{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Ou tput" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warni ng" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 1 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 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "Module 4 : Trigonometry" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 53 "402 : Polar Graphs - Roses Rings Bracelets and Heart s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 9 "S E T U \+ P" }}{PARA 0 "" 0 "" {TEXT -1 251 "In this project we will use the fol lowing command package. Type and execute this line before begining the project below. If you re-enter the worksheet for this project, be sur e to re-execute this statement before jumping to any point in the work sheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________________________ __________________________" }}{PARA 4 "" 0 "" {TEXT -1 23 "A. Cardioid s & Limacons" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________ __________________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "We're going to lo ok at a variety of cardioids, which are graph of the form" }}{PARA 0 " " 0 "" {TEXT -1 65 " y = a +- b sin(theta) or y=a \+ +- b cos(theta)" }}{PARA 0 "" 0 "" {TEXT -1 97 "and see how the relati onship among the components effects the graph.\n\n COMPARING \+ a AND b" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "In particular, there are three cases : |a| = |b|. |a| > |b|, and \+ |a| < |b|. Each of these cases creates a distinctive version of the li macon." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "When |a| = |b|, the graph passes through the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "This shape is known a cardioid, or heart shaped curve. Note the reference circles of radius 1 and 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "polarplot( \{1,2, 1+sin(theta)\}, theta = 0..2*Pi, scaling = constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"\"\"\"\"!$F*F*7$$\"3#*p#fVPi j!**!#=$\"3+9&o'z\"y_O\"F/7$$\"3[#y/-5-Qn*F/$\"3K!y=V&*)GLDF/7$$\"3'4g ??['e[#*F/$\"3R]F<<.6.QF/7$$\"35D5SC42`')F/$\"3I,*y(H4U7]F/7$$\"3FQ^ZT ?D/zF/$\"3%=pD*eceDhF/7$$\"3=_>U!=uD3(F/$\"3w>82*oU&fqF/7$$\"3')e\\phd X;hF/$\"3@)*QjP!>8\"zF/7$$\"3[yHev:x5]F/$\"3[[%3w7ESl)F/7$$\"39g)pymR, #QF/$\"3Pi.Z,bcT#*F/7$$\"39.oK.jQDDF/$\"3!e$pSF\"oen*F/7$$\"3KlS\"H&o8 X8F/$\"3#ffI'ft64**F/7$$!3)R^L8JO5(f!#@$\"2-6Ot@)******!#<7$$!33@,ABB[ i8F/$\"3s;Nd#HZn!**F/7$$!3!p()*f7@=YEF/$\"3*p]U&fD`V'*F/7$$!3%R#pX#*)3 Jx$F/$\"3cBCb^l'3E*F/7$$!3nht,N_JU]F/$\"3;2*>c4&oN')F/7$$!3L_)*oQ%*[Rg F/$\"3Se*Rp1I-(zF/7$$!3k\\\"Q8J;$*3(F/$\"3c:s(f4sF0(F/7$$!3UtA:G>r1$f')F/$\"3?Pawd/k,]F/7$$!3(H:T1DO4B*F/$\"3 i\"*R%R(HvXQF/7$$!3E3Gi$\\BOm*F/$\"3!\\o6f)Q%=d#F/7$$!3Zw5r5+.2**F/$\" 3Kh5Xo]Ug8F/7$$!3'p)f24>&*****F/$\"3^)pb)>hJ,J!#?7$$!38)zW\"G!Q%3**F/$ !3)[9aqyJ,N\"F/7$$!3;&3vb[lSn*F/$!3w\"*=GeHGKDF/7$$!3eDXLTAEj#*F/$!39* )zsmMAnPF/7$$!35CZ!4<'=u')F/$!3#35.)=3zv\\F/7$$!37)oJ.#y1YzF/$!3!fgI[? 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N ote the reference circles of radius 1 and 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 257 "display( polarplo t( 8 + 8*cos(theta) , theta = 0..2*Pi, scaling = constrained, color = \+ green, thickness = 3), polarplot(\{8 + a*cos(theta) $ a = 9..15\}, the ta = 0..2*Pi, color = blue), polarplot(\{ 8 + a*cos(theta) $ a = 1..7 \}, theta = 0..2*Pi, color = red));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "63-%'CURVESG6%7`o7$$\"#;\"\"!$F*F*7$$\"3a.+HX&zVf \"!#;$\"3Wr'[nY0N4\"!#<7$$\"3oV2z8rfx:F/$\"3+q/Yzx@u@F27$$\"3#zO!fAI4a :F/$\"3-a&4/MBV4$F27$$\"3$\\S$GWPcA:F/$\"3YCj,_S:()RF27$$\"3#R'3ztj$zZ \"F/$\"3-@9PBa0\\\\F27$$\"3\">lkJtxTU\"F/$\"3o8<8Q)fj&eF27$$\"3AG7zE=c h8F/$\"3kDC)\\z/\\q'F27$$\"3Qs&4Tw]7H\"F/$\"3#R5z1Yj(zuF27$$\"3,2xGrhb 97F/$\"3q+sCzECq\")F27$$\"3%)zOAkx:K6F/$\"3%**p/[PARx)F27$$\"3y3xPZ/f^ 5F/$\"3k+nbf`p^#*F27$$\"3?qOD$)y3z'*F2$\"3eGOxUHhZ'*F27$$\"3?Efq'R#R*y )F2$\"3)fE.jZw#p**F27$$\"3a#3!zoq/')yF2$\"36uH&oQ>+-\"F/7$$\"3/Kl.'fI \"[pF2$\"3KCt(oI$QM5F/7$$\"3n3f?sPC\"o!*eBUF2$\"3o'y&=2yv@5F/7$$\"3%\\M)\\ >ZB`LF2$\"3m,1KZNu'***F27$$\"3\\\"R2&)3:0`#F2$\"3?SHPv)=bp*F27$$\"3Z2l \"Q!4'3A\"F2$\"3]\"zTEQBO**)F27$$!3sxTHDN+uZ!#?$\"39K_))=F2$\"3\"Ql)3>./DMF27$$!3(**f$GI pb8>F2$\"3\"f6NaR%HDDF27$$!3m'\\^id!y];F2$\"3%ox)on5FU;F27$$!3mbgfMyCD 8F2$\"3gkY5#[[q-\"F27$$!3_VK=Z$zvG*Fjr$\"3sl,_pC`k`Fjr7$$!3%pdE@gU$zcF jr$\"3-s'p&eK5mBFjr7$$!3-]\"G+h#\\+EFjr$\"3c?I11C'3#p!#>7$$!3e%)\\K36X otFju$\"39!yM/\\H=,\"Fju7$$!3er$fu$)Qs%Q!#A$\"3E[h1\\N:$>\"!#C7$$!37)f'*))*Fdr7$$!3fnAE1'*[ADFjr$!3cw>#[)y'Gg'Fju7$$!3o O*QBCw'faFjr$!3t.7>F2$!3j'[j9P7t^#F27$$!3cJcbjI****>F2$!3y5u]HmDF2$!3/\"yA&e6l5(*F27$$\"3;&==)[mq XUF2$!3!H]Z)Gt>A5F/7$$\"3IoRuzLS7^F2$!3e*ed&y!H\\.\"F/7$$\"3O'QgKz\"G0 gF2$!3Sp![NLI#R5F/7$$\"3&**>I\"yPwNpF2$!3$z3\"fg)3X.\"F/7$$\"3Yx())QI8 M(yF2$!3_Zpn\")pF?5F/7$$\"3k.*yI%em*z)F2$!37z,!RX4h'**F27$$\"3y3!eteX9 r*F2$!3Y&[#>/'\\Sj*F27$$\"3%fl(Gm'=F0\"F/$!3e:+;T+pX#*F27$$\"3V'y=2#oJ J6F/$!3wa\\K-$[%z()F27$$\"3I9DnGzo;7F/$!3[h>n&3mG:)F27$$\"3%o&y0_+!fH \"F/$!3\"G-d0gtJV(F27$$\"3A)>:()GC1O\"F/$!3uU')4H>L;nF27$$\"3'4$>&p@0) =9F/$!3U?jKl>jOfF27$$\"3oy2!R,))GZ\"F/$!3\"G1k<.^P/&F27$$\"3g'zHoCz!=: F/$!3![Dv_AEj4%F27$$\"3!*RrNd*eCb\"F/$!3]?\"*G`DjZJF27$$\"3qQ+\"oeQxd \"F/$!3]`'\\>D\")\\ U>m78!#D-%'COLOURG6&%$RGBGF+$\"*++++\"!\")F+-%*THICKNESSG6#\"\"$-F$6$7 go7$$\"#Yhh\" F/$\"3$oLfSS>AB%F27$$\"3KFZG$o`yc\"F/$\"3!ya9l![:]_F27$$\"3$eP+e39(4:F /$\"3]G:&Rx$43iF27$$\"3h&=n-xW?W\"F/$\"3nbB/UnB,rF27$$\"3%=^cGm#>-MG\"F/$\"3if*4Z\\lLj)F27$$\"39'[Mj'\\j% >\"F/$\"3%R.AE4/\"e#*F27$$\"3=&p$Re(fz5\"F/$\"3_O8YY>iZ(*F27$$\"3GtViZ ;2=5F/$\"3ILD5G5w95F/7$$\"3SlLy2AuE#*F2$\"3!*H!3\"*eLl/\"F/7$$\"3!yR&) pSd,E)F2$\"3u(z.#>'3%o5F/7$$\"3iXCUIw/fsF2$\"3yvD%o>q13\"F/7$$\"3q6n!4 7A$oiF2$\"3WJ)>MA#f#3\"F/7$$\"3&GktBK.EI&F2$\"3#e-[9bdX2\"F/7$$\"3UW)y ;xB&pVF2$\"3)**31)y=1d5F/7$$\"3=**)3!\\0PaMF2$\"3/Cz/,_#)H5F/7$$\"3-)> a4)3H%f#F2$\"3cLIoP>()R**F27$$\"3>Kt5m[&*Q7F2$\"3.V0L#f9p7*F27$$!3!y** \\VD?Ox%Fdr$\"3yzm9UYi%*zF27$$!35,Cy7j8H#*Fjr$\"3s%>_!3()f5nF27$$!35#) f]50u'[\"F2$\"36vP6Xr:=aF27$$!3!H;v'\\b@Plo-!*QE%F27$$!3=_z PCtgX [6F2$\"3/3sOtxFU6F27$$!3x/OINT$[+(Fjr$\"3W%z3\\D['GaFjr7$$!3qjcIP-A*y \"Fjr$\"3=:=?a(eM.\"Fjr7$$\"3uK;$>lv;%GFjr$!33@'>Gp')Q=\"Fjr7$$\"3dMV! 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How does the choice of one of these effect the graph? Lets take a look at all four at once!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Can you decide which graph belongs to which? 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The Rose Garden" }}{PARA 0 "" 0 "" {TEXT -1 83 "____________________________________________________ _______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "We're go ing to look polar functions of the form f = a sin(n ) and r = a cos(n \+ ) which are sometimes called multi-petaled roses." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "EVEN AND ODD NUMBER PETAL S" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The \+ first distinction to be made is between when n is an even or odd numbe r." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Whe n n is an odd number, the resulting rose has exactly n petals" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "polarplot( \{9, 9*sin(5* theta)\}, theta = 0..2*Pi, scaling = cons trained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%' CURVESG6$7S7$$\"\"*\"\"!$F*F*7$$\"3:VL#p8Ed\"*)!#<$\"3ji;qh.vG7F/7$$\" 3z.V=!*=U1()F/$\"3Y-p))e+'*zAF/7$$\"3?T&=Q$ysB$)F/$\"3=vaX&G*zAMF/7$$ \"3OA4'>$Qw(y(F/$\"3%3,,o$)y6^%F/7$$\"3bCwKPo#Q6(F/$\"3mAJ.$4FI^&F/7$$ \"3)pvzBw;VP'F/$\"37)=k,U)e`jF/7$$\"3`ia_&=5[]&F/$\"3130(Q8(=?rF/7$$\" 3I!oC!=Wp4XF/$\"3'RgZ[^B')y(F/7$$\"3IuG3,d7QMF/$\"3DELK^*3uJ)F/7$$\"3s AT*HnZGF#F/$\"3yUim98G3()F/7$$\"3()eEinJi57F/$\"3xNvmBc?=*)F/7$$!3Oi,? !oKRP&!#?$\"3#)*\\-cR)*****)F/7$$!3')3\")*34MiA\"F/$\"3/lhJjD2;*)F/7$$ !3E*))R8!Rc\"Q#F/$\"3Gc#)e.$z\"z')F/7$$!3OJ7@.!)z&R$F/$\"3`\"=(R'*)zZL )F/7$$!3SDc^6P3QXF/$\"3W;z0'e;@x(F/7$$!3Vn3#[\\SbV&F/$\"3aifCgq?trF/7$ $!3!\\L/-o%Q!Q'F/$\"3a$\\zj)[\\ZjF/7$$!3&e/bev$y8rF/$\"34dT[$o#38bF/7$ $!3=P`tSgP$z(F/$\"3.$*)))>Tw9]%F/7$$!37QqdDE%yI)F/$\"3_#f\\lnx6Y$F/7$$ !33G0EW6E(p)F/$\"3d;0K(\\fYJ#F/7$$!3gp*R'4qK;*)F/$\"3;bfghDQC7F/7$$!3% zQo\"=n&****)F/$\"3iG,(y]%=\"z#!#>7$$!34>.LDUf<*)F/$!3UI([$3'=^@\"F/7$ $!3Kxv,P*emq)F/$!3[-PXiY0zAF/7$$!3Cs5?b+7,0R$F/7$$!3._U \"Qbnn!yF/$!3=\"zApt6#yWF/7$$!3K>&)HQ5Y^rF/$!3KXvM%y>TY&F/7$$!3#pQEiR4 &*R'F/$!3IQ=UxH@GjF/7$$!3R$z7\"\\ysVaF/$!3mR$H>o&*p;(F/7$$!3AtU,Y'>;\\ %F/$!3Q!ew..h!*z(F/7$$!3er`*H #F/$!3#Goi3pr7q)F/7$$\"3]qxic%=AX$F/$!3E(p!\\5&p:J)F/7$$\"3K1]fJ/(H]%F /$!3%G?7(**G^#z(F/7$$\"3IAl1y_T)\\&F/$!3[%=ObiE^7(F/7$$\"33V%fNFq$*Q'F /$!3\"**p(QH*\\%QjF/7$$\"3$4kw#)[g,6(F/$!3Eoyp\"yax^&F/7$$\"3qX)R'4P!p !yF/$!3G_/7rV(zZ%F/7$$\"3?pTM93]-$)F/$!3')4v6B=(RZ$F/7$$\"31%*[C_*>#*o )F/$!317D4nbmWBF/7$$\"3@YjLm*fi\"*)F/$!3z1(RApq[A\"F/7$$\"3A)********* *****)F/$!3\"yPMayi.8\"!#C-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-F$6$7a[l7$F +F+7$$\"3:\"o%*G]OB`\"F/$\"3;!Hq9=,'[_F`s7$$\"3&=T^[uiX,$F/$\"3?gbVP$Q v1#!#=7$$\"35mocQuS)R%F/$\"3t#z3V'R&Q`%F_\\l7$$\"3!357I$\\?RcF/$\"3Wnj *=Fd=x(F_\\l7$$\"3uFddR2erlF/$\"3\\z5JM\"fg5\"F/7$$\"3sttinSxVtF/$\"3! 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