{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Tim es" 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 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 55 "High School Modul es > Trigonometry by Gregory A. Moore " }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 256 14 "The Sine Graph" }}{PARA 0 "" 0 "" {TEXT -1 58 "\n An exploration of the many variations of the sine graph.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execute the Code Resource section first. Although there will be no output immediately, these definition s are used later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 " 0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "SolidPlot := proc(f,a,b)\nlocal c;\nc := COLOR( RGB, .6, .8, .5);\nplot( f(x), x = a..b, filled = true, color=c, style =patchnogrid);\011\nend:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "SolidPlotRCol := proc(f,a,b)\nlocal c;\nc := COLOR(RGB, 5*evalf (rand()/10^13,2), \n 5*evalf(rand()/10^13,2), \n \+ 5*evalf(rand()/10^13,2) );\nplot( f(x), x = a..b, filled = t rue, color=c, style=patchnogrid);\011\011\nend:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "SolidPlotCol := proc(f,a,b, COL)\nlocal \+ box,i,n,x1,x2,xmid,delta,y1,y2,A,B, m,M,slope, concav, col: \nplot( f (x), x = a..b, filled = true, color=COL, style=patchnogrid, numpoints= 1000);\011\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 699 "SinePl ot := proc(f)\nlocal A, B,G,start, endpt, freq, period, lift, amp : \+ \nlift := ( maximize(f(x), x = 0..10) + minimize(f(x), x = 0..10) )/2 ;\nG := f(x) - lift;\nstart := solve(G = 0, x);\nif (nops(G) = 2) the n\n freq := subs( x = 1, op(1, op(1, op(2,G))));\n period := 2*Pi/ freq;\n amp := evalf(subs(x = start + period/4, G)); \nelse\n amp \+ := 1; \n freq := subs( x= 1, op(1,G));\n period := 2*Pi/freq;\n \+ \nfi;\nendpt := start + period;\n\nplots[display](\n SolidPlot( x-> sin(x), 0, 2*Pi),\n plot( f(x), x = start..endpt, color = blue, thi ckness = 2),\n plot( [[start,lift],[endpt,lift]], \n \+ color = red, thickness = 1, linestyle = 2), \n plot( 0, x = 0. .2*Pi) );\011\nend:" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 24 " 1. The Basic Sine Graph" }}{PARA 0 "" 0 "" {TEXT -1 117 "\nFor each value of x, there is a value of sin(x). In other words, y = sin(x) is a functi on of x, and can be plotted.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sin( 3*Pi/7): % = evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "n := 12;\narray( [seq( [ 2*i*Pi/n, evalf(sin(2*i*Pi/n )) ], i = 1..n ) ]);" }}}{PARA 0 "" 0 "" {TEXT -1 42 "\nThis is the \+ basic shape of the sine graph" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f := x-> sin(x);\nSolidPlot( f, 0, 2*Pi);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 " 2. Altering the Amplitude" }}{PARA 0 " " 0 "" {TEXT -1 121 "\nThe amplitude of the sine function is the heigh t of its hill (and depth of its valley). For sin(x), the amplitude is \+ 1.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 469 "display( [ plot(sin (x), x = 0..2*Pi, filled = true, color = COLOR(RGB, .9, .8, .5) ),\n \+ plot(sin(x), x = 0..2*Pi, thickness = 2, color = maroon),\n \+ plot( [[Pi/2,0],[Pi/2,1]], color = blue, linestyle = 2, thickne ss = 3),\n plottools[arrow]( [Pi/2,0],[Pi/2,1], .1, .2, .1, c olor = red),\n plot( [[3*Pi/2,0],[3*Pi/2,-1]], color=red, lin estyle=2,thickness=3),\n plottools[arrow]( [3*Pi/2,0],[3*Pi/2 ,-1], .1, .2, .1, color = red) ] );" }}}{PARA 0 "" 0 "" {TEXT -1 127 " \nA constant in front of the sin(x) function alters the amplitude. In \+ fact, the amplitude is the absolute value of this number.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f := x -> 3*sin(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "SinePlot( f );" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 275 "Below are plots of sine functions with amplitudes ranging from 1/16 to 1, that is, the graphs of 1/16sin(x), 1/8sin(x) , 3/16sin(x),1/4sin(x),5/16sin(x),3/8sin(x),7/16sin(x),1/2sin(x),9/16s in(x), 5/8sin(x), 11/16sin(x), 3/4sin(x), 13/16sin(x), 7/8sin(x), 15/1 6sin(x), sin(x)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "displa y( [ seq( \n SolidPlotCol( x-> (k/16)*sin(x),0,2*Pi,\n COLOR(RGB, \+ 1 - k/20, .5 - k/32, 0 )),k = 1..16) ]);" }}}{PARA 0 "" 0 "" {TEXT -1 102 "\nIf the constant is negative, the stretching factor is the same, but the sine is flipped upside down.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f := x -> -.4*sin(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "SinePlot( f );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 36 " 3. Altering the Period & Freque ncy" }}{PARA 0 "" 0 "" {TEXT -1 91 "\n\nThe sine graph is periodic - i t repeats. Each section that repeats is called a \"period\".\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot( sin(x), x=0..10*Pi, co lor = blue);" }}}{PARA 0 "" 0 "" {TEXT -1 52 "\nThe fundamental \"piec e\" that repeats has length of " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\" \"%#PiGF%" }{TEXT -1 19 ". The frequency is " }{XPPEDIT 18 0 "1/(2*Pi) " "6#*&\"\"\"F$*&\"\"#F$%#PiGF$!\"\"" }{TEXT -1 75 " - that is, the si n(x) function completes 1 cycle every \"time\" interval of " } {XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 65 " - or, ano ther way to think of it is that the function completes " }{XPPEDIT 18 0 "1/(2*Pi)" "6#*&\"\"\"F$*&\"\"#F$%#PiGF$!\"\"" }{TEXT -1 37 " of a p eriod every one unit of time.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "display( plot(sin(x), x = -2*Pi..6*Pi, color = blue),\n \+ plot(sin(x), x = -0..2*Pi, color = red, thickness = 4),\n \+ plot( [[0,-1],[2*Pi,-1],[2*Pi,1],[0,1]], color = red, linestyle = 2) );\n `Fundamental Period`;" }}}{PARA 0 "" 0 "" {TEXT -1 227 " \n\nIf there is a constant times x inside of the sin(x) function, the \+ period and frequency change. For example, sin(2x) will complete its cy cles twice as fast as sin(x) - each one taking half as long. So the pe riod of sin(2x) is " }{XPPEDIT 18 0 "2*Pi/2 = Pi" "6#/*(\"\"#\"\"\"%#P iGF&F%!\"\"F'" }{TEXT -1 39 ", and the frequency is 1 over that, or " }{XPPEDIT 18 0 "1/Pi" "6#*&\"\"\"F$%#PiG!\"\"" }{TEXT -1 2 ".\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f := x -> sin(2*x);\nSinePlo t( f );" }}}{PARA 0 "" 0 "" {TEXT -1 88 "If the constant is less than \+ one, the graph has a smaller frequency and a longer period." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f := x -> sin((1/2)*x);\nSinePlot( \+ f );" }}}{PARA 0 "" 0 "" {TEXT -1 102 "\nIf the constant is negative, \+ the stretching factor is the same, but the sine is flipped upside down .\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f := x -> -.4*sin(x); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "SinePlot( f );" }}} {PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 " 4. Amplitude & Period Changes" }}{PARA 0 "" 0 "" {TEXT -1 233 " \nThe changes to period and amplitude can both operate independently o n a single sine function. The period is a property of the x coordiate, and the amplitude is a property of the y coordinate, so neither has a ny impact on the other.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f := x -> 10*sin(2*x); \nSinePlot(f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f := x -> (1/10)*sin(2*x); \nSinePlot(f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f := x -> 3*sin((1/2)*x); \nSinePlo t(f);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 " 5. Shifting & Lifting" }}{PARA 0 "" 0 "" {TEXT -1 233 " \nIn algebra we learn that y = f(x) and (y-b) = f(x-a), or y = f(x-a) \+ + b, have the same graphs except the latter is shifted a units right, \+ and b units up. The same concept applies to all other functions, inclu ding the sine function.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot( \{x^2, (x-3)^2 + 4\} , x = -3..6, y = 0..9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f := x -> sin(x - Pi/3) + 4 ;\nSinePlot(f );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f := x -> sin(x - Pi /2) - 2 ;\nSinePlot(f);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 38 " 6. Shifting/Lifting/Amplitude/Period " }}{PARA 0 "" 0 "" {TEXT -1 87 "\nHere are some examples where all fo ur transformation take place in the same function.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "f := x -> 3*sin( 2*x - Pi/3) + 1;\nSinePl ot( f );" }}}{PARA 0 "" 0 "" {TEXT -1 16 "\nAnother exmaple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f := x -> 10*sin( (1/2)*x + Pi) + 5 ;\nSinePlot( f );" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 " 7. Compos ite Functions of Sine" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wit h(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "f := x -> abs (sin(x));\ndisplay(SolidPlotCol( x->sin(x), 0, 2*Pi, blue),\n p lot( f(x), x = 0..2*Pi, thickness = 3) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "f := x -> sin(x)^2;\ndisplay(SolidPlotCol( x->sin(x), 0, 2*Pi, blue),\n plot(f(x), x = 0..2*Pi,thickness \+ = 3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "f := x -> sin(x) ^3;\ndisplay(SolidPlotCol( x->sin(x), 0, 2*Pi, blue),\n plot(f( x), x = 0..2*Pi,thickness = 3));" }}}{PARA 0 "" 0 "" {TEXT -1 53 "\nWe can look at families of similar sine functions. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 432 "restart; with(plots): \nSolidPlotCol := pr oc(f,a,b, COL)\nlocal box,i,n,x1,x2,xmid,delta,y1,y2,A,B, m,M,slope, \+ concav, col: \nn:= 60; delta := (b-a)/n; x2 := a;\ncol := COL;\nfo r i from 1 to n do\n\011\011x1 := evalf(x2);\011 y1 := evalf( f(x1)); \n\011\011x2 := evalf(a + i*delta);\011 y2 := evalf( f(x2)); \011\n B[i]:=polygonplot([[x1,0],[x1,y1],[x2,y2],[x2,0]],\n\011\011 \011\011\011color=col, style=patchnogrid);\nod:\ndisplay(\{ seq( B[i], i=1..n ) \} );\011\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot( \{ sin(x)^k $ k = 1..10\}, x = 0..2*Pi, color = aquamarine); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 223 "display( [ seq( SolidPlotCol( x-> sin(x)^(16-k),0, 2*Pi,\n COLOR(RGB, .4 + k/24, .5 + k/24, .2 + k/24)), k = 1.. 12),\n plot( [sin(x)^(16-k) $ k = 1..12], \n x = 0.. 2*Pi, color =gray, axes = none)]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Below are plots of the followin g 10 functions superimposed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 256 "" 0 "" {XPPEDIT 18 0 "sin(x)2*sin(2*x)3*sin(3*x)4*si n(4*x)5*sin(5*x)6*sin(6*x)7*sin(7*x)8*sin(8*x)9*sin(9*x)10*sin(10*x); " "6#%#%?G" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "display( plot( sin(x), x = \+ 0..2*Pi, thickness=3),\n seq( SolidPlotCol( x-> k*sin( k*x),0,2*Pi,\n COLOR(RGB, .2 + k/15, .2 + k /15, .75)), k = 1..10));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Below is a superimposition of the plots of the following 8 functions." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "for i from 1 to 8 do\n prin t(sum( sin((2*j+1)*x)/(2*j+1), j= 1..i));\nend do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "display( [plot( \{sum( sin((2*j+1)*x)/(2*j +1), j= 1..k) $ k = 1..8\}, \n x = 0..2*Pi, color=b lack)]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "How about\n " } {XPPEDIT 18 0 "sin(x), 2*sin(1/2*x)^2, 3*sin(1/3*x)^3, 4*sin(1/4*x)^4, 5*sin(1/5*x)^5, 6*sin(1/6*x)^6, 7*sin(1/7*x)^7, 8*sin(1/8*x)^8, 9*sin (1/9*x)^9, 10*sin(1/10*x)^10, 11*sin(1/11*x)^11, 12*sin(1/12*x)^12" "6 .-%$sinG6#%\"xG*&\"\"#\"\"\"*$-F$6#*(F)F)F(!\"\"F&F)F(F)*&\"\"$F)*$-F$ 6#*(F)F)F0F.F&F)F0F)*&\"\"%F)*$-F$6#*(F)F)F6F.F&F)F6F)*&\"\"&F)*$-F$6# *(F)F)F \+ " 0 "" {MPLTEXT 1 0 227 "display( [ seq( SolidPlotCol( x-> k*(sin(x/ k)^k),0,12*Pi,\n COLOR(RGB, .2 + k/25, .5 + k/60, .1 + k/25)) , k = 1..12),\n plot( [k*(sin(x/k)^k) $ k = 1..12], \n \+ x = 0..12*Pi, color =blue, thickness=2)]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Or\n " }{XPPEDIT 18 0 "1/2*sin(1/2*x)^2,1/3*sin(1/3*x)^ 3,1/4*sin(1/4*x)^4,1/5*sin(1/5*x)^5,1/6*sin(1/6*x)^6,1/7*sin(1/7*x)^7, 1/8*sin(1/8*x)^8,1/9*sin(1/9*x)^9,1/10*sin(1/10*x)^10,1/11*sin(1/11*x) ^11,1/12*sin(1/12*x)^12;" "6-*(\"\"\"F$\"\"#!\"\"-%$sinG6#*(F$F$F%F&% \"xGF$F%*(F$F$\"\"$F&-F(6#*(F$F$F-F&F+F$F-*(F$F$\"\"%F&-F(6#*(F$F$F2F& F+F$F2*(F$F$\"\"&F&-F(6#*(F$F$F7F&F+F$F7*(F$F$\"\"'F&-F(6#*(F$F$F " 0 "" {MPLTEXT 1 0 231 "display( [ seq( SolidPlotC ol( x-> (1/k)*(sin(x/k)^k),0,6*Pi,\n COLOR(RGB, .2+k/25, .6 \+ + k/60, .3 + k/25)), k = 2..12),\n plot( [(1/k)*(sin(x/k)^k) \+ $ k = 2..12], \n x = 0..6*Pi, color =blue, thickness=2)]);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "restart:\ncol := j -> COL OR(RGB,.5-j/20,.6-j/20,.7-j/20):\nplot( [x/(2+sin(j*x/2)) $ j = 1..10] , x = 0..5*Pi, \n color = [ col(j)$ j = 1..10] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 259 34 " \n \+ \251 2002 Waterloo Maple Inc" }}}{MARK "12 6 0 0" 428 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }