{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 " Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } } {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 54 "High School Modul es > Trigonometry by Gregory A. Moore" }{TEXT 260 1 " " }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 256 17 "Other Trig Graphs" }}{PARA 0 "" 0 "" {TEXT -1 74 "\nAn exploration of the many variations of the other trigonometric graphs.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execute the Code Resource section first. Although there will be no \+ output immediately, these definitions are used later in this worksheet .]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 " 0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; w ith(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 497 "SolidPlotB ound := proc(f,a,b, M)\nlocal n,delta,x1,x2,y1,y2,B,i: \nn:= 60; de lta := (b-a)/n; x2 := a;\nfor i from 1 to n do\n\011\011x1 := evalf( x2);\011 y1 := evalf( f(x1));\n\011\011x2 := evalf(a + i*delt a);\011 y2 := evalf( f(x2));\011\n if(y1 > M) then y1 := M; else if ( y1 < -M) then y1 := -M; fi;fi;\n if(y2 > M) then y2 := M; else if (y2 < -M) then y2 := -M;fi;fi;\n\n B[i]:=polygonplot([[x1,0],[x1,y1],[x2 ,y2],[x2,0]],\n\011\011\011\011\011color=red, style=patchnogrid);\nod; \011\ndisplay(\{ seq( B[i],i=1..n ) \} );\011\nend:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 " 1. The Tangent Graph" }}{PARA 0 "" 0 "" {TEXT -1 120 "\ny = ta n(x) is also a function. It too is periodic. However, it differs from \+ sine and cosine because its period is only " }{XPPEDIT 18 0 "Pi" "6#%# PiG" }{TEXT -1 33 " and it has vertical asymptotes.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := x -> tan(x):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "SolidPlotBound( x -> tan(x), -1.56, 1.56, 20); " }}}{PARA 0 "" 0 "" {TEXT -1 79 "\nThis is one period of the tangent. The tangent is continuous on the interval (" }{XPPEDIT 18 0 "-Pi/2,Pi /2" "6$,$*&%#PiG\"\"\"\"\"#!\"\"F(*&F%F&F'F(" }{TEXT -1 52 "), and thi s constitutes a complete period of length " }{XPPEDIT 18 0 "Pi" "6#%#P iG" }{TEXT -1 55 ". Of course, it repeats this same pattern indefinite ly." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot( tan(x), x = -2* Pi..2*Pi, y = -20..20, discont = true);" }}}{PARA 0 "" 0 "" {TEXT -1 76 "\n\nNotice that there are vertical asymptotes for all values of x \+ of the form " }{XPPEDIT 18 0 "x = Pi/2 + k*Pi" "6#/%\"xG,&*&%#PiG\"\" \"\"\"#!\"\"F(*&%\"kGF(F'F(F(" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "display( plot( tan(x), x = -2*Pi..2*Pi, y = -20 ..20, \n discont = true, color = blue, thickness = 2), \n seq( plot( [[(2*k+1)*Pi/2 ,-20],[(2*k+1)*Pi/2 ,20]], \n \+ color = green, linestyle = 2), k = -2. .1) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "display( plot( t an(x), x = -6*Pi..6*Pi, y = -20..20, \n discont = true, color = blue, thickness = 1),\n seq( plot( [[(2*k+1)*Pi/2 \+ ,-20],[(2*k+1)*Pi/2 ,20]], \n color = green, li nestyle = 2), k = -6..5) );" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 " 2. The Secant Graph" }} {PARA 0 "" 0 "" {TEXT -1 159 "\nThe secant is the reciprocal of the co sine. So this helps us to understand the graph. Since -1 <= cos(x) <= \+ 1, this means secant is entirely in the intervals " }{XPPEDIT 18 0 "(- infinity,-1] union [1, infinity)" "6#-%&unionG6$7$,$%)infinityG!\"\",$ \"\"\"F)7$F+F(" }{TEXT -1 75 ". Here is the cosine, and then the cosin e with its reciprocal, the secant.\n" }}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "display( [SolidPlotBound( x -> cos(x), 0, 2*Pi, 5 ), \n plot(sec(x), x = 0..2*Pi, y = -5..5, \n \+ color = blue, thickness = 2, discont = true )]);" }} }{EXCHG }{PARA 0 "" 0 "" {TEXT -1 197 "\nLike the tangent function, th e secant is undefined whenever cosine is zero. This means there are ve rtical asymptotes in the exact same places as the tangent function - a ll values of x of the form " }{XPPEDIT 18 0 "x = Pi/2 + k*Pi" "6#/%\"x G,&*&%#PiG\"\"\"\"\"#!\"\"F(*&%\"kGF(F'F(F(" }{TEXT -1 2 ".\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "display( plot( sec(x), x = \+ -2*Pi..2*Pi, y = -20..20, \n discont = true, color = bl ue, thickness = 2),\n seq( plot( [[(2*k+1)*Pi/2 ,-20],[(2*k +1)*Pi/2 ,20]], \n color = green, linestyle = 2 ), k = -2..1) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "display( plot( sec(x), x = -6*Pi..6*Pi, y = -20..20, \n \+ discont = true, color = blue, thickness = 1),\n seq( \+ plot( [[(2*k+1)*Pi/2 ,-20],[(2*k+1)*Pi/2 ,20]], \n \+ color = green, linestyle = 2), k = -6..5) );" }}}{PARA 0 "" 0 "" {TEXT -1 229 "\nJust like the cosine, when we alter the functi on in various ways, we affect its period and height. The secant doesn' t have an amplitude, but its minimal positive value is the reciprocal \+ of the amplitude of the associate cosine.\n" }}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "display( [SolidPlotBound( x -> 2*cos(3*x), 0, 2*Pi, 5), \n plot( (1/2)*sec(3*x), x = 0..2*Pi, y = -5..5, \n color = blue, thickness = 2, dis cont = true )]);\n`Notice that the period is 2*Pi/3.`;\n`The amplitude of the cosine is 2, while the minimal positive value of the secant is 1/2`;" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "display( [SolidPlotBound( x -> (1/2)*cos((1/2)*x), 0, 4*Pi, 5), \n \+ plot( 2*sec((1/2)*x), x = 0..4*Pi, y = -5..5, \n \+ color = blue, thickness = 2, discont = true )]);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 " 3. \+ The Cosecant Graph" }}{PARA 0 "" 0 "" {TEXT -1 236 "\nThe cosecant is \+ the reciprocal of the sine, and this helps us to understand it in the \+ same way we understood secant. Like the cosine, the sine is limited to the same range -1 <= sin(x) <= 1, this means csc is entirely in the i ntervals " }{XPPEDIT 18 0 "(-infinity,-1] union [1, infinity)" "6#-%&u nionG6$7$,$%)infinityG!\"\",$\"\"\"F)7$F+F(" }{TEXT -1 43 ". Also, if \+ you recall, the cosine \"trails\" " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\" \"\"\"\"#!\"\"" }{TEXT -1 37 " behind the sine, so sec will follow " } {XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 121 " behin d the csc in the same way - thus the graph of csc will look the same a s the graph of the sec - excepted shifted by " }{XPPEDIT 18 0 "Pi/2" " 6#*&%#PiG\"\"\"\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 72 "\nHere is th e sine, and then the cosine with its reciprocal, the secant.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "SolidPlotBound( x -> sin(x), 0, 2*Pi, 10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "display( [SolidPlotBound( x -> sin(x), 0, 2*Pi, 5), \n plot (csc(x), x = 0..2*Pi, y = -5..5, \n color = blu e, thickness = 2, discont = true )]);" }}}{EXCHG }{PARA 0 "" 0 "" {TEXT -1 248 "\nLike the tangent and secant functions, the cosecant i s undefined at certain values. Those two are undefined when cos(x) = 0 , while csc is undefined when sin(x) = 0. This means there are vertica l asymptotes but in different places - when x = kPi.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "display( plot( csc(x), x = -2*Pi.. 2*Pi, y = -20..20, \n discont = true, color = blue, thi ckness = 2),\n seq( plot( [[ k*Pi,-20],[k*Pi ,20]], \n \+ color = green, linestyle = 2), k = -2..1) \+ );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "display( plot( csc(x ), x = -6*Pi..6*Pi, y = -20..20, \n discont = true, col or = blue, thickness = 1),\n seq( plot( [[k*Pi ,-20],[k*Pi \+ ,20]], \n color = green, linestyle = 2), \+ k = -6..5) );" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 42 " 4. Secants & Cosecants - Together at Last" }} {PARA 0 "" 0 "" {TEXT -1 95 "\nTo help see the differences in the seca nt and cosecant, let's look at them on the same graph.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 388 "display( plot( sin(x), x=0..2*Pi, \+ color = blue),\n plot( cos(x), x=0..2*Pi, color = green ),\n \+ plot( csc(x), x=0..2*Pi, y = -10..10, color = blue, \n \+ thickness = 3, discont = true) ,\n plot( sec(x), x=0..2*Pi, y = -10..10, color = green, \n \+ thickness = 3, discont = true) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 259 33 "\n \251 20 02 Waterloo Maple Inc" }}}{MARK "0 1" 28 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }