{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 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 268 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 128 0 128 1 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 }{PSTYLE "Normal " -1 256 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 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 53 "High School Modul es > Precalculus by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " \+ " }{TEXT 263 47 "The Polar & Exponential Form of Complex Numbers" }} {PARA 0 "" 0 "" {TEXT -1 152 "\nThis is a further development of compl ex numbers. Also see worksheets on Complex Numbers and Complex Number \+ Operations in the Algebra I & II PowerTool." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 153 "[Directions : Execute the Cod e 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 1 " " }{TEXT 258 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; w ith(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 917 "ComplexPlo t := proc()\n local k,A,Pt,c,cg,shade,r,u,Lines; \n u:= 0;\n c g := COLOR(RGB, .7,.8,.7);\n for k from 1 to nargs do\n shade \+ := evalf(rand()/10^12,2)/4 + .3;\n c := COLOR(RGB, shade,shade+. 2, shade); \n A||k := complexplot( [0,args[k]], linestyle = 2, \+ scaling = constrained,\n color = c);\n \+ Lines||k := plot( [[ 0, Im(args[k])], [Re(args[k]), Im(args[k])],\n \+ [Re(args[k]), 0]], color = cg); \n\n u \+ := max( u, abs(Re(args[k])),abs(Im(args[k])) ); \n od;\n\n r := ev alf(u/40,2);\n for k from 1 to nargs do \n shade := evalf(rand ()/10^12,2)/4 + .1;\n c := COLOR(RGB, shade,.8, shade); \n \+ Pt||k := plottools[disk]( [Re(args[k]), Im(args[k])],r, \n \+ color = c ); od;\n\n display( [seq( A||k, k = 1..narg s), seq( Pt||k, k = 1..nargs),\n seq( Lines||k, k = 1..nar gs)] ); \nend proc:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "reset := proc()\nglobal a,b,z,R,theta;\na := 'a': b := 'b': z := 'z' : theta := 'theta': R := 'R':\nreturn (a,b,z,R,theta):\nend proc:\n" } }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 259 36 "1. The Polar \+ Form of Complex Numbers" }}{PARA 0 "" 0 "" {TEXT -1 5 "\n " }{TEXT 265 21 "Concept of Polar Form" }{TEXT -1 84 "\n\nHere is a complex num ber in standard \"real part + imaginary part\" component form.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "z := 4 + 4*sqrt(3)*I; " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ComplexPlot(z);" }}}{PARA 0 "" 0 "" {TEXT -1 338 "\n\nWhen we look at the geometric representati on, we might notice that this point forms a right triangle by looking \+ at the real part of the number on the x-axis as one leg, and the diago nal line from the origin to the point as the hypotenuse. The angle tha t the hypotenuse makes with the x axis is called the argument of the c omplex number." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "argument(z );" }}}{PARA 0 "" 0 "" {TEXT -1 91 "\nAnd the distance from the origin to the point is the absolute value of the complex number." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "abs(z);" }}}{PARA 0 "" 0 "" {TEXT -1 165 "\nThese two numbers can identify the point completely. For exa mple, this same number can be expressed in this form - which is called the polar or trigonometric form.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "8*cos(Pi/3) + 8*sin(Pi/3)*I;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 267 28 "2. Converting To Polar Form " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" } {TEXT 268 45 " Converting from Component Form to Trig Form" }{TEXT -1 123 "\n\nAs above, the process involves finding two things : \n \+ - the argument and \n - the modulus. \n\n\n" }{TEXT 269 38 " Finding the Argument" }{TEXT -1 357 " \n\nWe can compute the argument directly using Maple, but to do it by \+ hand, we would use the two legs of the triangle to find the slope. Tho se two legs are none other than the real and imaginary parts of the co mplex number. Once we know the slope, we note that it is equal to the \+ tangent of the angle. Therefore the inverse tangent of the slope is th e angle.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "z := 11 + 11*I ;\narctan( Im(z) / Re(z) );\nargument(z);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 60 "z := 4*sqrt(3) - 4*I;\narctan( Im(z) / Re(z) );\na rgument(z);" }}}{PARA 0 "" 0 "" {TEXT -1 34 "\nSomething seems to go w rong here." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "z := -5 + 5*sq rt(3)*I;\narctan( Im(z) / Re(z) );\nargument(z);" }}}{PARA 0 "" 0 "" {TEXT -1 50 "\n What happened? Well, the range of the arctan is " } {XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F (F)" }{TEXT -1 96 " which is only quadrants I and IV. However, this po int is actually in quadrant II. So if we add " }{XPPEDIT 18 0 "Pi" "6# %#PiG" }{TEXT -1 46 " to our arctan, we get the actual argument of " } {XPPEDIT 18 0 "2*Pi/3" "6#*(\"\"#\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT -1 27 ". Here is another example.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "z := - 10*sqrt(3) + 10*I;arctan( Im(z) / Re(z) );\nargument(z );" }}}{PARA 0 "" 0 "" {TEXT 270 45 " \n\n Finding the \+ Absolute Value" }}{PARA 0 "" 0 "" {TEXT -1 166 "\nThe absolute value o f a complex number is essentially Pythagoras' theorem. We know the leg s of the triangle to be the real and imaginary parts of the complex nu mber." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "z := 9 + 40*I;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "a := Re(z);\nb := Im(z);\nc^ 2 = a^2 + b^2;\nc = sqrt(a^2 + b^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sqrt( Re(z)^2 + Im(z)^2);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "abs(z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "z := -15 + 8*I;\nabs(z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "z := -2 + 1*I;\nabs(z);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" } {TEXT 271 56 " Expressing a Complex Number in Polar Form " }}{PARA 0 "" 0 "" {TEXT -1 95 "\nOnce you know the argument and abso lute value, then just write it in the form :\n " } {XPPEDIT 18 0 "z = R*(cos(theta) + I*sin(theta))" "6#/%\"zG*&%\"RG\"\" \",&-%$cosG6#%&thetaGF'*&%\"IGF'-%$sinG6#F,F'F'F'" }{TEXT -1 20 "\nwhe re R = |z|, and " }{XPPEDIT 18 0 "theta = arg(z)" "6#/%&thetaG-%$argG6 #%\"zG" }{TEXT -1 8 ".\n \n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "z := -12 + 12*I;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "theta := argument(z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "R := abs(z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "z = R * ('co s'(theta) + I*'sin'(theta));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "z := 9 -40*I;\ntheta := argument(z);\nR := abs(z);\nz = R * ('co s'(theta) + I*'sin'(theta));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "z := sqrt(39) + sqrt(13)*I;\ntheta := argument(z); \ntheta := s implify(%);\nR := abs(z);\nz = R * ('cos'(theta) + I*'sin'(theta));" } }}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 260 30 "3. Converting from Polar Form " }}{PARA 0 "" 0 "" {TEXT 266 53 "\n Converting from Trig Form to Component Fo rm " }}{PARA 0 "" 0 "" {TEXT -1 65 "\n\nIt's even easier to convert fr om trig form to component form. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "reset():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "theta := 4*Pi/3;\nR := 60;\nz = R * ('cos'(theta) + I*'sin'(theta)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a := R*cos(theta);\nb : = R*sin(theta);\nz = a + b*I;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "theta := Pi/8;\nR := 256;\nz = R * ('cos'(theta) + I*'sin'(thet a));\na := R*cos(theta);\nb := R*sin(theta);\nz = a + b*I;\nevalf(%); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "theta := 1;\nR := 10; \nz = R * ('cos'(theta) + I*'sin'(theta));\na := R*cos(theta);\nb := R *sin(theta);\nz = a + b*I;\nevalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 2 " \n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 262 42 "4. The E xponential Form of Complex Numbers" }}{PARA 0 "" 0 "" {TEXT -1 5 "\n \+ " }{TEXT 272 23 "e to an imaginary Power" }{TEXT -1 216 "\n\nEuler d iscovered that e, when raised to an imaginary power, can be expressed \+ as a complex number in polar form. Obviously a revolutionary discovery made using Taylor series, which you will learn about in calculus.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "reset();" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "exp(theta*I): \n% = convert(%, trig);" } }}{PARA 0 "" 0 "" {TEXT -1 18 "\nHere are examples" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 45 "''exp( (7*Pi/6)*I)'': \n% = convert(%, trig );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "''exp( (2.75)*I)'': \+ \n% = convert(%, trig);" }}}{PARA 0 "" 0 "" {TEXT -1 23 "\nThe absolu te value of " }{XPPEDIT 18 0 "e^(i*theta)" "6#)%\"eG*&%\"iG\"\"\"%&the taGF'" }{TEXT -1 18 " for any value of " }{XPPEDIT 18 0 "theta" "6#%&t hetaG" }{TEXT -1 52 " is 1, and all such numbers lie on the unit circl e. " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 24 " can be any re al number." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "exp( (7*Pi/6) *I): z1 := convert(%, trig);\nexp( (2.75)*I): z2 := convert(%, tri g);\nexp( 1*I): z3 := convert(%, trig);\nexp( sqrt(2)*I): z4 := convert(%, trig);\nexp( (-4/7)*I): z5 := convert(%, trig);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "abs(z1); abs(z2); \nabs (z3): % = simplify(%); \nabs(z4): % = simplify(%);\nabs(z5): % = simpl ify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "display( polarpl ot( 1, color = gray), \n ComplexPlot( z1,z2,z3,z4,z5));" }}} {PARA 0 "" 0 "" {TEXT -1 3 "\n\n\n" }{TEXT 273 27 " e to a Compl ex Power" }{TEXT -1 122 "\n\nAlthough this might seem to be much more \+ complicated, it really isn't because of one of the simplest rules of e xponents." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "exp(a+b) = exp( a) * exp(b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "exp(a+I*b) \+ = exp(a) * exp(I*b);" }}}{PARA 0 "" 0 "" {TEXT -1 41 "\nSo it becomes \+ a product of two numbers. " }{XPPEDIT 18 0 "e^a" "6#)%\"eG%\"aG" } {TEXT -1 23 " is a real number, and " }{XPPEDIT 18 0 "e^(I*b)" "6#)%\" eG*&%\"IG\"\"\"%\"bGF'" }{TEXT -1 61 " is a polar form of a complex nu mber that we saw just above.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "exp( 3 + 2*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "exp ( 3 + 2*I ) = exp(3) * exp(2*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "exp( 3 + 2*I ) = exp(3) * convert( exp(2*I), trig) ; " }}}{PARA 0 "" 0 "" {TEXT -1 25 "\nHere is another example." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "exp( 5 + Pi*I ) = exp(5) * c onvert( exp(Pi*I), trig) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "exp( ln(8) + (3*Pi/4)*I ) = \n exp(ln(8)) * convert( exp(( 3*Pi/4)*I), trig) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "exp( 1 + 2*Pi*I ) = exp(1) * convert( exp(2*Pi*I), trig) ;" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " } {TEXT 261 43 "5. Developing Trig Identities at Warp Speed" }}{PARA 0 " " 0 "" {TEXT -1 134 "\nAs you are probably well aware, trigonometry ha s no shortage of formulas and identities. Here are some hopefully fami liar reminders.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "sin(a + b):\n% = expand(%, trig);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "cos(a - b):\n% = expand(%, trig);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "tan(2*a):\n% = expand(%, trig);" }}}{PARA 0 "" 0 "" {TEXT -1 384 "\nThese formulas are less easy to prove than they are to remember - using the geometry. However, they can be developed very qu ickly using the exponential form of complex numbers. What we do is exp ress an exponential expression in two different ways, using the standa rd rules of exponents. Then we expand each side into polar forms, and \+ equate the real and imaginary parts of each side." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "exp( (a+b)*I): \nL := %:\n% = expand(%);\ncon vert( %, trig):\nconvert( L, trig) = expand( rhs(%) );\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "`Equate the real parts of the left \+ and right sides`;\ncos(a+b) = cos(a)*cos(b)-sin(a)*sin(b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "`Equate the imaginary parts of the left and right sides`;\nI*sin(a+b) = I*cos(a)*sin(b)+I*sin(a)*cos(b); \nsin(a+b) = cos(a)*sin(b)+ sin(a)*cos(b);" }}}{PARA 0 "" 0 "" {TEXT -1 199 "\n\nVoila! You have just developed two trig identities for the price of one. You have a formula for the sine of a sum of angles, and a formula for the cosine of a sum. What formulas come out of these?\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "exp( 2*a*I): \nL := %:\n % = expand(%);\nconvert( %, trig):\nconvert( L, trig) = expand( rhs(%) );" }}}{EXCHG }{EXCHG }{PARA 0 "" 0 "" {TEXT -1 76 "\nYou can even am aze your friends with your own identities \"not in the book.\"" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "exp( 4*a*I): \nL := %:\n% = \+ expand(%);\nconvert( %, trig):\nconvert( L, trig) = expand( rhs(%) ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "exp( (2*a + b)*I): \nL \+ := %:\n% = expand(%);\nconvert( %, trig):\nconvert( L, trig) = expand( rhs(%) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "exp( (3*b)*I) : \nL := %:\n% = expand(%);\nconvert( %, trig):\nconvert( L, trig) = e xpand( rhs(%) );" }}}{EXCHG }{EXCHG }}{PARA 0 "" 0 "" {TEXT 257 35 "\n \251 2002 Waterloo Maple Inc" }}}{MARK "0 1" 47 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }