{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 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 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 55 "High School Modul es > Trigonometry by Gregory A. Moore " }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 256 6 "Angles" }}{PARA 0 "" 0 "" {TEXT -1 53 "\nExplorati on of Degree and Radian measure of angles.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execute the Code Resource section first. Althou gh there will be no output immediately, these definitions are used lat er 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 39 "restart; with(plots): with(plottools): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 366 "DegreePlot := proc( t )\nlocal A, B, C, theta;\ntheta := evalf(t*Pi/180);\nA := pieslice([0,0], 1, 0..t heta, color = COLOR(RGB,1,.85, 0) ):\nB := pieslice([0,0], 1, theta ..2*Pi, color = COLOR(RGB,.8,.45,.05) ):\nC := textplot( [ evalf(1.2*c os(theta)), evalf(1.2*sin(theta)), \n convert(t, strin g)]):\ndisplay( [A,B,C], scaling=constrained ); \nend proc:\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "AnglePlot := proc( theta \+ )\nlocal A, B, C;\nA := pieslice([0,0], 1, 0..theta, color = COLOR( RGB,.8,.3,.5) ):\nB := pieslice([0,0], 1, theta..2*Pi, color = COLOR( RGB,.4,.3,.5) ):\nC := textplot( [ evalf(1.2*cos(theta)), evalf(1.2*s in(theta)), \n convert(theta, string)]):\ndisplay( [A, B,C], scaling=constrained); \nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 494 "AngleSpectrum := proc( theta )\nlocal A, B, C, a1, a2,col,k,n;\nn := floor(2*Pi/theta); \nfor k from 1 to n do\n a1 := theta*(k-1); a2 := theta*(k); \n if( k mod 2 = 0) then col := CO LOR(RGB,.8,.3,.5)\n else col := COLOR(RGB,.4,.3,.5) ; fi;\n A||k := plottools[pieslice]([0,0], 1, a1..a2, color = col ): \n C||k : = textplot( [ evalf(1.2*cos(a2)), evalf(1.2*sin(a2)), \n \+ convert(a2, string)]):\nod:\ndisplay( seq( \{A||k, C||k\}, k = 1..n ), scaling=constrained); \nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 " 1. Degrees " }}{PARA 0 "" 0 "" {TEXT -1 180 "\nA degree is 1/360 of a rotation. I f you were to stand in one point and rotate, your view would sweep out 360 degrees in a complete rotation. One degree, is therefore, very sm all.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "DegreePlot(1);" }} }{PARA 0 "" 0 "" {TEXT -1 46 "\n30 degrees is a more reasonably sized \+ angled." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "DegreePlot(30);" }}}{PARA 0 "" 0 "" {TEXT -1 38 "\n90 degrees is called a \"right angle \"." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "DegreePlot(90);" }}} {PARA 0 "" 0 "" {TEXT -1 41 "\n135 degrees is into the second quadrant ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "DegreePlot(135);" }}} {PARA 0 "" 0 "" {TEXT -1 65 "\n\n\nAbove 180 degrees, angles start to \+ get into the 3rd quadarant." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "DegreePlot(210);" }}}{PARA 0 "" 0 "" {TEXT -1 58 "\n\nAnd above 27 0 degrees, angles get into the 4th quadrant." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "DegreePlot(300);" }}}{PARA 0 "" 0 "" {TEXT -1 467 "\nWhy is 360 used? The ancient Babylonians computed there were 360 da ys in a year. The passage of seasons is a universally known cycle, so the Babylonians assigned degrees of a circle to days of the year. 36 0 is also a convenient number to be divided! Since we often deal with \+ fractions of a circle, its convenient to divide 360 by various denomin ators. Here is the set of 24 whole numbers that divide 360. In fact, t he smaller integer that does NOT divide 360 is 7." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "numtheory[divisors](360); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "for k from 1 to 23 do cat(numtheory[divi sors](360)[k],` * `,\n 360/numtheory[divisors](360)[ k], ` = 360`); od;" }}}{PARA 0 "" 0 "" {TEXT -1 56 "\n\nSo 1/2 of a ci rcle has a degree measure of 1/2 of 360." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "360/2; DegreePlot(%);" }}}{PARA 0 "" 0 "" {TEXT -1 55 "\nSo 1/4 of a circle has a degree measure of 1/4 of 360." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "360/4; DegreePlot(%);" }}} {PARA 0 "" 0 "" {TEXT -1 55 "\nSo 1/3 of a circle has a degree measure of 1/3 of 360." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "360/3; De greePlot(%);" }}}{PARA 0 "" 0 "" {TEXT -1 55 "\nSo 1/8 of a circle has a degree measure of 1/8 of 360." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "360/8; DegreePlot(%);" }}}{PARA 0 "" 0 "" {TEXT -1 57 "\nSo 1/ 12 of a circle has a degree measure of 1/12 of 360." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "360/12; DegreePlot(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 " 2. Minutes & S econds" }}{PARA 0 "" 0 "" {TEXT -1 248 "\nEach degree can be further b roken into 60 minutes, and each minute can be broken in 60 seconds. Fr actional degrees can be represented in this system. You can think of a giant clock with 360 hours, 60 minutes per hour, and 60 seconds per m inute.\n\n\n" }{TEXT 267 12 " " }{TEXT 268 10 "Convert : " }{TEXT 269 43 "Degrees/Minutes/Seconds to Decimal Degrees " }{TEXT -1 2 "\n\n" }{TEXT 275 25 " Example 2.1 : " }{TEXT -1 62 " Expr ess 34 degrees, 42 minutes, 29 seconds in decimal form\n. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "34 + 42/60 + 29/60^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n \n" }{TEXT 276 23 " Example 2.2 : " }{TEXT -1 62 " Express 119 degrees, 7 minutes, 57 seconds in decimal form\n. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "119 + 7/60 + 57/60^2;\nevalf(%, 15);" }}} {PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" } {TEXT 271 8 " " }{TEXT 272 11 "Convert : " }{TEXT 273 54 "Frac tional/Decimal Degrees to Degrees/Minutes/Seconds " }{TEXT -1 1 "\n" } {TEXT 274 23 "\n Example 2.3 : " }{TEXT -1 56 "Express 72.8921 \+ degrees in Degree/Minute/Second form\n. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "72.8921; \na := evalf(%);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "degrees := floor(a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "fractional_degree := a - degrees;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "minutes := floor( 60*fractional_degree );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "fractional_min := fractio nal_degree - minutes/60;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "seconds :=floor( 60^2 * fractional_min); " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 155 "`Compare & Check`;\nevalf(a); \nprint(cat(degrees, ` degrees, `, minutes, ` minutes, `, seconds, ` seconds`));\nevalf( degrees + minutes/60 + seconds/60^2);" }}}{PARA 0 "" 0 "" {TEXT -1 177 "\nNote that these two values are slightly different. This is beca use we rounded off the seconds. If we left the decimals to be a decima l number too, then these both would agree.\n" }{TEXT 270 5 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "seconds := 60^2 * fractional _min ; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "`Compare & Chec k`;\nevalf(a); \nprint(cat(degrees, ` degrees, `, minutes, ` minutes , `, seconds, ` seconds`));\nevalf( degrees + minutes/60 + seconds/60^ 2);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 277 23 " Example 2.4 : " }{TEXT -1 54 "Express 18 5/7 degrees in D egree/Minute/Second form\n. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "18 + 5/7; \na := evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "degrees := floor(a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "fractional_degree := a - degrees;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "minutes := floor( 60*fractional_degree );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "fractional_min := fractional_degree - minutes/60;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "seconds : =floor( 60^2 * fractional_min); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "`Compare & Check`;\nevalf(a); \nprint(cat(degrees, ` degrees, `, minutes, ` minutes, `, seconds, ` seconds`));\nevalf( d egrees + minutes/60 + seconds/60^2);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 12 " 3. Radians" }}{PARA 0 "" 0 "" {TEXT -1 162 "\nAnother w ay of measuring angles is to compute the arc length that the angle cor responds to on a unit circle. Since the circumference of an entire uni t circle is " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 113 ", that is the radian measure of a complete rotation. Therefore , fractions of a circle correspond to fractions of " }{XPPEDIT 18 0 "2 *Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 38 " radians.\n\n\n1/2 of a ci rcle is 1/2 of " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" } {TEXT -1 20 " radians, or simply " }{XPPEDIT 18 0 "Pi" "6#%#PiG" } {TEXT -1 9 " radians\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "(2 *Pi)/2; AnglePlot(%);" }}}{PARA 0 "" 0 "" {TEXT -1 28 "\n\n1/4 of a ci rcle is 1/4 of " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" } {TEXT -1 14 " radians, or " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\" \"#!\"\"" }{TEXT -1 9 " radians\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "(2*Pi)/4; AnglePlot(%);" }}}{PARA 0 "" 0 "" {TEXT -1 28 "\n\n1/8 of a circle is 1/8 of " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\" \"\"%#PiGF%" }{TEXT -1 20 " radians, or simply " }{XPPEDIT 18 0 "Pi/4 " "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 9 " radians\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "(2*Pi)/8; AnglePlot(%);" }}}{PARA 0 "" 0 "" {TEXT -1 30 "\n\n1/12 of a circle is 1/12 of " }{XPPEDIT 18 0 "2*Pi " "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 20 " radians, or simply " } {XPPEDIT 18 0 "Pi/6" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 9 " radians \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "(2*Pi)/12; AnglePlot(% );" }}}{PARA 0 "" 0 "" {TEXT -1 85 "\n\n\nSo what does 1 radian look l ike? (We'll find out how many degrees this is later.)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "AnglePlot(1);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 32 " 4. Radian & Degree Conversions" }}{PARA 0 "" 0 "" {TEXT -1 114 "\nThe key to conv ersion between the two types of angle measuring systems is that half o f circle is 180 degrees and " }{XPPEDIT 18 0 "Pi " "6#%#PiG" }{TEXT -1 12 " radians. \n\n" }{TEXT 260 12 " " }{TEXT 261 10 "Con vert : " }{TEXT 262 19 "Radians to Degrees " }{TEXT -1 14 ": Multiply \+ by " }{XPPEDIT 18 0 "180/Pi" "6#*&\"$!=\"\"\"%#PiG!\"\"" }{TEXT -1 1 " \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Pi/4; % * 180/Pi;" }}} {PARA 0 "" 0 "" {TEXT -1 25 "Note that 45 degrees and " }{XPPEDIT 18 0 "Pi/4" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 33 " radians represent \+ the same angle" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "DegreePlot (45);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "AnglePlot(Pi/4);" }}}{PARA 0 "" 0 "" {TEXT -1 24 "\nHere are other examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "2*Pi/3; % * 180/Pi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Pi/6; % * 180/Pi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "7*Pi/6; % * 180/Pi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "5*Pi/3; % * 180/Pi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "11*Pi/6; % * 180/Pi;" }}}{PARA 0 "" 0 "" {TEXT -1 69 "\nThere is also the built-in Maple function to make these conve rsions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "convert(Pi/2, uni ts, rad, degree);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "conver t(11*Pi/6, units, rad, degree);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "convert(4*Pi/3, units, rad, degree);" }}}{PARA 0 "" 0 "" {TEXT -1 4 "\n \n\n" }{TEXT 263 12 " " }{TEXT 264 10 "Conve rt : " }{TEXT 265 19 " Degrees to Radians" }{TEXT -1 15 " : Multiply b y " }{XPPEDIT 18 0 "Pi/180" "6#*&%#PiG\"\"\"\"$!=!\"\"" }{TEXT -1 1 " \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "60; % * Pi/180;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "135; % * Pi/180;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "240; % * Pi/180;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "315; % * Pi/180;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "330; % * Pi/180;" }}}{PARA 0 "" 0 "" {TEXT -1 134 "\nThere is also the built-in Maple function to make these convers ions - note to \"multiply\" the number of degrees by the word \"degree s.\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "with(Units[Standard] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "convert(45*Unit(degre e),units,radians);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "conve rt(120*Unit(degree), units, radians);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "convert(115*Unit(degree), units, radians);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "convert(252*Unit(degree), un its, radians);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 " 5. Famous A ngles" }}{PARA 0 "" 0 "" {TEXT -1 156 "\nThere are various angles whic h are used frequently in geometry and trignometry - in particular the \+ multiples of 30, 45, 60, and 90 degrees - or in radians " }{XPPEDIT 18 0 "Pi/6, Pi/4, Pi/3, Pi/2" "6&*&%#PiG\"\"\"\"\"'!\"\"*&F$F%\"\"%F'* &F$F%\"\"$F'*&F$F%\"\"#F'" }{TEXT -1 102 ". It's a good idea to become familiar with all multiples of these angles in both degrees and radia ns.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "AngleSpectrum(Pi/2) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "AngleSpectrum(Pi/4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "AngleSpectrum(Pi/8);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "AngleSpectrum(Pi/6);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "AngleSpectrum(Pi/12);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "AngleSpectrum(Pi/3);" }}} {PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "6. Angles & Arclength" }}{PARA 0 "" 0 "" {TEXT -1 381 "\nThere i s another advantage of using radians when it comes to finding the leng th of an arc. Since a radian measure is already an arclength on a unit circle, any larger or smaller circle can be scaled appropriately - in a since all circles are \"similar\" circles to the unit circle. So th e arclength of an arc with central angle theta is \n Arclength \+ = radius * theta\n\n " }{TEXT 266 7 "Example" }{TEXT -1 68 " : \+ Find the arclength on a circle with radius 100 and central angle " } {XPPEDIT 18 0 "Pi/7" "6#*&%#PiG\"\"\"\"\"(!\"\"" }{TEXT -1 1 "\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ang := Pi/7; \nradius := 10 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "arclength := radius*a ng;\nevalf(%);" }}}}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 259 34 " \+ \251 2002 Waterloo Maple Inc" }}}{MARK "0 1" 46 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }