Section 7-6 Complex Numbers in Rectangular and Polar Forms
Contents
1. C Convert the results from part B back to rectangular form and com pare with the results in part A Multiplication and Division in Polar Form There is a particular advantage in representing complex numbers in polar form multiplication and division become very easy Theorem 1 provides the reason The exponential polar form of a complex number obeys the product and quotient rules for exponents bb b and b b b PRODUCTS AND QUOTIENTS IN POLAR FORM THEOREM If z re and z ne then 1 E BEE E FEE ees One en E Pc Ay 1 ieee Serr sears ea Saree eae AA i0 Z2 FEEF F3 We establish the multiplication property and leave the quotient property for Problem 32 in Exercise 7 6 ZZ nen e Write in trigonometric form r r cos 0 i sin 8 cos 8 i sin 0 Multiply r r cos 8 cos 8 i cos 8 sin 0 i sin 8 cos 0 sin 8 sin 0 r r cos 8 cos 8 sin 0 sin 8 Use sum identities i cos sin sin 6 cos 8 r r cos 0 0 i sin 0 8 Write in exponential form pre t EXAMPLE Products and Quotients 4 If z 8e and z 2 find A 222 B z Z Solutions A 227 8e 222 B 26530 160757 7 6 Complex Numbers in Rectangular and Polar Forms 559 MATCHED PROBLEM Ifz 9e and z 3e find 4 A zz B za Historical Note There is hardly an area in mathematics that does not have some imprint of the fam2. magni tude of 20 pounds in a direction of 0 Force v has a magnitude of 10 pounds in a direction of 60 A Convert the polar forms of these complex numbers to rectangular form and add B Convert the sum from part A back to polar form C The vector going from the pole to the complex number in part B is the resultant of the two original forces What is its magnitude and direction 34 Forces and Complex Numbers Repeat Problem 33 with forces u and v associated with the complex numbers 8e and 6e respectively
3. parabola A e 0 6 B e 1 C e 2 Astronomy A The planet Mercury travels around the sun in an elliptical orbit given approximately by Section 7 6 Complex Numbers in Rectangular and Polar Forms Rectangular Form Polar Form Multiplication and Division in Polar Form Historical Note Utilizing polar concepts studied in the last two sections we now show how com plex numbers can be written in polar form which can be very useful in many applications A brief review of Section 2 4 on complex numbers should prove helpful before proceeding further Rectangular Form Recall from Section 2 4 that a complex number is any number that can be writ ten in the form a bi 554 7 ADDITIONAL TOPICS IN TRIGONOMETRY FIGURE where a and b are real numbers and 7 is the imaginary unit Thus associated with Complex plane each complex number a bi is a unique ordered pair of real numbers a b and vice versa For example imaginarse a13 fa b 3 Si corresponds to 3 3 Associating these ordered pairs of real numbers with points in a rectangular coordinate system we obtain a complex plane see Fig 1 When complex num bers are associated with points in a rectangular coordinate system we refer to the x axis as the real axis and the y axis as the imaginary axis The complex num ber a bi is said to be in rectangular form EXAMPLE Plotting in the Complex Plane 1 Plot the following complex numbers in a
4. 65 66 67 68 x 69 7 6 Complex Numbers in Rectangular and Polar Forms 553 Sailboat Racing Referring to the figure estimate to the 3 442 x 10 nearest knot the speed of the sailboat sailing at the follow ro 1 0 206 cos 0 He gne Osto ae pay pao a where r is measured in miles and the sun is at the Sailboat Racing Referring to the figure estimate to the pole Graph the orbit Use TRACE to find the distance nearest knot the speed of the sailboat sailing at the follow from Mercury to the sun at aphelion greatest ing angles to the wind 45 90 120 and 150 distance from the sun and at perihelion shortest distance from the sun Conic Sections Using a graphing utility graph the equation B Johannes Kepler 1571 1630 showed that a line joining a planet to the sun sweeps out equal areas in P 8 space in equal intervals in time see the figure Use I e cos 0 this information to determine whether a planet travels for the following values of e called the eccentricity of the faster or slower at aphelion than at perihelion Explain conic and identify each curve as a hyperbola an ellipse your answer or a parabola A e 0 4 B e 1 C e 1 6 It is instructive to explore the graph for other positive val ues of e Conic Sections Using a graphing utility graph the equation 8 r 1 e cos 0 for the following values of e and identify each curve as a hyperbola an ellipse or a
5. angular form Compute the exact values for parts A and B for part C compute a and b for a bi to two decimal places A Zz PF B o Je ONE C T19e 2 Solutions A x iy 2e07 2 cos 57 6 i sin 57 6 4 A a5 2 2 V3 i B x iy 3e 3 cos 60 i sin 60 Ea 3 3v3 p 2 FIGURE 8 7 19e 7 3 81 6 09 i C x iy 7 19e0 7 19 cos 2 13 i sin 2 13 3 81 6 09 i Ta i 2 DS FR a 81 6 851 er Figure 8 shows the same conversion done by a graphing calculator with a built in conversion routine Explore Oiscuse 2 If your calculator has a built in polar to rectangular conversion routine try it on V2e and V2e then reverse the process to see if you get back where you started For complex numbers in exponential polar form some calculators require 0 to be in radian mode for calculations Check your user s manual MATCHED PROBLEM Write parts A C in rectangular form Compute the exact values for parts A and 4 B for part C compute a and b for a bi to two decimal places A z Ve Di B z 30 120 O z 6 49 e208 558 7 ADDITIONAL TOPICS IN TRIGONOMETRY Explore Digcuss Let z V3 iand z 1 iV3 A Find z z and z z using the rectangular forms of z and z 3 B Find z z and z z using the exponential polar forms of z and z 0 in degrees Assume the product and quotient exponent laws hold for e
6. complex plane A 2 3i B 3 Si C 4 D 3i Solution MATCHED PROBLEM Plot the following complex numbers in a complex plane 1 A 4 2i B 2 3i C 5 D 4i Explore Discuse On a real number line there is a one to one correspondence between the set of real numbers and the set of points on the line each real number is associated with exactly one point on the line and each point on the line 1 is associated with exactly one real number Does such a correspondence exist between the set of complex numbers and the set of points in an extended plane Explain how a one to one correspondence can be established Polar Form Complex numbers also can be written in polar form Using the polar rectangu lar relationships from Section 7 5 x rcos 8 and y rsin 0 FIGURE 2 Rectangular polar relationship F FIGURE 3 A D 141e citi Polar 1 412 C 794 DEFINITION 1 DEFINITION 2 7 6 Complex Numbers in Rectangular and Polar Forms 555 we can write the complex number z x iy in polar form as follows z x iy r cos 0 ir sin 8 r cos 0 i sin 8 1 This rectangular polar relationship is illustrated in Figure 2 In a more advanced treatment of the subject the following famous equation is established i0 e cos 0 7 sin 8 2 where e obeys all the basic laws of exponents Thus equation 1 takes on the form z x yi r cos 0 i sin 0 re 3 We will freel
7. d B exactly compute the modulus and argu ment for part C to two decimal places A z 1 i B z V3 i C z 5 2i Locate in a complex plane first then if x and y are associated with special angles r and can often be determined by inspection A A sketch shows that z is associated with a special 45 triangle Fig 4 Thus by inspection r V2 0 n 4 not 77 4 and z V2 cos 7 4 isin 7 4 4 2 oo mi B A sketch shows that z is associated with a special 30 60 triangle Fig 5 Thus by inspection r 2 0 57 6 and Z gt 2 cos 57 6 i sin 57 6 J oman C A sketch shows that z is not associated with a special triangle Fig 6 So we proceed as follows r V 5 2 5 39 To two decimal places 0 v tan 2 76 To two decimal places Thus 5 39 cos 2 76 i sin 2 76 5 39e 77 To two decimal places 3 Figure 7 shows the same conversion done by a graphing calculator with a built in conversion routine with numbers displayed to two decimal places I A Polar Je Re PEL Write parts A C in polar form 0 in radians m lt 0 lt m Compute the modu lus and arguments for parts A and B exactly compute the modulus and argument for part C to two decimal places A 1 i B1 iV3 3 7i 7 6 Complex Numbers in Rectangular and Polar Forms 557 EXAMPLE From Polar to Rectangular Form 3 Write parts A C in rect
8. ous Swiss mathematician Leonhard Euler 1707 1783 who spent most of his productive life at the New St Petersburg Academy in Russia and the Prus sian Academy in Berlin One of the most prolific writers in the history of the sub ject he is credited with making the following familiar notations standard f x function notation e natural logarithmic base i imaginary unit V 1 For our immediate interest he is also responsible for the extraordinary relationship e cos 0 isin 0 If we let 0 m we obtain an equation that relates five of the most important numbers in the history of mathematics eb 1 0 Answers to Matched Problems 1 2 A V2 cos 37 4 isin 3 4 V2e B 2 cos 7 3 i sin w 3 2e C 5 83 cos 2 11 i sin 2 11 5 83 3 A iV2 B 24 NS C 3 16 5 67 4 A 1 27e B z z 3e0 560 7 ADDITIONAL TOPICS IN TRIGONOMETRY EXERCISE 7 6 A In Problems 1 8 plot each set of complex numbers in a complex plane 1A 34 4 B 2 i1 C 2i 2 A 4 iB 3 2i C 3i A 3 3i B 4 C 2 3i A 3 B 2 1 C 4 4 4i sAn BSN 20 CS ae A rer B Ag C VV2eC TAi A Age B Jen C Z 5a onan Bb Q A 2 B ge C Ae Sh In Problems 9 12 change parts A C to polar form For Problems 9 and 10 choose 9 in degrees 180 lt 0 S 180 for Problems 11 and 12 choose 9 in radians m lt 9 m Compute the modulus and arguments for pa
9. rts A and B exactly compute the modulus and argument for part C to two decimal places 9 A V3 i B 1l i C 5 6i 10 A 1 iV3 3i O 7 4 11 A iV3 B VvV3 i 8 5i 12 A V3 i B2 2 65i In Problems 13 16 change parts A C to rectangular form Compute the exact values for parts A and B for part C compute a and b for a bi to two decimal places 13 A 2e B Ve C 3 0864 14 A 26 BV e CC 5 710 04 15 A 6e B VIe CC 4 090 8 16 A V3e B V2e C 6 83e 18 In Problems 17 22 find z Z and Z Z gt VW am ee 19 z 3 2 222 18 z 6e z 3e 20 3 36 2 22 21 z 3 05e z 11 94e 22 2 TAle 2 662 Simplify Problems 23 26 directly and by using polar forms Write answers in both rectangular and polar forms 0 is in degrees 23 1 i 24 1 i 25 1 0 d 26 1 iV3 V3 i 27 1 i 28 1 i Hooo 29 Show that r e is a cube root of re 30 Show that r e is a square root of re 31 If z re show that z re and 2 ree What do you think z will be for n a natural number 32 Prove 10 ot o Pi Ai 0 0 a Te ff APPLICATIONS de 33 Forces and Complex Numbers An object is located at the pole and two forces u and v act on the object Let the forces be vectors going from the pole to the complex num bers 20e and 10e respectively Force u has a
10. y use re as a polar form for a complex number In fact some graph ing calculators display the polar form of x iy this way see Fig 3 where is in radians and numbers are displayed to two decimal places Since cos and sin 0 are both periodic with period 2m we have cos 0 2km cos 8 k any integer sin 2kr sin 8 Thus we can write a more general polar form for a complex number z x iy as given below and observe that re is periodic with period 2k k any integer GENERAL POLAR FORM OF A COMPLEX NUMBER For k any integer z x iy r cos 0 2kr i sin 0 2kr z pe Vram The number r is called the modulus or absolute value of z and is denoted by mod z or z The polar angle that the line joining z to the origin makes with the polar axis is called the argument of z and is denoted by arg z From Figure 2 we see the following relationships MODULUS AND ARGUMENT FOR z x y modz r Vx y Never negative arg z 0 2km k any integer where sin 0 y r and cos 0 x r The argument is usually chosen so that 160 lt 0 180 or rmn lt 05 556 7 ADDITIONAL TOPICS IN TRIGONOMETRY EXAMPLE Solutions FIGURE 4 i i FIGURE 5 FIGURE 6 FIGURE 5 20 5 39e amp 7 MATCHED PROBLEM 2 From Rectangular to Polar Form Write parts A C in polar form 0 in radians m lt 0 lt m Compute the mod ulus and arguments for parts A an
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