OUiF_1q'y(FR7$$\"3BS%oEzW>r\"F]]l$\"3xX@.:wK\\D75FR7$$\"3+NymoSe78Fat$\"3 _R1!)[!RFW(F*7$$\"3!R\"o4nPB[;Fat$\"3%e%ym;tsXeF*7$$\"3=%f#p.SUf>Fat$ \"3FdB\"zap&R[F*7$$\"3_db%)oMk\"G#Fat$\"3E[lgVxFuSF*7$$\"38hkk+L)[h#Fa t$\"3sT183R3pMF*7$$\"3!*)\\!3GW0ZHFat$\"3%eYOlXl3*HF*7$$\"3a^zmhKt)G$F at$\"3;i.r2[**)e#F*7$$\"35Y9FkTo*e$Fat$\"33Yu++!Q)*G#F*7$$\"3nkbd+O[GR Fat$\"3s<>`Hh!))*>F*7$$\"3#pK;*zTnoUFat$\"3*p`5dfkPu\"F*7$$\"3+>E*)Q3^ 'f%Fat$\"3FNz!HYH^_\"F*7$$\"3)*3jbKz@%*[Fat$\"3`:S\")H]xW8F*7$$\"3YE!p k%p*eFat$\"3_!\\>K28SD)Fat7$$\"3y&H5'z$Gd?'Fat$\"3tmF/m:pUoFat7$$ \"3B(*fZH6^WlFat$\"39F)eS;95O&Fat7$$\"3[Q'*[AC6noFat$\"3mF9'oF[&**RFat 7$$\"3T)Q^'GIr.sFat$\"3')z5L`\"Reh#Fat7$$\"3Pq]Wgu\"G^(Fat$\"3?tLDl%yn O\"Fat7$$\"3%>#Gp[$Gi%yFat$\"36V)3%)f;85$Fev7$$\"3Ex\"f\"e&[D>)Fat$!3! 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"" {TEXT 294 8 "Question" }{TEXT -1 22 ": Verify the identity " } {XPPEDIT 18 0 "1/(csc*theta-1)-1/(csc*theta+1)=2*tan^2*theta" "6#/,&*& \"\"\"F&,&*&%$cscGF&%&thetaGF&F&F&!\"\"F+F&*&F&F&,&*&F)F&F*F&F&F&F&F+F +*(\"\"#F&*$%$tanGF0F&F*F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 295 8 "Solution" }{TEXT -1 2 ": " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(csc*theta-1)-1/(cs c*theta+1)=(``(csc*theta+1)-``(csc*theta-1))/((csc*theta+1)*(csc*theta -1))" "6#/,&*&\"\"\"F&,&*&%$cscGF&%&thetaGF&F&F&!\"\"F+F&*&F&F&,&*&F)F &F*F&F&F&F&F+F+*&,&-%!G6#,&*&F)F&F*F&F&F&F&F&-F26#,&*&F)F&F*F&F&F&F+F+ F&*&,&*&F)F&F*F&F&F&F&F&,&*&F)F&F*F&F&F&F+F&F+" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=(csc*theta+1-csc*theta+1)/( csc^2*theta-1)" "6#/%!G* &,**&%$cscG\"\"\"%&thetaGF)F)F)F)*&F(F)F*F)!\"\"F)F)F),&*&F(\"\"#F*F)F )F)F,F," }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2/(csc^2*theta-1)" "6#/%!G*&\"\"#\"\"\",&*&%$cscGF&%&thetaGF' F'F'!\"\"F," }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=2/(cot^2*theta)" "6#/%!G*&\"\"#\"\"\"*&%$cotGF&%&the taGF'!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "( using the identity " }{XPPEDIT 18 0 "cot^2*theta+1 = csc^2*theta" "6#/,&*&%$cot G\"\"#%&thetaG\"\"\"F)F)F)*&%$cscGF'F(F)" }{TEXT -1 4 " ), " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*tan^2*theta" "6#/%!G*( \"\"#\"\"\"*$%$tanGF&F'%&thetaGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "1/(csc(theta )-1)-1/(csc(theta)+1);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*&\"\"\"F%,&-%$cscG6#%&thetaGF%F%!\"\"F+F%*&F%F%,&F'F%F%F%F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"-%$sinG6#%&thetaGF%-%$c osGF)!\"#F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The graphs of the left and right sides coincide, except where \+ " }{XPPEDIT 18 0 "theta = ` . . . `,-Pi,0,Pi,2*Pi,3*Pi,` . . . `;" "6) /%&thetaG%(~.~.~.~G,$%#PiG!\"\"\"\"!F'*&\"\"#\"\"\"F'F,*&\"\"$F,F'F,F% " }{TEXT -1 21 ". At these values of " }{XPPEDIT 18 0 "theta" "6#%&the taG" }{TEXT -1 82 " the left side of the identity is not defined but t he right side has the value 0. 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in*theta+tan*theta)=csc*theta" "6#/*&,&\"\"\"F&*&%$secGF&%&thetaGF&F&F &,&*&%$sinGF&F)F&F&*&%$tanGF&F)F&F&!\"\"*&%$cscGF&F)F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 299 8 "Solu tion" }{TEXT -1 2 ": " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1+sec*theta)/(sin*theta+tan*theta) = (1+sec*theta)/(sin*theta+s in*theta/(cos*theta));" "6#/*&,&\"\"\"F&*&%$secGF&%&thetaGF&F&F&,&*&%$ sinGF&F)F&F&*&%$tanGF&F)F&F&!\"\"*&,&F&F&*&F(F&F)F&F&F&,&*&F,F&F)F&F&* (F,F&F)F&*&%$cosGF&F)F&F/F&F/" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(1+sec*theta)/(sin*theta*(1+1/(cos*t heta))" "6#/%!G*&,&\"\"\"F'*&%$secGF'%&thetaGF'F'F'*(%$sinGF'F*F',&F'F '*&F'F'*&%$cosGF'F*F'!\"\"F'F'F1" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(1+sec*theta)/(sin*theta*(1+sec*thet a))" "6#/%!G*&,&\"\"\"F'*&%$secGF'%&thetaGF'F'F'*(%$sinGF'F*F',&F'F'*& F)F'F*F'F'F'!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(sin*theta)" "6#/%!G*&\"\"\"F&*&%$sinGF&%&thetaGF &!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=csc*theta" "6#/%!G*&%$cscG\"\"\"%&thetaGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "(1+sec(theta))/(sin(theta)+tan(theta));\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%-%$secG6#%&thetaGF%F%,&-%$sinGF( F%-%$tanGF(F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$-%$sin G6#%&thetaG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The graphs of the left and right sides coincide, except w here " }{XPPEDIT 18 0 "theta = ` . . . `,-3*Pi/2,-Pi/2,Pi/2,3*Pi/2,5*P i/2,7*Pi/2,` . . . `;" "6*/%&thetaG%(~.~.~.~G,$*(\"\"$\"\"\"%#PiGF)\" \"#!\"\"F,,$*&F*F)F+F,F,*&F*F)F+F,*(F(F)F*F)F+F,*(\"\"&F)F*F)F+F,*(\" \"(F)F*F)F+F,F%" }{TEXT -1 21 ". At these values of " }{XPPEDIT 18 0 " theta" "6#%&thetaG" }{TEXT -1 82 " the left side of the identity is no t defined but the right side has the value 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 594 "pi := evalf (Pi):\nf := theta->(1+sec(theta))/(sin(theta)+tan(theta)):'f(theta)'=f (theta);\ng := theta->csc(theta):'g(theta)'=g(theta);\np1:=plot([f(the ta),g(theta)],theta=-2..10.3,color=[red,green],thickness=[1,2],\n th ickness=2,discont=true):\np2:=plot([[[pi,-8],[pi,8]],[[2*pi,-8],[2*pi, 8]],\n [[3*pi,-8],[3*pi,8]]],color=black,linestyle=3):\nt1:=plots[te xtplot]([10.3,-.6,`q`],font=[SYMBOL,11]):\nplots[display]([p1,p2,t1],x tickmarks=[-pi/2=`-p/2`,pi/2=`p/2`,\n pi=`p`,3*pi/2=`3p/2`,2*pi=`2p` ,5*pi/2=`5p/2`,3*pi=`3p`],\n ytickmarks=3,font=[SYMBOL,11],\n labe ls=[``,``],view=[-2..10.3,-8..8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"fG6#%&thetaG*&,&\"\"\"F*-%$secGF&F*F*,&-%$sinGF&F*-%$tanGF&F*!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%&thetaG-%$cscGF&" }} {PARA 13 "" 1 "" {GLPLOT2D 459 258 258 {PLOTDATA 2 "6,-%'CURVESG6-7S7$ $!\"#\"\"!$!3S;YHq,v*4\"!#<7$$!373zX\"eW1*>F-$!3y-:Im$4^4\"F-7$$!3)\\e 2=[/D)>F-$!336QDy0=\"4\"F-7$$!3o]VS)3]L(>F-$!3[tnV\"\\\")o3\"F-7$$!351 $RP,NT'>F-$!3m\"[x$f#zE3\"F-7$$!3mxbrOP'\\&>F-$!3mFpBJ(>'y5F-7$$!3)QS4 *)ygk%>F-$!3]G?NF-$!3Ik#R%)G&Gr5F-7$$!3Oxz-\\ 5bG>F-$!3c\\Md'G%fn5F-7$$!3Qv\">.#[Z>>F-$!3q_2))Qy-k5F-7$$!3R`=[.)Q,\" >F-$!3)*Go]@\\Zg5F-7$$!3FS)pZj:>!>F-$!3'pOwW@Tu0\"F-7$$!3!eK_WHeE*=F-$ !3+Ka,3<8a5F-7$$!3aSggUHO$)=F-$!3+uX+C(>40\"F-7$$!39K`r[^Su=F-$!3]d4am #Gz/\"F-7$$!3yx0p81Fm=F-$!3%*=dLF*)HX5F-7$$!3+sqvkyfc=F-$!3er1v?!zA/\" F-7$$!3CEQjxPS[=F-$!3<#4//I5)R5F-7$$!3?2oL1@()Q=F-$!3l[GOU1/P5F-7$$!3 \\)Rf@'\\VI=F-$!3\"3?1fazY.\"F-7$$!3([O\"yx!y6#=F-$!3c9Af8d=K5F-7$$!3J 65zMLO7=F-$!3mN&GD)R!*H5F-7$$!3XSy?vg;.=F-$!3qZqF*))=w-\"F-7$$!3#oxO)> ,s%z\"F-$!3)GC6zi0c-\"F-7$$!3!p9gf,5cy\"F-$!3+g\\JfX_B5F-7$$!3Sy.Qyr9w 5F-7$$!3a!Q7E58!f#)\\C;5F-7$$!3>Z*\\!*GI3u\"F -$!3*=%z0()>j95F-7$$!3pVZL^.8Kb?IBXq\"F-$!3? 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 8 "Solution " }{TEXT -1 2 ": " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " (sec*theta-tan*theta)^2=(1/(cos*theta)-sin*theta/(cos*theta))^2" "6#/* $,&*&%$secG\"\"\"%&thetaGF(F(*&%$tanGF(F)F(!\"\"\"\"#*$,&*&F(F(*&%$cos GF(F)F(F,F(*(%$sinGF(F)F(*&F2F(F)F(F,F,F-" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(1-sin*theta)^2/(cos^2*t heta)" "6#/%!G*&,&\"\"\"F'*&%$sinGF'%&thetaGF'!\"\"\"\"#*&%$cosGF,F*F' F+" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(1-sin*theta)^2/(1-sin^2*theta)" "6#/%!G*&,&\"\"\"F'*&%$sinGF'%&th etaGF'!\"\"\"\"#,&F'F'*&F)F,F*F'F+F+" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(1-sin*theta)^2/((1-sin*theta)* (1+sin*theta))" "6#/%!G*&,&\"\"\"F'*&%$sinGF'%&thetaGF'!\"\"\"\"#*&,&F 'F'*&F)F'F*F'F+F',&F'F'*&F)F'F*F'F'F'F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``=(1-sin*theta)/(1+sin*theta)" "6#/%!G*&,&\"\"\"F'*&%$sinGF'%&thetaGF '!\"\"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 39 "(sec(theta)- tan(theta))^2;\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&- %$secG6#%&thetaG\"\"\"-%$tanGF(!\"\"\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"\"\"!\"\"-%$sinG6#%&thetaGF&F&,&F(F&F&F&F'F'" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The gra phs of the left and right sides coincide, except where " }{XPPEDIT 18 0 "theta = ` . . . `,-3*Pi/2,Pi/2,5*Pi/2,9*Pi/2,` . . . `;" "6(/%&thet aG%(~.~.~.~G,$*(\"\"$\"\"\"%#PiGF)\"\"#!\"\"F,*&F*F)F+F,*(\"\"&F)F*F)F +F,*(\"\"*F)F*F)F+F,F%" }{TEXT -1 21 ". At these values of " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 82 " the left side of the identity is not defined but the right side has the value 0. 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" }}{PARA 0 "" 0 "" {TEXT -1 71 "The p oints P and Q are in the same vertical line, and so have the same " } {TEXT 281 1 "x" }{TEXT -1 22 " coordinate, that is, " }{XPPEDIT 18 0 " cos(-theta) = cos*theta;" "6#/-%$cosG6#,$%&thetaG!\"\"*&F%\"\"\"F(F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 282 1 "x" } {TEXT -1 67 " axis is the perpendicular bisector of the line segment P Q, so the " }{TEXT 283 1 "y" }{TEXT -1 78 " coordinates of P and Q hav e the same magnitude, but opposite signs, that is, " }{XPPEDIT 18 0 "s in(-theta) = -sin*theta;" "6#/-%$sinG6#,$%&thetaG!\"\",$*&F%\"\"\"F(F, F)" }{TEXT -1 2 ". 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}{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 20 " radians are called " }{TEXT 259 13 "supplementary" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "The following identities hold." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(Pi-theta) = -cos*theta;" "6#/-%$cos G6#,&%#PiG\"\"\"%&thetaG!\"\",$*&F%F)F*F)F+" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi-theta) = sin*theta; " "6#/-%$sinG6#,&%#PiG\"\"\"%&thetaG!\"\"*&F%F)F*F)" }{TEXT -1 1 " " } }{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 279 10 "__________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 84 "These identities can be appreciated by referring to the following \+ picture, in which " }{XPPEDIT 18 0 "phi = Pi-theta;" "6#/%$phiG,&%#PiG \"\"\"%&thetaG!\"\"" }{TEXT -1 51 ", and the points P and Q are symmet rical about the " }{TEXT 285 1 "y" }{TEXT -1 7 " axis. 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),$*&F%F)F(F)!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 278 10 "__________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "These identities can be appreci ated by referring to the following picture, in which " }{XPPEDIT 18 0 "phi = theta+Pi;" "6#/%$phiG,&%&thetaG\"\"\"%#PiGF'" }{TEXT -1 59 ", a nd the points P and Q are symmetrical about the origin. 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!3MjfuqPJs=F/$\"3c'3SvnZZ`$F/7$$!3g#RS6&))pV@F/$\"3u8..3J1xLF/7$$!3#Hz $R`[V'R#F/$\"3.O0Xf2n-KF/7$$!3yYZOfBaFEF/$\"3kvwol5'f,$F/7$$!3qL\"HDH7 r'GF/$\"3%HJ02Y0#*y#F/7$$!3)oiF7J*ylIF/$\"3[9OSe'G#pDF/7$$!3w>dIb*4$fK F/$\"3.hbh`S\")=BF/7$$!3wAY3`#yqT$F/$\"3/GC3%4@$z?F/7$$!3;:)y'H^VpNF/$ \"3gs='*eoI0=F/7$$!3'3#*[\"GAy#p$F/$\"3hshMd@KP:F/7$$!3R1:_xVB+QF/$\"3 q]%[cH'G[7F/7$$!39RANp!eM)QF/$\"3c6-7:E;&e*F27$$!3)>g$ed:%p%RF/$\"3o.L `siZ$\\'F27$$!3Y,!y\"3i&[)RF/$\"37dgq(pptZ$F27$$!3/jW&zqE)**RF/$\"3+U& y&*[8Ls$Fd`l7$$!3_`')R(G93*RF/$!3151EiyE4FF27$$!3#3i&za(G;'RF/$!3cpLN9 Q=FbF27$$!3Oy<+QFv.RF/$!3yt!3+@&*=s)F27$$!3_]GAh$F/$!3-ADB!\\#3=Fa\\p$\"#HFa\\pQ\"fFe\\pF_\\nF]_p-F\\\\p6&7$$\"$ N\"Fa\\pFb\\pF\\_pF_\\nF]_p-F\\\\p6&7$$\"$h\"Fa\\pFb\\pF\\_pF_\\nF]_p- F\\\\p6&7$$!$N\"Fa\\pF[]pFi_pF_\\nF]_p-F\\\\p6&7$$!$4\"Fa\\pF[]pFi_pF_ \\nF]_p-%*AXESTICKSG6$F*F*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q !Fe\\pFhap-F^_p6#%(DEFAULTG-%%VIEWG6$F[bpF[bp" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" }}{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "Addition formulas for the sine a nd cosine functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 25 " b e two angles such that " }{XPPEDIT 18 0 "alpha+beta;" "6#,&%&alphaG\" \"\"%%betaGF%" }{TEXT -1 14 " is less than " }{XPPEDIT 18 0 "Pi/2" "6# *&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 10 " radians. " }}{PARA 0 "" 0 "" {TEXT -1 94 "Then we can draw the following picture in which the lengt h of the line OP is 1 unit, angle POQ" }{XPPEDIT 18 0 "`` = alpha" "6# /%!G%&alphaG" }{TEXT -1 14 " and angle QOR" }{XPPEDIT 18 0 "`` = beta " "6#/%!G%%betaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "P is the point" }{XPPEDIT 18 0 "``(cos(al pha+beta),sin(alpha+beta));" "6#-%!G6$-%$cosG6#,&%&alphaG\"\"\"%%betaG F+-%$sinG6#,&F*F+F,F+" }{TEXT -1 13 ", that is, OS" }{XPPEDIT 18 0 "`` = cos(alpha+beta);" "6#/%!G-%$cosG6#,&%&alphaG\"\"\"%%betaGF*" } {TEXT -1 7 " and SP" }{XPPEDIT 18 0 "`` = sin(alpha+beta);" "6#/%!G-%$ sinG6#,&%&alphaG\"\"\"%%betaGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 477 477 477 {PLOTDATA 2 "6G-%'CURVESG6%7$7$$\"\"!F)F(7$$\"3*)*****fh#=EU!#=$\" 3S+++ry2j!*F--%*LINESTYLEG6#\"\"\"-%'COLOURG6&%$RGBGF)F)F)-F$6%7$7$F+F (7$F+$\"3v******4PWPKF--F16#\"\"$F4-F$6%7$7$$\"3x******R/sUpF-F=F*-F56 &F7$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-F$6%7$F'FE-%&COLORG6&F7$\"#& )!\"#F)F)FM-F$6%7$7$FFF(FE-F56&F7$\")p:#R%FL$\")`B)e)FL$\")fqkdFLFM-F$ 6%7$FFYQ$cosF]qFT-Fep6&7$$\"$N$FisF]tF\\qFTFdq-Fep6%7$$\"\")F\\p$ \"# " 0 "" {MPLTEXT 1 0 163 "sin(alpha+b eta)=expand(sin(alpha+beta));\nsin(alpha-beta)=expand(sin(alpha-beta)) ;\ncos(alpha+beta)=expand(cos(alpha+beta));\ncos(alpha-beta)=expand(co s(alpha-beta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#,&%&alpha G\"\"\"%%betaGF),&*&-F%6#F(F)-%$cosG6#F*F)F)*&-F0F.F)-F%F1F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#,&%&alphaG\"\"\"%%betaG!\"\" ,&*&-F%6#F(F)-%$cosG6#F*F)F)*&-F1F/F)-F%F2F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,&%&alphaG\"\"\"%%betaGF),&*&-F%6#F(F)-F%6#F* F)F)*&-%$sinGF.F)-F3F0F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$c osG6#,&%&alphaG\"\"\"%%betaG!\"\",&*&-F%6#F(F)-F%6#F*F)F)*&-%$sinGF/F) -F4F1F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 85 "Examples to illustrate the addition formulas involving \+ the sine and cosine functions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 258 "" 0 "" {TEXT -1 18 "Using the formula " }{XPPEDIT 18 0 "sin(alpha+beta) = sin*alpha*cos*b eta+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 10 ", we have " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(x+Pi/3) = sin*x*c os(Pi/3)+cos*x*sin(Pi/3);" "6#/-%$sinG6#,&%\"xG\"\"\"*&%#PiGF)\"\"$!\" \"F),&*(F%F)F(F)-%$cosG6#*&F+F)F,F-F)F)*(F1F)F(F)-F%6#*&F+F)F,F-F)F)" }{TEXT -1 1 "," }}{PARA 258 "" 0 "" {TEXT -1 10 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(x+Pi/3) = sin*x*``(1/2) +cos*x*``(sqrt(3)/2);" "6#/-%$sinG6#,&%\"xG\"\"\"*&%#PiGF)\"\"$!\"\"F) ,&*(F%F)F(F)-%!G6#*&F)F)\"\"#F-F)F)*(%$cosGF)F(F)-F16#*&-%%sqrtG6#F,F) F4F-F)F)" }{TEXT -1 21 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 6 "expand" }{TEXT -1 48 " automatically substitutes the the exact valu es " }{XPPEDIT 18 0 "cos(Pi/3);" "6#-%$cosG6#*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "sin(Pi/3);" "6#-%$sinG6#*&%#PiG\"\" \"\"\"$!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(3)/2;" "6#*&-%%sq rtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 5 " and " } {TEXT 270 7 "warning" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 33 " The brackets around the argument " }{TEXT 271 1 "x" }{TEXT -1 149 " ar e usually omitted in the mathematical notation for trigonometric funct ions, but when working with trigonometric functions with Maple the arg ument " }{TEXT 259 28 "must be enclosed in brackets" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "sin*x" "6#*&%$sinG \"\"\"%\"xGF%" }{TEXT -1 21 " must be replaced by " }{XPPEDIT 18 0 "si n(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 4 " .. " }{TEXT 272 6 "sin(x)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "expand(sin(x+Pi/3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%$sinG6#%\"xGF&F&*&F%F&*&-%$cosGF*F& \"\"$F%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 45 "\nAs an illustration of the formula (iii) let \+ " }{XPPEDIT 18 0 "f(x) = sin*x*``(1/2);" "6#/-%\"fG6#%\"xG*(%$sinG\"\" \"F'F*-%!G6#*&F*F*\"\"#!\"\"F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g( x) = cos*x*``(sqrt(3)/2);" "6#/-%\"gG6#%\"xG*(%$cosG\"\"\"F'F*-%!G6#*& -%%sqrtG6#\"\"$F*\"\"#!\"\"F*" }{TEXT -1 29 ", and consider the graphs of " }{XPPEDIT 18 0 "y = f(x), y = g(x)" "6$/%\"yG-%\"fG6#%\"xG/F$-% \"gG6#F(" }{TEXT -1 15 ", and the sum " }{XPPEDIT 18 0 "y = f(x) + g( x)" "6#/%\"yG,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F)F*" }{TEXT -1 10 ", that i s," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = sin*x*``(1/ 2)+cos*x*``(sqrt(3)/2);" "6#/%\"yG,&*(%$sinG\"\"\"%\"xGF(-%!G6#*&F(F( \"\"#!\"\"F(F(*(%$cosGF(F)F(-F+6#*&-%%sqrtG6#\"\"$F(F.F/F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The graph of this last equation, plotted in " }{TEXT 260 3 "red" } {TEXT -1 70 " below, certainly looks like a horizontal translation of \+ the graph of " }{XPPEDIT 18 0 "y = sin*x" "6#/%\"yG*&%$sinG\"\"\"%\"xG F'" }{TEXT -1 64 ", which is to be expected, since it can be written i n the form " }{XPPEDIT 18 0 "y = sin(x+Pi/3);" "6#/%\"yG-%$sinG6#,&% \"xG\"\"\"*&%#PiGF*\"\"$!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 455 "f := x -> s in(x)*(1/2):\n'f(x)'=f(x);\ng := x -> cos(x)*sqrt(3)/2:\n'g(x)'=g(x); \np1 := plot([f(x),g(x),f(x)+g(x)],x=-Pi..10.1,y,color=[blue,magenta,r ed],thickness=[1$2,2],\nlegend=[`y = f(x)`,`y = g(x)`,`y = f(x)+g(x)`] ):\nt1 := plots[textplot]([[10.1,-.1,`x`],[-.2,1.25,`y`]],font=[HELVET ICA,10]):\nplots[display]([p1,t1],xtickmarks=[-pi=`-p`,-pi/2=`-p/2`,pi /2=`p/2`,pi=`p`,\n 3*pi/2=`3p/2`,2*pi=`2p`,5*pi/2=`5p/2`,3*pi=`3p`], font=[SYMBOL,11],labels=[``,``]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"fG6#%\"xG,$*&#\"\"\"\"\"#F+-%$sinGF&F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,$*&#\"\"\"\"\"#F+*&-%$cosGF&F+\"\"$F*F+F +" }}{PARA 13 "" 1 "" {GLPLOT2D 760 355 355 {PLOTDATA 2 "6+-%'CURVESG6 &7ar7$$!3*****4tk#fTJ!#<$!340m?Em*)RJ!#E7$$!3mnsCH\"ys*HF*$!3?q3F!o02> (!#>7$$!3MNX=6O'H&GF*$!3`vDND7>B9!#=7$$!3(ooj1K'RFFF*$!37QY,dz#zahgCF*$!3qvj()QFwZJF97$$!3gZ bTG>S>BF*$!3Su')4mJ>jOF97$$!3S9mqWF97$$!3U=x%GsGO*=F*$!3*HxKH<.ve:7Fl'\\F97$$!3(fG%4F-*4i\"F*$!3+ j)>Kv-P*\\F97$$!3mofd,\"*>)e\"F*$!3%pP$oxGC**\\F97$$!3O^w0wzSb:F*$!3'> 9O$3!3%**\\F97$$!3/M$R0&ohA:F*$!3kFgjnz>%*\\F97$$!3u;5-Dd#)*[\"F*$!3ih &=sN=O)\\F97$$!3c6'o/[7SN\"F*$!3m'G/[IrH)[F97$$!3Q1i\"fB*>=7F*$!3Qcz^X ]R#p%F97$$!3eII=Bt#GP*F9$!3nspfXDwHSF97$$!3a!=U\">$\\F(zF9$!3A/@X#Gusd $F97$$!3_I85:8nslF9$!3ib?;,\"yZ0$F97$$!3(=?$['**=D8&F9$!3)yj)>\"pj]X#F 97$$!3?t]'ynmBp$F9$!3)ykEfD=X!=F97$$!3Tw&F37$$\"34k`@a2BEFF3$\"3a+NBJl%HO\"F37$$\" 3%z&zWO*Q1q\"F9$\"3Yq:BmgEi%)F37$$\"3/'p+yO=X8$F9$\"3@;Nti0sT:F97$$\"3 9MM:*z(RoXF9$\"3Z=:ke3d0AF97$$\"3sB%fEK1-&fF9$\"3^Yvz/#HE!GF97$$\"3K8a ;Y[,KtF9$\"3=2#y))QfiM$F97$$\"3+!>J^OHoe)F9$\"3cbUR@I\"\\y$F97$$\"3nmp 4%)QkT)*F9$\"3;v?p.\"\\S;%F97$$\"30,1arRPL6F*$\"3*Gc`Y)GBHXF97$$\"3U0: naNe#G\"F*$\"3Y(pGJ.nPz%F97$$\"3'Q1a!*f$yX8F*$\"3#)f,vOj&R([F97$$\"3IA mVVO)*39F*$\"3X#)=tSVoM\\F97$$\"3u!=>yo$=s9F*$\"3'HoPKc3d(\\F97$$\"3=R lo*\\F97$$\"3M!)RDq?9s:F*$\"3MRqVsa****\\F97$$\"3s@ iI3/!*3;F*$\"3+99fO1P'*\\F97$$\"35j%ejueck\"F*$\"3G.@o(e&*f)\\F97$$\"3 C/2T%3mV]%H=F*$\"3r%))\\Z_vO$[F97$$\"3!QNnr')*ff>F*$\"30Hw8*Q;oi%F97$ $\"3EQ]r(H\\(*3#F*$\"3o@2FfZpTVF97$$\"3`C4g]QaKAF*$\"3[5zYZzfWRF97$$\" 3!3\"o[.%Q`P#F*$\"33g=YPi?nMF97$$\"3(eR/ee78^#F*$\"3mKAa'=bo%HF97$$\"3 %4)>7onGZEF*$\"3c/N3*o.@P#F97$$\"3q^>ui:;*y#F*$\"3@WychC!fs\"F97$$\"3Z A>Odj.JHF*$\"3eP`N7(>]/\"F97$$\"3_(pStq@81$F*$\"3iwpzN&Q#4SF37$$\"3es% >t02;>$F*$!3_gIr\\xn*\\#F37$$\"3N,5@3u8KLF*$!3-Yci84pp%*F37$$\"37ID5fx msMF*$!3ofWK\"o*HD;F97$$\"3hLKM)HR'=OF*$!3]4,()HV9nXF97$$\"3+Y&R&)H.]V%F*$!3n rz$ywXF*$!3%eVCP3)4a\\F97$$\"388U*))=f9h%F*$!3e+ xmaXbu\\F97$$\"3![9I8=Nhk%F*$!3K+m\"*)4I!*)\\F97$$\"3Oxgwt6\"3o%F*$!3- /oY7t](*\\F97$$\"3.4??mr[:ZF*$!3#))zIJ+w***\\F97$$\"3qSzjeJ;]ZF*$!3)\\ *>z-KV'*\\F97$$\"3EtQ2^\"R[y%F*$!3o4L(4<$)o)\\F97$$\"3#\\!)4N9:&>[F*$! 35>fr*QP8(\\F97$$\"3eOd%f8\">a[F*$!3kgVx\\X\")\\\\F97$$\"3/P!)peVR))\\ F*$!3w#G]$>&e2\"[F97$$\"3]P.X\"e(fA^F*$!3E^Jmi#)=&e%F97$$\"3kiOHztgr_F *$!3$[-W%R,NQUF97$$\"3w()p8xrh?aF*$!3Wgl5J!yvz$F97$$\"3q5x(z54Xb&F*$!3 'H()zC%HTHLF97$$\"3iL%=)Q5S)o&F*$!3SV47=/l,GF97$$\"3d=8l`#f8$eF*$!3=< \"zIOXI=#F97$$\"3a.U[ouJufF*$!35jAtH7!*>:F97$$\"3_78*)e)eQ5'F*$!3__>6j JM=*)F37$$\"3_@%)H\\-SLiF*$!3mlE.JZA)[#F37$$\"39C%y`?B]P'F*$\"3O^G80GW &e%F37$$\"3zE%e9;Ym^'F*$\"35Kj3X$Hn:\"F97$$\"3%zl%>EB!*\\mF*$\"35([#*) yLw#z\"F97$$\"33*)3$4\\eJy'F*$\"3#y'zzC+,(R#F97$$\"3>D))4z&\\C#pF*$\"3 _Tq1xd-$)HF97$$\"3IhnEn1uhqF*$\"3fWy4e$e7^$F97$$\"3SZQ/mJ#z>(F*$\"3y_^ DUN,iRF97$$\"3^L4#[m0TL(F*$\"3.(HKt+/%RVF97$$\"3AEMI?hnwuF*$\"3+(*ol;N I[YF97$$\"3\"*=fyvlC>wF*$\"3iR$4_)*zG'[F97$$\"3OE5bAF!zo(F*$\"3+wBsPG? 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" }}{PARA 0 "" 0 " " {TEXT -1 27 "Whenever one of the graphs " }{XPPEDIT 18 0 "y=f(x)" "6 #/%\"yG-%\"fG6#%\"xG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "y=g(x)" "6#/% \"yG-%\"gG6#%\"xG" }{TEXT -1 11 " meets the " }{TEXT 275 1 "x" }{TEXT -1 48 " axis, the other graph intersects the graph of " }{XPPEDIT 18 0 "y = f(x) + g(x)" "6#/%\"yG,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F)F*" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "For example, when " } {XPPEDIT 18 0 "x=Pi/2" "6#/%\"xG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 14 ", we see that " }{XPPEDIT 18 0 "f(x)=1/2" "6#/-%\"fG6#%\"xG*&\"\"\"F) \"\"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "g(x)=0" "6#/-%\"gG6#%\"xG \"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)+g(x)=1/2" "6#/,&-%\"fG 6#%\"xG\"\"\"-%\"gG6#F(F)*&F)F)\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 35 "For another example (where neither " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " nor " }{XPPEDIT 18 0 "g(x)" " 6#-%\"gG6#%\"xG" }{TEXT -1 11 " is 0) let " }{XPPEDIT 18 0 "x=Pi/3" "6 #/%\"xG*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 15 ". In this case " } {XPPEDIT 18 0 "f(x)=g(x)" "6#/-%\"fG6#%\"xG-%\"gG6#F'" }{XPPEDIT 18 0 "``=sqrt(3)/4" "6#/%!G*&-%%sqrtG6#\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "f(x)+g(x)=sqrt(3)/2" "6#/,&-%\"fG6#%\"xG\" \"\"-%\"gG6#F(F)*&-%%sqrtG6#\"\"$F)\"\"#!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 54 "Note that this checks the identity (iii) \+ for the case " }{XPPEDIT 18 0 "x=Pi/3" "6#/%\"xG*&%#PiG\"\"\"\"\"$!\" \"" }{TEXT -1 8 ", since " }{XPPEDIT 18 0 "sin(x+Pi/3)=sin(2*Pi/3)" "6 #/-%$sinG6#,&%\"xG\"\"\"*&%#PiGF)\"\"$!\"\"F)-F%6#*(\"\"#F)F+F)F,F-" } {XPPEDIT 18 0 "``=sqrt(3)/2" "6#/%!G*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 103 "More generally, not ice that the resulting graph appears to be a horizontal translation of the graph of " }{XPPEDIT 18 0 "y = sin*x" "6#/%\"yG*&%$sinG\"\"\"%\"x GF'" }{TEXT -1 93 ". Indeed, since the identity (iii) means that the t hick curve is also given by the equation " }{XPPEDIT 18 0 "y = sin(x+ Pi/3);" "6#/%\"yG-%$sinG6#,&%\"xG\"\"\"*&%#PiGF*\"\"$!\"\"F*" }{TEXT -1 48 ", we would expect its graph to be obtained as a " }{TEXT 259 22 "horizontal translation" }{TEXT -1 17 " of the graph of " } {XPPEDIT 18 0 "y = sin*x" "6#/%\"yG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 9 " through " }{XPPEDIT 18 0 "Pi/3" "6#*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 19 " units to the left." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 258 "" 0 "" {TEXT -1 18 "Using the formula " }{XPPEDIT 18 0 "c os(alpha-beta) = cos*alpha*cos*beta+sin*alpha*sin*beta" "6#/-%$cosG6#, &%&alphaG\"\"\"%%betaG!\"\",&**F%F)F(F)F%F)F*F)F)**%$sinGF)F(F)F/F)F*F )F)" }{TEXT -1 10 ", we have " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "cos(Pi-x) = cos*Pi*cos*x+sin*Pi*sin*x;" "6#/-%$cosG6#,& %#PiG\"\"\"%\"xG!\"\",&**F%F)F(F)F%F)F*F)F)**%$sinGF)F(F)F/F)F*F)F)" } {TEXT -1 1 "," }}{PARA 258 "" 0 "" {TEXT -1 10 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(Pi-x) = (-1)*`.`*cos*x+ 0*`.`*sin*x;" "6#/-%$cosG6#,&%#PiG\"\"\"%\"xG!\"\",&**,$F)F+F)%\".GF)F %F)F*F)F)**\"\"!F)F/F)%$sinGF)F*F)F)" }{TEXT -1 1 "," }}{PARA 258 "" 0 "" {TEXT -1 2 "or" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(Pi-x) = -cos*x;" "6#/-%$cosG6#,&%#PiG\"\"\"%\"xG!\"\",$*&F%F)F* F)F+" }{TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT -1 19 "This is a stand ard " }{TEXT 259 16 "symmetry formula" }{TEXT -1 25 " for the cosine f unction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Similarly, using the formula " }{XPPEDIT 18 0 "sin(alpha-beta) = s in*alpha*cos*beta-cos*alpha*sin*beta" "6#/-%$sinG6#,&%&alphaG\"\"\"%%b etaG!\"\",&**F%F)F(F)%$cosGF)F*F)F)**F.F)F(F)F%F)F*F)F+" }{TEXT -1 10 ", we have " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi -x) = sin*Pi*cos*x-cos*Pi*sin*x;" "6#/-%$sinG6#,&%#PiG\"\"\"%\"xG!\"\" ,&**F%F)F(F)%$cosGF)F*F)F)**F.F)F(F)F%F)F*F)F+" }{TEXT -1 1 "," }} {PARA 258 "" 0 "" {TEXT -1 10 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi-x) = 0*`.`*cos*x-(-1)*`.`*sin*x;" "6#/ -%$sinG6#,&%#PiG\"\"\"%\"xG!\"\",&**\"\"!F)%\".GF)%$cosGF)F*F)F)**,$F) F+F)F/F)F%F)F*F)F+" }{TEXT -1 1 "," }}{PARA 258 "" 0 "" {TEXT -1 2 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi-x) = sin*x ;" "6#/-%$sinG6#,&%#PiG\"\"\"%\"xG!\"\"*&F%F)F*F)" }{TEXT -1 1 "." }} {PARA 258 "" 0 "" {TEXT -1 19 "This is a standard " }{TEXT 259 16 "sym metry formula" }{TEXT -1 23 " for the sine function." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Note: Other symmetry fo rmulas can be obtained in a similar way. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 77 " Other examples involving addition formulas for the sine and cosine fun ctions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT 300 8 " Question" }{TEXT -1 85 ": Use appropriate addition formulas and the ex act values of the sines and cosines of " }{XPPEDIT 18 0 "Pi/4" "6#*&%# PiG\"\"\"\"\"%!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Pi/6" "6#*&%# PiG\"\"\"\"\"'!\"\"" }{TEXT -1 52 " to find the exact values of the si ne and cosine of " }{XPPEDIT 18 0 "Pi/12" "6#*&%#PiG\"\"\"\"#7!\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "5*Pi/12" "6#*(\"\"&\"\"\"%#PiGF%\"# 7!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "Pi/4+Pi/6=5*Pi/12" "6#/,&*&%#PiG\" \"\"\"\"%!\"\"F'*&F&F'\"\"'F)F'*(\"\"&F'F&F'\"#7F)" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "Pi/4-Pi/6=Pi/12" "6#/,&*&%#PiG\"\"\"\"\"%!\"\"F'*&F& F'\"\"'F)F)*&F&F'\"#7F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(5*Pi/1 2)=sin(Pi/4+Pi/6)" "6#/-%$sinG6#*(\"\"&\"\"\"%#PiGF)\"#7!\"\"-F%6#,&*& F*F)\"\"%F,F)*&F*F)\"\"'F,F)" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=sin(Pi/4)*cos(Pi/6)+cos(Pi/4)*sin(Pi /6)" "6#/%!G,&*&-%$sinG6#*&%#PiG\"\"\"\"\"%!\"\"F,-%$cosG6#*&F+F,\"\"' F.F,F,*&-F06#*&F+F,F-F.F,-F(6#*&F+F,F3F.F,F," }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(sqrt(2)/2)*``(sqrt(3) /2)+``(sqrt(2)/2)*``(1/2)" "6#/%!G,&*&-F$6#*&-%%sqrtG6#\"\"#\"\"\"F-! \"\"F.-F$6#*&-F+6#\"\"$F.F-F/F.F.*&-F$6#*&-F+6#F-F.F-F/F.-F$6#*&F.F.F- F/F.F." }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (sqrt(6)+sqrt(2))/4;" "6#/%!G*&,&-%%sqrtG6#\"\"'\"\"\"-F(6# \"\"#F+F+\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "wh ile " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(5*Pi/12) \+ = cos(Pi/4+Pi/6);" "6#/-%$cosG6#*(\"\"&\"\"\"%#PiGF)\"#7!\"\"-F%6#,&*& F*F)\"\"%F,F)*&F*F)\"\"'F,F)" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=cos(Pi/4)*cos(Pi/6)-sin(Pi/4)*sin(Pi /6)" "6#/%!G,&*&-%$cosG6#*&%#PiG\"\"\"\"\"%!\"\"F,-F(6#*&F+F,\"\"'F.F, F,*&-%$sinG6#*&F+F,F-F.F,-F56#*&F+F,F2F.F,F." }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(sqrt(2)/2)*``(sqrt(3) /2)-``(sqrt(2)/2)*``(1/2)" "6#/%!G,&*&-F$6#*&-%%sqrtG6#\"\"#\"\"\"F-! \"\"F.-F$6#*&-F+6#\"\"$F.F-F/F.F.*&-F$6#*&-F+6#F-F.F-F/F.-F$6#*&F.F.F- F/F.F/" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (sqrt(6)-sqrt(2))/4;" "6#/%!G*&,&-%%sqrtG6#\"\"'\"\"\"-F(6# \"\"#!\"\"F+\"\"%F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "S imilarly " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi/1 2) = sin(Pi/4-Pi/6);" "6#/-%$sinG6#*&%#PiG\"\"\"\"#7!\"\"-F%6#,&*&F(F) \"\"%F+F)*&F(F)\"\"'F+F+" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=sin(Pi/4)*cos(Pi/6)-cos(Pi/4)*sin(Pi/6)" " 6#/%!G,&*&-%$sinG6#*&%#PiG\"\"\"\"\"%!\"\"F,-%$cosG6#*&F+F,\"\"'F.F,F, *&-F06#*&F+F,F-F.F,-F(6#*&F+F,F3F.F,F." }{TEXT -1 1 " " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(sqrt(2)/2)*``(sqrt(3)/2)-` `(sqrt(2)/2)*``(1/2)" "6#/%!G,&*&-F$6#*&-%%sqrtG6#\"\"#\"\"\"F-!\"\"F. -F$6#*&-F+6#\"\"$F.F-F/F.F.*&-F$6#*&-F+6#F-F.F-F/F.-F$6#*&F.F.F-F/F.F/ " }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` ` = (sqrt(6)-sqrt(2))/4;" "6#/%!G*&,&-%%sqrtG6#\"\"'\"\"\"-F(6#\"\"#! \"\"F+\"\"%F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "while " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(Pi/12) = cos(Pi /4-Pi/6);" "6#/-%$cosG6#*&%#PiG\"\"\"\"#7!\"\"-F%6#,&*&F(F)\"\"%F+F)*& F(F)\"\"'F+F+" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = cos(Pi/4)*cos(Pi/6)+sin(Pi/4)*sin(Pi/6);" "6#/%!G, &*&-%$cosG6#*&%#PiG\"\"\"\"\"%!\"\"F,-F(6#*&F+F,\"\"'F.F,F,*&-%$sinG6# *&F+F,F-F.F,-F56#*&F+F,F2F.F,F," }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(sqrt(2)/2)*``(sqrt(3)/2)+``(sqr t(2)/2)*``(1/2);" "6#/%!G,&*&-F$6#*&-%%sqrtG6#\"\"#\"\"\"F-!\"\"F.-F$6 #*&-F+6#\"\"$F.F-F/F.F.*&-F$6#*&-F+6#F-F.F-F/F.-F$6#*&F.F.F-F/F.F." } {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (sqrt(6)+sqrt(2))/4;" "6#/%!G*&,&-%%sqrtG6#\"\"'\"\"\"-F(6#\"\"#F+F+ \"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 89 ": The calculation could hav e been shortened by making use of the co-function identities: " } {XPPEDIT 18 0 "cos(Pi/2-theta)=sin*theta" "6#/-%$cosG6#,&*&%#PiG\"\"\" \"\"#!\"\"F*%&thetaGF,*&%$sinGF*F-F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin(Pi/2-theta)=cos*theta" "6#/-%$sinG6#,&*&%#PiG\"\"\"\"\"#!\" \"F*%&thetaGF,*&%$cosGF*F-F*" }{TEXT -1 8 ". Since " }{XPPEDIT 18 0 "P i/12+5*Pi/12=Pi/2" "6#/,&*&%#PiG\"\"\"\"#7!\"\"F'*(\"\"&F'F&F'F(F)F'*& F&F'\"\"#F)" }{TEXT -1 19 ", it folllows that " }{XPPEDIT 18 0 "sin(Pi /12)=cos(5*Pi/12)" "6#/-%$sinG6#*&%#PiG\"\"\"\"#7!\"\"-%$cosG6#*(\"\"& F)F(F)F*F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(Pi/12)=sin(5*Pi/12 )" "6#/-%$cosG6#*&%#PiG\"\"\"\"#7!\"\"-%$sinG6#*(\"\"&F)F(F)F*F+" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT 296 8 "Question" } {TEXT -1 6 ": Let " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 27 " be real numbers such that " }{XPPEDIT 18 0 "0<=alpha" "6#1\"\"!%&alphaG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 286 "alpha := arcsin(3/5):\nbeta := Pi-arcsin(5/13):\ncos('alpha')=simplify(cos(alpha));\ncos('beta')= simplify(cos(beta));\nsin('alpha+beta')=expand(sin(alpha+beta));\ncos( 'alpha+beta')=expand(cos(alpha+beta));\nsin('alpha-beta')=expand(sin(a lpha-beta));\ncos('alpha-beta')=expand(cos(alpha-beta));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$cosG6#%&alphaG#\"\"%\"\"&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$cosG6#%%betaG#!#7\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#,&%&alphaG\"\"\"%%betaGF)#!#;\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,&%&alphaG\"\"\"%%betaGF)#!#j\"#l " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#,&%&alphaG\"\"\"%%betaG! \"\"#!#c\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,&%&alphaG\" \"\"%%betaG!\"\"#!#L\"#l" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT 303 8 "Question" } {TEXT -1 22 ": Verify the identity " }{XPPEDIT 18 0 "cos(alpha-beta)/( cos*alpha*sin*beta)=tan*alpha+cot*beta" "6#/*&-%$cosG6#,&%&alphaG\"\" \"%%betaG!\"\"F***F&F*F)F*%$sinGF*F+F*F,,&*&%$tanGF*F)F*F**&%$cotGF*F+ F*F*" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 304 8 "Solution" }{TEXT -1 2 ": " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(alpha-beta)/(cos*alpha*sin*beta)=(c os*alpha*cos*beta+sin*alpha*sin*beta)/(cos*alpha*sin*beta)" "6#/*&-%$c osG6#,&%&alphaG\"\"\"%%betaG!\"\"F***F&F*F)F*%$sinGF*F+F*F,*&,&**F&F*F )F*F&F*F+F*F***F.F*F)F*F.F*F+F*F*F***F&F*F)F*F.F*F+F*F," }{TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=cos*alpha*co s*beta/(cos*alpha*sin*beta) + sin*alpha*sin*beta/(cos*alpha*sin*beta) " "6#/%!G,&*,%$cosG\"\"\"%&alphaGF(F'F(%%betaGF(**F'F(F)F(%$sinGF(F*F( !\"\"F(*,F,F(F)F(F,F(F*F(**F'F(F)F(F,F(F*F(F-F(" }{TEXT -1 1 " " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=cos*beta/(sin*beta ) + sin*alpha/(cos*alpha)" "6#/%!G,&*(%$cosG\"\"\"%%betaGF(*&%$sinGF(F )F(!\"\"F(*(F+F(%&alphaGF(*&F'F(F.F(F,F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=cot*beta+tan*alpha" "6#/%!G, &*&%$cotG\"\"\"%%betaGF(F(*&%$tanGF(%&alphaGF(F(" }{TEXT -1 1 " " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=tan*alpha+cot*beta " "6#/%!G,&*&%$tanG\"\"\"%&alphaGF(F(*&%$cotGF(%%betaGF(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "alpha:='alpha':beta:='beta':\ncos(alpha-beta)/(cos(a lpha)*sin(beta));\n``=map(expand,%);\n``=expand(rhs(%));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*(-%$cosG6#,&%&alphaG\"\"\"%%betaG!\"\"F)-F%6#F( F+-%$sinG6#F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(,&*&-%$cosG6# %&alphaG\"\"\"-F)6#%%betaGF,F,*&-%$sinGF*F,-F2F.F,F,F,F(!\"\"F3F4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&-%$sinG6#%%betaG!\"\"-%$cosGF) \"\"\"F.*&-F-6#%&alphaGF+-F(F1F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Additi on formulas for the tangent function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(alpha+beta) = sin(alpha+beta)/cos(alpha+beta);" "6#/-%$tanG6#,&%& alphaG\"\"\"%%betaGF)*&-%$sinG6#,&F(F)F*F)F)-%$cosG6#,&F(F)F*F)!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 17 "so, by expanding " } {XPPEDIT 18 0 "sin(alpha+beta);" "6#-%$sinG6#,&%&alphaG\"\"\"%%betaGF( " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(alpha+beta);" "6#-%$cosG6#,& %&alphaG\"\"\"%%betaGF(" }{TEXT -1 10 ", we have " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(alpha+beta) = (sin*alpha*cos*beta +cos*alpha*sin*beta)/(cos*alpha*cos*beta-sin*alpha*sin*beta);" "6#/-%$ tanG6#,&%&alphaG\"\"\"%%betaGF)*&,&**%$sinGF)F(F)%$cosGF)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)!\"\"F4" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Dividing numerator and denominator of the right hand side by " }{XPPEDIT 18 0 "cos*alpha*cos*beta;" "6#**%$cosG\"\"\"%&alphaGF% F$F%%%betaGF%" }{TEXT -1 8 " gives: " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(alpha+beta) = (sin*alpha*cos*beta/(cos*alpha*cos*beta)+cos*alpha*sin*beta/(cos*alph a*cos*beta))/(1-sin*alpha*sin*beta/(cos*alpha*cos*beta));" "6#/-%$tanG 6#,&%&alphaG\"\"\"%%betaGF)*&,&*,%$sinGF)F(F)%$cosGF)F*F)**F/F)F(F)F/F )F*F)!\"\"F)*,F/F)F(F)F.F)F*F)**F/F)F(F)F/F)F*F)F1F)F),&F)F)*,F.F)F(F) F.F)F*F)**F/F)F(F)F/F)F*F)F1F1F1" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan (alpha+beta) = (tan*alpha+tan*beta)/(1-tan*alpha*tan*beta);" "6#/-%$ta nG6#,&%&alphaG\"\"\"%%betaGF)*&,&*&F%F)F(F)F)*&F%F)F*F)F)F),&F)F)**F%F )F(F)F%F)F*F)!\"\"F1" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 263 15 "_______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "Similarly, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "tan(alpha-beta) = (tan*alpha-tan*beta)/(1+tan*alpha*tan *beta);" "6#/-%$tanG6#,&%&alphaG\"\"\"%%betaG!\"\"*&,&*&F%F)F(F)F)*&F% F)F*F)F+F),&F)F)**F%F)F(F)F%F)F*F)F)F+" }{TEXT -1 1 "." }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{TEXT 264 14 "______________" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 6 "expand" }{TEXT -1 33 " brings these formulas into play." }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "'tan(alpha+beta)'=expan d(tan(alpha+beta));\n'tan(alpha-beta)'=expand(tan(alpha-beta));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$tanG6#,&%&alphaG\"\"\"%%betaGF)*&, &-F%6#F(F)-F%6#F*F)F),&F)F)*&F-F)F/F)!\"\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$tanG6#,&%&alphaG\"\"\"%%betaG!\"\"*&,&-F%6#F(F)-F%6 #F*F+F),&F)F)*&F.F)F0F)F)F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 76 "Examples to illustrate the addition formulas involving \+ the tangent function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Using the fo rmula " }{XPPEDIT 18 0 "tan(alpha+beta) = (tan*alpha+tan*beta)/(1-tan* alpha*tan*beta);" "6#/-%$tanG6#,&%&alphaG\"\"\"%%betaGF)*&,&*&F%F)F(F) F)*&F%F)F*F)F)F),&F)F)**F%F)F(F)F%F)F*F)!\"\"F1" }{TEXT -1 14 ", we se e that " }{XPPEDIT 18 0 "tan(x+Pi/4) = (tan*x+tan(Pi/4))/(1-tan*x*tan( Pi/4));" "6#/-%$tanG6#,&%\"xG\"\"\"*&%#PiGF)\"\"%!\"\"F)*&,&*&F%F)F(F) F)-F%6#*&F+F)F,F-F)F),&F)F)*(F%F)F(F)-F%6#*&F+F)F,F-F)F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "tan(Pi/4)= 1" "6#/-%$tanG6#*&%#PiG\"\"\"\"\"%!\"\"F)" }{TEXT -1 10 ", we have " } }{PARA 257 "" 0 "" {TEXT -1 2 " t" }{XPPEDIT 18 0 "an(x+Pi/4) = (1+tan *x)/(1-tan*x);" "6#/-%#anG6#,&%\"xG\"\"\"*&%#PiGF)\"\"%!\"\"F)*&,&F)F) *&%$tanGF)F(F)F)F),&F)F)*&F1F)F(F)F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "expand( tan(x+Pi/4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%-%$tanG6# %\"xGF%F%,&F%F%F&!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph of " }{XPPEDIT 18 0 "y = (1+tan*x)/(1-tan*x);" "6#/%\"yG*&,&\" \"\"F'*&%$tanGF'%\"xGF'F'F',&F'F'*&F)F'F*F'!\"\"F-" }{TEXT -1 58 " app ears to be just a horizontal translation of the graph " }{XPPEDIT 18 0 "y = tan*x;" "6#/%\"yG*&%$tanG\"\"\"%\"xGF'" }{TEXT -1 54 ". Indeed \+ the formula above implies that the graph of " }{XPPEDIT 18 0 "y = (1+ tan*x)/(1-tan*x);" "6#/%\"yG*&,&\"\"\"F'*&%$tanGF'%\"xGF'F'F',&F'F'*&F )F'F*F'!\"\"F-" }{TEXT -1 29 " is the same as the graph of " } {XPPEDIT 18 0 "y=tan(x+Pi/4)" "6#/%\"yG-%$tanG6#,&%\"xG\"\"\"*&%#PiGF* \"\"%!\"\"F*" }{TEXT -1 53 ", which can be obtained by translating the graph of " }{XPPEDIT 18 0 "y=tan*x" "6#/%\"yG*&%$tanG\"\"\"%\"xGF'" }{TEXT -1 9 " through " }{XPPEDIT 18 0 "Pi/4" "6#*&%#PiG\"\"\"\"\"%!\" \"" }{TEXT -1 20 " units to the left. " }}{PARA 0 "" 0 "" {TEXT -1 28 "The vertical asymptotes are " }{XPPEDIT 18 0 "x=-3*Pi/4,x=Pi/4,x=5*Pi /4,x=9*Pi/4" "6&/%\"xG,$*(\"\"$\"\"\"%#PiGF(\"\"%!\"\"F+/F$*&F)F(F*F+/ F$*(\"\"&F(F)F(F*F+/F$*(\"\"*F(F)F(F*F+" }{TEXT -1 5 " etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 406 "p 1 := plot((1+tan(x))/(1-tan(x)),x=-3.5..9.6,y=-5..5,discont=true):\np2 := plots[implicitplot](\{x=-3*Pi/4,x=Pi/4,x=5*Pi/4,x=9*Pi/4\},\n x=- Pi..3*Pi,y=-5..5,linestyle=3,color=black):\nt1 := plots[textplot]([[9. 6,-.3,`x`],[-.3,5,`y`]],font=[HELVETICA,10]):\nplots[display]([p1,p2,t 1],xtickmarks=[-pi=`-p`,-pi/2=`-p/2`,pi/2=`p/2`,pi=`p`,\n 3*pi/2=`3p /2`,2*pi=`2p`,5*pi/2=`5p/2`],font=[SYMBOL,11],labels=[``,``]);" }} {PARA 13 "" 1 "" {GLPLOT2D 509 360 360 {PLOTDATA 2 "6--%'CURVESG6,7gn7 $$!3++++++++N!#<$\"3ChQ\\x\"R)\\X!#=7$$!3SFW6&Ho]Z$F*$\"3erMn**RGa[F-7 $$!3E#f-?NvLX$F*$\"3hAH%pYE_7&F-7$$!3pQ#z'o$z*GMF*$\"3:DI&egKsV&F-7$$! 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{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 258 "" 0 "" {TEXT -1 19 "Using the \+ formula " }{XPPEDIT 18 0 "tan(alpha+beta) = (tan*alpha+tan*beta)/(1-t an*alpha*tan*beta);" "6#/-%$tanG6#,&%&alphaG\"\"\"%%betaGF)*&,&*&F%F)F (F)F)*&F%F)F*F)F)F),&F)F)**F%F)F(F)F%F)F*F)!\"\"F1" }{TEXT -1 10 ", we have " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(x+Pi) = (tan*x+tan*Pi)/(1-tan*x*tan*Pi);" "6#/-%$tanG6#,&%\"xG\"\"\"%#PiGF)*& ,&*&F%F)F(F)F)*&F%F)F*F)F)F),&F)F)**F%F)F(F)F%F)F*F)!\"\"F1" }{TEXT -1 1 "," }}{PARA 258 "" 0 "" {TEXT -1 10 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(x+Pi) = (tan*x+0)/(1-tan*x*`.` *0);" "6#/-%$tanG6#,&%\"xG\"\"\"%#PiGF)*&,&*&F%F)F(F)F)\"\"!F)F),&F)F) **F%F)F(F)%\".GF)F.F)!\"\"F2" }{TEXT -1 1 "," }}{PARA 258 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(x+Pi) = tan*x;" "6#/-%$tanG6#,&%\"xG\"\"\"%#PiGF)*&F%F)F(F)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 90 "This formula also follows from the fact that the tangent \+ function is periodic with period " }{XPPEDIT 18 0 "Pi" "6#%#PiG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 86 "Other examples involving ad dition formulas for the sine, cosine and tangent functions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT 308 8 "Question" }{TEXT -1 85 ": Use appropriate addition formulas and the exact values of the si nes and cosines of " }{XPPEDIT 18 0 "3*Pi/4;" "6#*(\"\"$\"\"\"%#PiGF% \"\"%!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Pi/6;" "6#*&%#PiG\"\" \"\"\"'!\"\"" }{TEXT -1 29 " to find the exact values of " }{XPPEDIT 18 0 "sin(11*Pi/12);" "6#-%$sinG6#*(\"#6\"\"\"%#PiGF(\"#7!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "cos(11*Pi/12);" "6#-%$cosG6#*(\"#6\"\" \"%#PiGF(\"#7!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "tan(11*Pi/12); " "6#-%$tanG6#*(\"#6\"\"\"%#PiGF(\"#7!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 307 8 "Solution" }{TEXT -1 2 ": " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(11*Pi/12)=sin(3*Pi/4+Pi/6)" "6#/-%$ sinG6#*(\"#6\"\"\"%#PiGF)\"#7!\"\"-F%6#,&*(\"\"$F)F*F)\"\"%F,F)*&F*F) \"\"'F,F)" }{XPPEDIT 18 0 "``=sin(3*Pi/4)*cos(Pi/6)+cos(3*Pi/4)*sin(Pi /6)" "6#/%!G,&*&-%$sinG6#*(\"\"$\"\"\"%#PiGF,\"\"%!\"\"F,-%$cosG6#*&F- F,\"\"'F/F,F,*&-F16#*(F+F,F-F,F.F/F,-F(6#*&F-F,F4F/F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(sqrt(2)/2)*``(sqrt(3)/2)+``(-sqrt(2)/2)*``(1/2) \+ " "6#/%!G,&*&-F$6#*&-%%sqrtG6#\"\"#\"\"\"F-!\"\"F.-F$6#*&-F+6#\"\"$F.F -F/F.F.*&-F$6#,$*&-F+6#F-F.F-F/F/F.-F$6#*&F.F.F-F/F.F." }{XPPEDIT 18 0 "`` = (sqrt(6)-sqrt(2))/4;" "6#/%!G*&,&-%%sqrtG6#\"\"'\"\"\"-F(6#\" \"#!\"\"F+\"\"%F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(11*Pi/12) = cos(3 *Pi/4+Pi/6);" "6#/-%$cosG6#*(\"#6\"\"\"%#PiGF)\"#7!\"\"-F%6#,&*(\"\"$F )F*F)\"\"%F,F)*&F*F)\"\"'F,F)" }{XPPEDIT 18 0 "`` = cos(3*Pi/4)*cos(Pi /6)-sin(3*Pi/4)*sin(Pi/6);" "6#/%!G,&*&-%$cosG6#*(\"\"$\"\"\"%#PiGF,\" \"%!\"\"F,-F(6#*&F-F,\"\"'F/F,F,*&-%$sinG6#*(F+F,F-F,F.F/F,-F66#*&F-F, F3F/F,F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(-sqrt(2)/2)*``(sqrt(3)/ 2)-``(sqrt(2)/2)*``(1/2);" "6#/%!G,&*&-F$6#,$*&-%%sqrtG6#\"\"#\"\"\"F. !\"\"F0F/-F$6#*&-F,6#\"\"$F/F.F0F/F/*&-F$6#*&-F,6#F.F/F.F0F/-F$6#*&F/F /F.F0F/F0" }{XPPEDIT 18 0 "`` = (-sqrt(6)-sqrt(2))/4;" "6#/%!G*&,&-%%s qrtG6#\"\"'!\"\"-F(6#\"\"#F+\"\"\"\"\"%F+" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(11*Pi/12)=sin(11*Pi/12) /cos(11*Pi/12)" "6#/-%$tanG6#*(\"#6\"\"\"%#PiGF)\"#7!\"\"*&-%$sinG6#*( F(F)F*F)F+F,F)-%$cosG6#*(F(F)F*F)F+F,F," }{XPPEDIT 18 0 "`` = (sqrt(6) -sqrt(2))/(-sqrt(6)-sqrt(2));" "6#/%!G*&,&-%%sqrtG6#\"\"'\"\"\"-F(6#\" \"#!\"\"F+,&-F(6#F*F/-F(6#F.F/F/" }{XPPEDIT 18 0 "``=(-sqrt(6)+sqrt(2) )/(sqrt(6)+sqrt(2))" "6#/%!G*&,&-%%sqrtG6#\"\"'!\"\"-F(6#\"\"#\"\"\"F/ ,&-F(6#F*F/-F(6#F.F/F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 80 "The last expression can be simplified further by rationalising the denominator. " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(11*Pi/12) = -(sqrt(6)-sqrt(2))^2/(s qrt(6)+sqrt(2)*(sqrt(6)-sqrt(2)));" "6#/-%$tanG6#*(\"#6\"\"\"%#PiGF)\" #7!\"\",$*&,&-%%sqrtG6#\"\"'F)-F16#\"\"#F,F6,&-F16#F3F)*&-F16#F6F),&-F 16#F3F)-F16#F6F,F)F)F,F," }{XPPEDIT 18 0 "``=-(6-2*sqrt(12)+2)/(6-2)" "6#/%!G,$*&,(\"\"'\"\"\"*&\"\"#F)-%%sqrtG6#\"#7F)!\"\"F+F)F),&F(F)F+F0 F0F0" }{XPPEDIT 18 0 "``=-(8-4*sqrt(3))/4" "6#/%!G,$*&,&\"\")\"\"\"*& \"\"%F)-%%sqrtG6#\"\"$F)!\"\"F)F+F0F0" }{XPPEDIT 18 0 "`` = -2+sqrt(3) ;" "6#/%!G,&\"\"#!\"\"-%%sqrtG6#\"\"$\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "Alternatively, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(11*Pi/12)=tan(3*Pi/4+Pi/6)" "6#/-%$tanG6#*( \"#6\"\"\"%#PiGF)\"#7!\"\"-F%6#,&*(\"\"$F)F*F)\"\"%F,F)*&F*F)\"\"'F,F) " }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` `=(tan(3*Pi/4)+tan(Pi/6))/(1-tan(3*Pi/4)*tan(Pi/6))" "6#/%!G*&,&-%$tan G6#*(\"\"$\"\"\"%#PiGF,\"\"%!\"\"F,-F(6#*&F-F,\"\"'F/F,F,,&F,F,*&-F(6# *(F+F,F-F,F.F/F,-F(6#*&F-F,F3F/F,F/F/" }{XPPEDIT 18 0 "``=(-1+sqrt(3)/ 3)/(1-(-1)*(sqrt(3)/3)" "6#/%!G*&,&\"\"\"!\"\"*&-%%sqrtG6#\"\"$F'F-F(F 'F',&F'F'*&,$F'F(F'*&-F+6#F-F'F-F(F'F(F(" }{XPPEDIT 18 0 "``=(sqrt(3)- 3)/(sqrt(3)+3)" "6#/%!G*&,&-%%sqrtG6#\"\"$\"\"\"F*!\"\"F+,&-F(6#F*F+F* F+F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "Rationalising th e denominator in the last expression gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(11*Pi/12)=(sqrt(3)-3)^2/((sqrt(3)+3 )*(sqrt(3)-3))" "6#/-%$tanG6#*(\"#6\"\"\"%#PiGF)\"#7!\"\"*&,&-%%sqrtG6 #\"\"$F)F2F,\"\"#*&,&-F06#F2F)F2F)F),&-F06#F2F)F2F,F)F," }{XPPEDIT 18 0 "``=(3-6*sqrt(3)+9)/(3-9)" "6#/%!G*&,(\"\"$\"\"\"*&\"\"'F(-%%sqrtG6# F'F(!\"\"\"\"*F(F(,&F'F(F/F.F." }{XPPEDIT 18 0 "``=(12-6*sqrt(3))/(-6) " "6#/%!G*&,&\"#7\"\"\"*&\"\"'F(-%%sqrtG6#\"\"$F(!\"\"F(,$F*F/F/" } {XPPEDIT 18 0 "``=-2+sqrt(3)" "6#/%!G,&\"\"#!\"\"-%%sqrtG6#\"\"$\"\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 33 "Verify the following \+ identities: " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "(1+ cos*theta)/(sin*theta)+sin*theta/(1+cos*theta)=2*csc*theta" "6#/,&*&,& \"\"\"F'*&%$cosGF'%&thetaGF'F'F'*&%$sinGF'F*F'!\"\"F'*(F,F'F*F',&F'F'* &F)F'F*F'F'F-F'*(\"\"#F'%$cscGF'F*F'" }{TEXT -1 14 " (b) " } {XPPEDIT 18 0 "sin^2*theta/(1-cos*theta) = 1+cos*theta;" "6#/*(%$sinG \"\"#%&thetaG\"\"\",&F(F(*&%$cosGF(F'F(!\"\"F,,&F(F(*&F+F(F'F(F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(c) " }{XPPEDIT 18 0 "sin*theta/(1-cos*theta)=(1+cos*thet a)/(sin*theta)" "6#/*(%$sinG\"\"\"%&thetaGF&,&F&F&*&%$cosGF&F'F&!\"\"F +*&,&F&F&*&F*F&F'F&F&F&*&F%F&F'F&F+" }{TEXT -1 30 " \+ (d) " }{XPPEDIT 18 0 "sec^2*theta+csc^2*theta=sec^2*theta*csc^2 *theta" "6#/,&*&%$secG\"\"#%&thetaG\"\"\"F)*&%$cscGF'F(F)F)**F&F'F(F)F +F'F(F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(e) " }{XPPEDIT 18 0 "(cot*theta-csc*theta)^2=(1-cos *theta)/(1+cos*theta)" "6#/*$,&*&%$cotG\"\"\"%&thetaGF(F(*&%$cscGF(F)F (!\"\"\"\"#*&,&F(F(*&%$cosGF(F)F(F,F(,&F(F(*&F1F(F)F(F(F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 33 "_________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 83 "Use appropriate addition formulas and t he exact values of the sines and cosines of " }{XPPEDIT 18 0 "Pi/3;" " 6#*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 29 " to find the exact values of " }{XPPEDIT 18 0 "sin(7*Pi/12);" "6#-%$sinG6#*(\"\"(\"\"\"%#PiGF(\"#7 !\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cos(7*Pi/12)" "6#-%$cosG6#*(\" \"(\"\"\"%#PiGF(\"#7!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "tan(7*P i/12)" "6#-%$tanG6#*(\"\"(\"\"\"%#PiGF(\"#7!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "sin(7*Pi/12)=(sq rt(6)+sqrt(2))/4" "6#/-%$sinG6#*(\"\"(\"\"\"%#PiGF)\"#7!\"\"*&,&-%%sqr tG6#\"\"'F)-F06#\"\"#F)F)\"\"%F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cos (7*Pi/12)=(sqrt(2)-sqrt(6))/4" "6#/-%$cosG6#*(\"\"(\"\"\"%#PiGF)\"#7! \"\"*&,&-%%sqrtG6#\"\"#F)-F06#\"\"'F,F)\"\"%F," }{TEXT -1 2 ", " } {XPPEDIT 18 0 "tan(7*Pi/12)=-2-sqrt(3) " "6#/-%$tanG6#*(\"\"(\"\"\"%#P iGF)\"#7!\"\",&\"\"#F,-%%sqrtG6#\"\"$F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "___________________ ______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 " alpha" "6#%&alphaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta" "6#%%bet aG" }{TEXT -1 27 " be real numbers such that " }{XPPEDIT 18 0 "0<=alph a" "6#1\"\"!%&alphaG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 287 "alpha := arcsin(2/3):\nbeta := 2*Pi-arccos(3/5):\nco s('alpha')=simplify(cos(alpha));\nsin('beta')=simplify(sin(beta));\nsi n('alpha+beta')=expand(sin(alpha+beta));\ncos('alpha+beta')=expand(cos (alpha+beta));\nsin('alpha-beta')=expand(sin(alpha-beta));\ncos('alpha -beta')=expand(cos(alpha-beta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %$cosG6#%&alphaG,$*&\"\"$!\"\"\"\"&#\"\"\"\"\"#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%%betaG#!\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#,&%&alphaG\"\"\"%%betaGF),&#\"\"#\"\"&F)*(\" \"%F)\"#:!\"\"F.#F)F-F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#, &%&alphaG\"\"\"%%betaGF),&*&\"\"&!\"\"F-#F)\"\"#F)#\"\")\"#:F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#,&%&alphaG\"\"\"%%betaG!\"\" ,&#\"\"#\"\"&F)*(\"\"%F)\"#:F+F/#F)F.F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,&%&alphaG\"\"\"%%betaG!\"\",&*&\"\"&F+F.#F)\"\"#F)#\" \")\"#:F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 33 "_________________ ________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 85 "Use appropriate addit ion formulas to derive the following pairs of symmetry formulas:" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([cos(theta+ Pi) = -cos*theta, ``],[sin(theta+Pi) = -sin*theta, ``]);" "6#-%*PIECEW ISEG6$7$/-%$cosG6#,&%&thetaG\"\"\"%#PiGF-,$*&F)F-F,F-!\"\"%!G7$/-%$sin G6#,&F,F-F.F-,$*&F6F-F,F-F1F2" }{TEXT -1 4 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECE WISE([cos(theta+Pi/2) = -sin*theta, ``],[sin(theta+Pi/2) = cos*theta, \+ ``]);" "6#-%*PIECEWISEG6$7$/-%$cosG6#,&%&thetaG\"\"\"*&%#PiGF-\"\"#!\" \"F-,$*&%$sinGF-F,F-F1%!G7$/-F46#,&F,F-*&F/F-F0F1F-*&F)F-F,F-F5" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 "________________________ _________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 10 "Simplify " } {XPPEDIT 18 0 "sin*x+sin(x+2*Pi/3)+sin(x+4*Pi/3);" "6#,(*&%$sinG\"\"\" %\"xGF&F&-F%6#,&F'F&*(\"\"#F&%#PiGF&\"\"$!\"\"F&F&-F%6#,&F'F&*(\"\"%F& F-F&F.F/F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 64 "Use an appropriate addition formula t o establish the identity: " }{XPPEDIT 18 0 "tan(theta+Pi/3)=(tan*thet a+sqrt(3))/(1-sqrt(3)*tan*theta)" "6#/-%$tanG6#,&%&thetaG\"\"\"*&%#PiG F)\"\"$!\"\"F)*&,&*&F%F)F(F)F)-%%sqrtG6#F,F)F),&F)F)*(-F26#F,F)F%F)F(F )F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "_______________ __________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q7" }}{PARA 0 "" 0 "" {TEXT -1 23 "Illustrate the formul a " }{XPPEDIT 18 0 "cos(x-Pi/4) = cos*x*cos(Pi/4)+sin*x*sin(Pi/4);" "6 #/-%$cosG6#,&%\"xG\"\"\"*&%#PiGF)\"\"%!\"\"F-,&*(F%F)F(F)-F%6#*&F+F)F, F-F)F)*(%$sinGF)F(F)-F46#*&F+F)F,F-F)F)" }{TEXT -1 39 " graphically by plotting the graphs of " }{XPPEDIT 18 0 "f(x) = cos*x*cos(Pi/4);" "6# /-%\"fG6#%\"xG*(%$cosG\"\"\"F'F*-F)6#*&%#PiGF*\"\"%!\"\"F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x) = sin*x*sin(Pi/4);" "6#/-%\"gG6#%\"xG*( %$sinG\"\"\"F'F*-F)6#*&%#PiGF*\"\"%!\"\"F*" }{TEXT -1 23 " together wi th the sum " }{XPPEDIT 18 0 "f(x)+g(x)" "6#,&-%\"fG6#%\"xG\"\"\"-%\"gG 6#F'F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "______________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q8" }}{PARA 0 "" 0 "" {TEXT -1 23 "(a) Plot the graph of " }{XPPEDIT 18 0 "y = (tan*x-1)/(tan*x+1);" "6#/%\"yG*&,&*&%$tanG\" \"\"%\"xGF)F)F)!\"\"F),&*&F(F)F*F)F)F)F)F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "(b) By using the formula " }{XPPEDIT 18 0 "tan(x+ phi) = (tan*x+tan*phi)/(1-tan*x*tan*phi);" "6#/-%$tanG6#,&%\"xG\"\"\"% $phiGF)*&,&*&F%F)F(F)F)*&F%F)F*F)F)F),&F)F)**F%F)F(F)F%F)F*F)!\"\"F1" }{TEXT -1 12 ", show that " }{XPPEDIT 18 0 "(tan*x-1)/(tan*x+1);" "6#* &,&*&%$tanG\"\"\"%\"xGF'F'F'!\"\"F',&*&F&F'F(F'F'F'F'F)" }{TEXT -1 14 " has the form " }{XPPEDIT 18 0 "tan(x+phi)" "6#-%$tanG6#,&%\"xG\"\"\" %$phiGF(" }{TEXT -1 26 " for a suitable choice of " }{XPPEDIT 18 0 "ph i" "6#%$phiG" }{TEXT -1 31 ", and hence that the graph of " } {XPPEDIT 18 0 "y = (tan*x-1)/(tan*x+1);" "6#/%\"yG*&,&*&%$tanG\"\"\"% \"xGF)F)F)!\"\"F),&*&F(F)F*F)F)F)F)F+" }{TEXT -1 33 " is a horizontal \+ translation of " }{XPPEDIT 18 0 "y = tan*x;" "6#/%\"yG*&%$tanG\"\"\"% \"xGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 "_______________ __________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "_______________________________ __" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 18 "Code for pictures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 30 "Code for \+ 1st symmetry picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 780 "darkgreen := COLOR(RGB,0,.7,0):\ncs := e valf(cos(Pi/3)):\nsn := evalf(sin(Pi/3)):\np1 := plot([[cos(t),sin(t), t=-Pi/3..Pi/3],\n[cos(t),sin(t),t=Pi/3..5*Pi/3]],color=[blue,red],thic kness=2):\np2 := plot([[[cs,-sn],[0,0],[cs,sn]],[[cs,-sn],[cs,sn]],[[0 ,0],[cs,0]]],\ncolor=[brown,darkgreen,coral],thickness=2):\np3 := plot ([0.25*cos(t),0.25*sin(t),t=-Pi/3..Pi/3],color=navy):\nt1 := plots[tex tplot]([[.85,.96,`P( cos ,sin )`],\n [.9,-.96,`Q( cos( ),sin( ))`],\n [.22,.54,`1`],[1.15,-.04,`x`],[-.05,1.15,`y`],\n [-.0 6,.09,`O`]],color=black):\nt2 := plots[textplot]([[.15,.1,`q`],[.15,-. 07,`-q`],\n [.857,.961,`q`],[1.08,.961,`q`],[.852,-.961,`-q`],\n [1.22,-.961,`-q`]],font=[SYMBOL,11],color=navy):\nplots[display] ([p1,p2,p3,t1,t2],tickmarks=[0,0],scaling=constrained);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 38 "Code fo r even and odd function graphs " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 819 "p1 := plot(cos(t*Pi),t=-2.2 ..2.2,color=red,thickness=2):\np2 := plot([[[1/3,1/2],[-1/3,1/2]]$4],s tyle=point,color=[green$3,black],\n symbol=[circle,diamond,c ross,circle],symbolsize=[10$3,12]):\np3 := plot([[1/3,1/2],[-1/3,1/2]] ,color=black,linestyle=2):\np4 := plot([[[4/3,-1/2],[-4/3,-1/2]]$4],st yle=point,color=[blue$3,black],\n symbol=[circle,diamond,cro ss,circle],symbolsize=[10$3,12]):\np5 := plot([[4/3,-1/2],[-4/3,-1/2]] ,color=black,linestyle=2):\nt1 := plots[textplot]([2.2,-.12,`q`],font= [SYMBOL,10],color=black):\nt2 := plots[textplot]([-.05,1.3,`y`],font=[ HELVETICA,10],color=COLOR(RGB,.01,.01,.01)):\nplots[display]([p||(1..5 ),t1,t2],font=[SYMBOL,10],\n xtickmarks=[-2=`-2p`,-1=`-p`,-.5=`-p/2`, 0=`0`,.5=`p/2`,\n 1=`p`,1.5=`3p/2`,2=`2p`,2.5=`5p/2`,3=`3p`],\n \+ ytickmarks=3,labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 867 "p1 := plot(sin(t*Pi),t=-2. 2..2.2,color=red,thickness=2):\np2 := plot([[[2/3,sqrt(3)/2],[-2/3,-sq rt(3)/2]]$4],style=point,color=[green$3,black],\n symbol=[ci rcle,diamond,cross,circle],symbolsize=[10$3,12]):\np3 := plot([[2/3,sq rt(3)/2],[-2/3,-sqrt(3)/2]],color=black,linestyle=2):\np4 := plot([[[5 /3,-sqrt(3)/2],[-5/3,sqrt(3)/2]]$4],style=point,color=[blue$3,black], \n symbol=[circle,diamond,cross,circle],symbolsize=[10$3,12] ):\np5 := plot([[-5/3,sqrt(3)/2],[5/3,-sqrt(3)/2]],color=black,linesty le=2):\nt1 := plots[textplot]([2.2,-.12,`q`],font=[SYMBOL,10],color=bl ack):\nt2 := plots[textplot]([-.05,1.3,`y`],font=[HELVETICA,10],color= COLOR(RGB,.01,.01,.01)):\nplots[display]([p||(1..5),t1,t2],font=[SYMBO L,10],\n xtickmarks=[-2=`-2p`,-1=`-p`,-.5=`-p/2`,0=`0`,.5=`p/2`,\n \+ 1=`p`,1.5=`3p/2`,2=`2p`,2.5=`5p/2`,3=`3p`],\n ytickmarks=3,la bels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 46 "Code for complementary angle triangle picture " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 594 "darkgreen := COLOR(RGB,0,.7,0):\np1 := plot([[[0,0],[4,3]],[[4, 3],[4,0]],[[4,0],[0,0]]],\n color=[brown,darkgreen,red],thickness= 2):\np2 := plot([[0.84*cos(t),0.84*sin(t),t=0..arctan(0.75)],\n [4+ 0.84*cos(t),3+0.84*sin(t),t=arctan(.75)+Pi..3*Pi/2],\n [[3.8,0],[3. 8,.2],[4,.2]]],color=navy):\nt1 := plots[textplot]([4.15,1.5,`a`],colo r=darkgreen):\nt2 := plots[textplot]([2,-.15,`b`],color=red):\nt3 := p lots[textplot]([1.94,1.75,`c`],color=brown):\nt4 := plots[textplot]([[ .63,.24,`q`],[3.75,2.5,`f`]],\n font=[SYMBOL,11],color=na vy):\nplots[display]([p1,p2,t1,t2,t3,t4],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 44 "Code for co mplementary angle circle picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 912 "darkgreen := COLOR(RGB,0,.7 ,0):\nmaroon := COLOR(RGB,.7,0,.9):\ncs := evalf(cos(Pi/8)):\nsn := ev alf(sin(Pi/8)):\np1 := plot([[cos(t),sin(t),t=0..Pi/8],[cos(t),sin(t), t=Pi/8..3*Pi/8],\n [cos(t),sin(t),t=3*Pi/8..2*Pi]],color=[blue,maroo n,red],thickness=2):\np2 := plot([[[cs,sn],[0,0],[sn,cs]],[[cs,sn],[sn ,cs]]],\n color=[brown,black],thickness=[2,1],linestyle=[1,2]):\np3 \+ := plot([[0.25*cos(t),0.25*sin(t),t=0..Pi/8],\n [0.5*cos(t),0.5*sin( t),t=0..3*Pi/8]],color=navy):\np4 := plot([[-.8,-.8],[.8,.8]],color=bl ack,linestyle=3):\nt1 := plots[textplot]([[1.37,.4,`P( cos ,sin )`] ,\n [.81,1,`Q( cos ,sin )`],[1.15,-.04,`x`],\n [-.05,1.15,`y`], [-.06,.09,`O`]],color=black):\nt2 := plots[textplot]([[.31,.07,`q`],[. 37,.25,`f`],[.82,1.01,`f`],\n [1.05,1.01,`f`],[1.37,.41,`q`],[1 .61,.41,`q`]],\n font=[SYMBOL,11],color=black):\nplots[disp lay]([p1,p2,p3,p4,t1,t2],tickmarks=[0,0],scaling=constrained);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 44 "Code for supplementary angle circle picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 909 "darkgreen : = COLOR(RGB,0,.7,0):\nmaroon := COLOR(RGB,.7,0,.9):\ncs := evalf(cos(P i/6)):\nsn := evalf(sin(Pi/6)):\np1 := plot([[cos(t),sin(t),t=0..Pi/6] ,[cos(t),sin(t),t=Pi/6..5*Pi/6],\n [cos(t),sin(t),t=5*Pi/6..2*Pi]],co lor=[blue,maroon,red],thickness=2):\np2 := plot([[[cs,sn],[0,0],[-cs,s n]],[[-cs,sn],[cs,sn]],\n [[-cs,0],[-cs,sn]],[[cs,0],[cs,sn]]],\nc olor=[brown,black$3],thickness=[2,1$3],linestyle=[1,3$3]):\np3 := plot ([[0.25*cos(t),0.25*sin(t),t=0..Pi/6],\n [0.4*cos(t),0.4*sin(t),t=0 ..5*Pi/6]],color=navy):\nt1 := plots[textplot]([[1.36,.55,`P( cos ,si n )`],\n [-1.36,.55,`Q( cos ,sin )`],[1.28,-.04,`x`],\n [ -.05,1.15,`y`],[-.06,-.09,`O`]],color=black):\nt2 := plots[textplot]([ [.31,.07,`q`],[.19,.29,`f`],[1.37,.55,`q`],\n [1.63,.55,`q`],[-1 .35,.56,`f`],[-1.08,.56,`f`]],\n font=[SYMBOL,11],color=bla ck):\nplots[display]([p1,p2,p3,t1,t2],tickmarks=[0,0],scaling=constrai ned);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 39 "Code for opposite angle circle picture " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 889 "darkgreen := COLOR(RGB,0,.7,0):\nmaroon := COLOR(RGB,.7,0,.9):\ncs := evalf(cos (Pi/6)):\nsn := evalf(sin(Pi/6)):\np1 := plot([[cos(t),sin(t),t=0..Pi/ 6],[cos(t),sin(t),t=Pi/6..7*Pi/6],\n [cos(t),sin(t),t=7*Pi/6..2*Pi]] ,color=[blue,maroon,red],thickness=2):\np2 := plot([[[cs,sn],[-cs,-sn] ],\n[[-cs,0],[-cs,-sn]],[[cs,0],[cs,sn]]],\n color=[brown,black$3],t hickness=[2,1$2],linestyle=[1,3$2]):\np3 := plot([[0.25*cos(t),0.25*si n(t),t=0..Pi/6],\n [0.4*cos(t),0.4*sin(t),t=0..7*Pi/6]],color=navy ):\nt1 := plots[textplot]([[1.34,.55,`P( cos ,sin )`],\n [-1.36 ,-.55,`Q( cos ,sin )`],[1.28,-.04,`x`],\n [-.05,1.15,`y`],[-.06 ,-.09,`O`]],color=black):\nt2 := plots[textplot]([[.31,.07,`q`],[.19,. 29,`f`],[1.35,.55,`q`],\n [1.61,.55,`q`],[-1.35,-.55,`f`],[-1.09 ,-.55,`f`]],\n font=[SYMBOL,11],color=black):\nplots[displa y]([p1,p2,p3,t1,t2],tickmarks=[0,0],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 34 "Code for addition formula picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1989 "purple := COLOR(RGB,.5,0,.9):\nda rkred := COLOR(RGB,.85,0,0):\ndarkgreen := COLOR(RGB,0,.85,0):\ndarkcy an := COLOR(RGB,0,.83,.83): \ncsA := evalf(cos(2*Pi/9)):\nsnA := evalf (sin(2*Pi/9)):\ncsB := evalf(cos(5*Pi/36)):\nsnB := evalf(sin(5*Pi/36) ):\ncAB := evalf(cos(13*Pi/36)):\nsAB := evalf(sin(13*Pi/36)):\ncAcB : = csA*csB:\ncAsB := csA*snB:\np1 := plot([[[0,0],[cAB,sAB]],[[cAB,0],[ cAB,cAsB]]],\n color=black,linestyle=[1,3]):\np2 := plot([[cAcB ,cAsB],[cAB,sAB]],color=red,thickness=2):\np3 := plot([[0,0],[cAcB,cAs B]],color=darkred,thickness=2):\np4 := plot([[cAcB,0],[cAcB,cAsB]],col or=aquamarine,thickness=2):\np5 := plot([[cAB,cAsB],[cAcB,cAsB]],color =cyan,thickness=2):\np6 := plot([[cAB,cAsB],[cAB,sAB]],color=purple,th ickness=2):\np7 := plot([[0,0],[cAcB,0]],color=magenta,thickness=2):\n t1 := plots[textplot]([[.06,.07,`a`],[.12,.03,`b`],\n [.45,.78,`b`], [.57,.3,`b`]],font=[SYMBOL,11],color=navy):\nt2 := plots[textplot]([.6 3,.6,`sin`],color=red):\nt3 := plots[textplot]([.665,.6,`a`],font=[SYM BOL,11],color=red):\nt4 := plots[textplot]([.3,.19,`cos`],color=darkre d):\nt5 := plots[textplot]([.335,.19,`a`],font=[SYMBOL,11],color=darkr ed):\nt6 := plots[textplot]([.8,.17,`cos sin`],color=darkgreen):\nt7 := plots[textplot]([.835,.17,`a b`],font=[SYMBOL,11],color=darkgr een):\nt8 := plots[textplot]([.54,.37,`sin sin`],color=darkcyan):\nt 9 := plots[textplot]([.571,.37,`a b`],font=[SYMBOL,11],color=darkc yan):\nt10 := plots[textplot]([.32,.46,`sin cos`],color=purple):\nt1 1 := plots[textplot]([.35,.46,`a b`],font=[SYMBOL,11],color=purpl e):\nt12 := plots[textplot]([.36,-.03,`cos cos`],color=magenta):\nt1 3 := plots[textplot]([.395,-.03,`a b`],font=[SYMBOL,11],color=mag enta):\nt14 := plots[textplot]([[.17,.45,`1`],[.4,.91,`P`],[.72,.33,`Q `],\n [.72,.035,`R`],[-.02,.035,`O`],[.4,.035,`S`],\n [.4,.33,`T`] ,[.81,-.02,`x`],[-.02,.92,`y`]],color=black):\nplots[display]([p1,p2,p 3,p4,p5,p6,p7,t1,t2,t3,t4,t5,t6,t7,t8,t9,t10,t11,\n t12,t13,t14],sca ling=constrained,tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }