SECTION 7-7 De Moivre s Theorem. De Moivre s Theorem, n a Natural Number nth-roots of z

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1 7-7 De Moivre s Theorem 563 B eactl; compute the modulus and argument for part C to two decimal places. 9. (A) 3 i (B) 1 i (C) 5 6i 10. (A) 1 i 3 (B) 3i (C) 7 4i 11. (A) i 3 (B) 3 i (C) 8 5i 12. (A) 3 i (B) 2 2i (C) 6 5i In Problems 13 16, change parts A C to rectangular form. Compute the eact values for parts A and B; for part C compute a and b for a bi to two decimal places. 13. (A) 2e ( /3)i (B) 2e 45 i (C) 3.08e 2.44i 14. (A) 2e 30 i (B) 2e ( 3 /4)i (C) 5.71e 0.48i 15. (A) 6e ( /6)i (B) 7e 90 i (C) 4.09e i 16. (A) 3e ( /2)i (B) 2e 135 i (C) 6.83e i In Problems 17 22, find z 1 z 2 and z 1 /z z 1 7e 82 i, z 2 2e 31 i 18. z 1 6e 132 i, z 2 3e 93 i 19. z 1 5e 52 i, z 2 2e 83 i 20. z 1 3e 67 i, z 2 2e 97 i 21. z e 1.76i, z e 2.59i 22. z e 0.79i, z e 1.07i Simplif Problems directl and b using polar forms. Write answers in both rectangular and polar forms, in degrees. 23. (1 i)(2 2i) 24. ( 3 i) ( 1 i) ( 3 i 3)(1 i 3) i 1 i i 3 i 29. The conjugate of a bi is a bi. What is the conjugate of re i? Eplain. 30. How is the product of a comple number z with its conjugate related to the modulus of z? Eplain. C 31. Show that r 1/3 e /3 is a cube root of re i. 32. Show that r 1/2 e /2 is a square root of re i. 33. If z re i, show that z 2 r 2 e 2 i and z 3 r 3 e 3 i. What do ou think z n will be for n a natural number? 34. Prove: APPLICATIONS z 1 z 2 r 1e i 1 r 2 e i 2 r 1 r 2 e i( 1 2) 35. Forces and Comple 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 comple numbers 20e 0 i and 10e 60 i, respectivel. Force u has a magnitude 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 comple 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 comple number in part B is the resultant of the two original forces. What is its magnitude and direction? 36. Forces and Comple Numbers. Repeat Problem 35 with forces u and v associated with the comple numbers 8e 0 i and 6e 30 i, respectivel. SECTION 7-7 De Moivre s Theorem De Moivre s Theorem, n a Natural Number nth-roots of z Abraham De Moivre ( ), of French birth, spent most of his life in London doing private tutoring, writing, and publishing mathematics. He belonged to man prestigious professional societies in England, German, and France, and he was a close friend of Isaac Newton. Using the polar form for a comple number, De Moivre established a theorem that still bears his name for raising comple numbers to natural number powers. More importantl, the theorem is the basis for the nth-root theorem, which enables us to find all n nth roots of an comple number, real or imaginar.

2 564 7 Additional Topics in Trigonometr De Moivre s Theorem, n a Natural Number We start with Eplore-Discuss 1 and generalize from this eploration. EXPLORE-DISCUSS 1 B repeated use of the product formula for the eponential polar form re i, discussed in the last section, establish the following: 1. ( i) 2 (re i ) 2 r 2 e 2 i 2. ( i) 3 (re i ) 3 r 3 e 3 i 3. ( i) 4 (re i ) 4 r 4 e 4 i Based on forms 1 3, and for n a natural number, what do ou think the polar form of ( i) n would be? If ou guessed r n e n i, ou have guessed De Moivre s theorem, which we now state without proof. A full proof of the theorem for all natural numbers n requires a method of proof, called mathematical induction, which is discussed in the second section in Sequence and Series. Theorem 1 De Moivre s Theorem If z i re i, and n is a natural number, then z n ( i) n (re i ) n r n e n i EXAMPLE 1 The Natural Number Power of a Comple Number Use De Moivre s theorem to find (1 i) 10. Write the answer in eact rectangular form. Solution (1 i) 10 ( 2e 45 i ) 10 ( 2) 10 e (10 45 )i 32e 450 i 32(cos 450 i sin 450 ) 32(0 i) 32i Convert 1 i to polar form. Use De Moivre s theorem. Change to rectangular form. Rectangular form Matched Problem 1 Use De Moivre s theorem to find (1 i 3) 5. Write the answer in eact polar and rectangular forms.

3 7-7 De Moivre s Theorem 565 EXAMPLE 2 The Natural Number Power of a Comple Number Use De Moivre s theorem to find ( 3 i) 6. Write the answer in eact rectangular form. Solution ( 3 i) 6 (2e 150 i ) e (6 150 )i 64e 900 i 64(cos 900 i sin 900 ) Convert 3 i to polar form. De Moivre s theorem Change to rectangular form. 64( 1 i0) 64 Rectangular form [Note: 3 i must be a sith root of 64, since ( 3 i) 6 64.] Matched Problem 2 Use De Moivre s theorem to find (1 i 3) 4. Write the answer in eact polar and rectangular forms. nth Roots of z We now consider roots of comple numbers. We sa w is an nth root of z, n a natural number, if w n z. For eample, if w 2 z, then w is a square root of z. If w 3 z, then w is a cube root of z. And so on. EXPLORE-DISCUSS 2 If z re i, then use De Moivre s theorem to show that r 1/2 e ( /2)i is a square root of z and r 1/3 e ( /3)i is a cube root of z. We can proceed in the same wa as in Eplore-Discuss 2 to show that r 1/n e ( /n)i is an nth root of re i, n a natural number: r 1/n e ( /n)i n (r 1/n ) n e n( /n)i re i But we can do even better than this. The nth-root theorem (Theorem 2) shows us how to find all the nth roots of a comple number. Theorem 2 nth-root Theorem For n a positive integer greater than 1, r 1/n e ( /n k360 /n)i k 0, 1,..., n 1 are the n distinct nth roots of re i, and there are no others.

4 566 7 Additional Topics in Trigonometr The proof of Theorem 2 is left to Problems 39 and 40 in Eercise 7-7. EXAMPLE 3 Finding all Sith Roots of a Comple Number Find si distinct sith roots of 1 i 3, and plot them in a comple plane. Solution First write 1 i 3 in polar form: 1 i 3 2e 120 i Using the nth-root theorem, all si roots are given b 2 1/6 e (120 /6 k360 /6)i 2 1/6 e (20 k60 )i k 0, 1, 2, 3, 4, 5 Thus, w 1 2 1/6 e ( )i 2 1/6 e 20 i w 3 w 2 radius 2 1/6 w 2 2 1/6 e ( )i 2 1/6 e 80 i w 3 2 1/6 e ( )i 2 1/6 e 140 i w 4 2 1/6 e ( )i 2 1/6 e 200 i w 4 w 1 w 5 2 1/6 e ( )i 2 1/6 e 260 i w 6 2 1/6 e ( )i 2 1/6 e 320 i FIGURE 1 w 5 w 6 All roots are easil graphed in the comple plane after the first root is located. The root points are equall spaced around a circle of radius 2 1/6 at an angular increment of 60 from one root to the net (Fig. 1). Matched Problem 3 Find five distinct fifth roots of 1 i. Leave the answers in polar form, and plot them in a comple plane. EXAMPLE 4 Solving a Cubic Equation Solve Write final answers in rectangular form, and plot them in a comple plane. Solution We see that is a cube root of 1, and there are a total of three roots. To find the three roots, we first write 1 in polar form: 1 1e 180 i

5 w 4 w De Moivre s Theorem 567 Using the nth-root theorem, all three cube roots of 1 are given b 1 1/3 e (180 /3 k360 /3)i 1e (60 k120 )i k 0, 1, 2 Thus, 1 w 1 w 1 1e 60 i cos 60 i sin i 3 2 w w 2 1e 180 i cos 180 i sin w 3 1e 300 i cos 300 i sin i 3 2 FIGURE 2 1 w 3 [Note: This problem also can be solved using factoring and the quadratic formula tr it.] The three roots are graphed in Figure 2. Matched Problem 4 Solve Write final answers in rectangular form, and plot them in a comple plane. Answers to Matched Problems 1. 32e 300 i 16 i e 240 i 8 i w 1 2 1/10 e 9 i, w 2 2 1/10 e 81 i, w 3 2 1/10 e 153 i, w 4 2 1/10 e 225 i, w 5 2 1/10 e 297 i w 3 w 2 radius 2 1/10 w , 1 3 i, i 2 2 w w 1 1 w 3 1

6 568 7 Additional Topics in Trigonometr EXERCISE 7-7 A In Problems 1 6, use De Moivre s theorem to evaluate. Epress answers in polar form. 1. (3e 40 i ) 4 2. ( 2e 15 i ) 5 3. (1 i) 6 4. (1 i) 12 3 i (1 i 3) B In Problems 7 12, find the value of each epression and write the final answer in eact rectangular form. (Verif the results in Problems 7 12 b evaluating each directl on a calculator.) 7. ( 3 i) 4 8. ( 1 i) 4 9. (1 i) ( 3 i) i i 3 For n and z as indicated in Problems 13 18, find all nth roots of z. Leave answers in polar form. 13. z 8e 30 i, n z 8e 45 i, n z 81e 60 i, n z 16e 90 i, n z 1 i, n z 1 i, n 3 For n and z as indicated in Problems 19 24, find all nth roots of z. Write answers in polar form, and plot in a comple plane. 19. z 8, n z 1, n z 16, n z 8, n z i, n z i, n 5 In Problems 25 28, determine whether the statement is true or false. If true, eplain wh. If false, give a countereample. 25. Ever negative real number has two square roots. 26. If w is a cube root of z, then so is the conjugate of w. 27. Ever twelfth root of 1 is a fourth root of Ever fourth root of 1 is a twelfth root of (A) Show that 1 i is a root of How man other roots does the equation have? (B) The root 1 i is located on a circle of radius 2 in the comple plane as indicated in the figure. Locate the other three roots of on the figure, and eplain geometricall how ou found their location. (C) Verif that each comple number found in part B is a root of Figure for (A) Show that 2 is a root of How man other roots does the equation have? (B) The root 2 is located on a circle of radius 2 in the comple plane as indicated in the figure. Locate the other two roots of on the figure, and eplain geometricall how ou found their location. (C) Verif that each comple number found in part B is a root of In Problems 31 34, solve each equation for all roots. Write final answers in polar and eact rectangular form C 2 For n and z as indicated in Problems 35 38, find all nth roots of z. Write answers in eact rectangular form. 35. z 4, n z 25i, n z 8i, n z 64i, n Show that r 1/n e ( /n k360 /n)i n re i for an natural number n and an integer k. 1 i

7 Chapter 7 Group Activit Show that r 1/n e ( /n k360 /n)i 43. Solve in the set of comple numbers. 44. Solve 3 i 0 in the set of comple numbers. is the same number for k 0 and k n. In Problems 41 44, write answers in polar form. 41. Find all comple zeros for P() Find all comple zeros for P() 6 1. In Problems 45 and 46, write answers using eact rectangular forms. 45. Write P() 6 64 as a product of linear factors. 46. Write P() 6 1 as a product of linear factors. CHAPTER 7 GROUP ACTIVITY Conic Sections and Planetar Orbits I Conic Sections in Polar Form (A) Introduction to Conics. To understand orbits of planets, comets, and other celestial bodies, one must know something of the nature and properties of conic sections. (Conic sections are treated in detail in Chapter 11. Here our treatment will be brief and limited to polar representations.) Conic sections get their name because the curves are formed b cutting a complete right circular cone of two nappes with a plane (Fig. 1). An plane perpendicular to the ais of the cone cuts a section that is a circle. Tilt the plane slightl and the section becomes an ellipse. If the plane is parallel to one edge of the cone, it will cut onl one nappe, and the section will be a parabola. Tilt the plane further to the vertical, then it will cut both nappes of the cone and produce a hperbola with two branches. Closed orbits are ellipses or circles. Open (or escape) orbits are parabolas or hperbolas. Circle Ellipse Parabola Hperbola FIGURE 1 Conic sections. (B) Conics and Eccentricit. Another wa of defining conic sections is in terms of their eccentricit. Let F be a fied point, called the focus, and let d be a fied line, called the directri (see Fig. 2). For positive values of eccentricit e, a conic section can be defined as the set of points {P} having the propert that the ratio of the distance from P to the focus F to the distance from P to the directri d is the constant e. As we will see, an ellipse, a parabola, or a hperbola can be obtained b choosing e appropriatel.

8 570 7 Additional Topics in Trigonometr FIGURE 2 Conic section. directri d point on conic section Q P(r, ) polar ais F focus p (C) Polar Representation of Conics. A unified treatment of conic sections can be obtained b use of the polar coordinate sstem. Polar equations of conics are used etensivel in celestial mechanics to describe and analze orbits of planets, comets, satellites, and other celestial bodies. Problem 1: Polar Equation of a Conic. Use the eccentricit definition of a conic section given in part B to show that the polar equation of a conic is given b r ep 1 e cos (1) where p is the distance between the focus F and the directri d, the pole of the polar ais is at F, and the polar ais is perpendicular to d and is pointing awa from d (see Fig. 2). Problem 2: Graphing Utilit Eploration, 0 e 1. For 0 e 1, use a graphing utilit to sstematicall eplore the nature of the changes in the graph of equation (1) as ou change the eccentricit e and the distance p. Summarize the results of holding e fied and changing p and the results of holding p fied and changing e. For 0 e 1, which conic section is produced? Problem 3: Graphing Utilit Eploration, e 1. For e 1, use a graphing utilit to sstematicall eplore the nature of the changes in the graph of equation (1) as ou change the distance p. Summarize the results of holding e to 1 and changing p. For e 1, which conic section is produced? Problem 4: Graphing Utilit Eploration, e 1. For e 1, use a graphing utilit to sstematicall eplore the nature of the changes in the graph of equation (1) as ou change the eccentricit e and the distance p. Summarize the results of holding e fied and changing p and the results of holding p fied and changing e. For e 1, which conic section is produced? II Planetar Orbits We are now interested in finding polar equations for the orbits of specific planets where the sun is at the pole. Then these equations can be graphed in a graphing utilit, and further questions about the orbits can be answered. The material in Table 1, found in the readil available World Almanac and rounded to 3 significant digits, gives us enough information to find the polar equation for an planet s orbit. Problem 5: Polar Equations for the Orbits of Mercur, Earth, and Mars. In all cases the polar ais intersects the planet s orbit at aphelion (the greatest distance from the sun). (A) Show that Mercur s orbit is given approimatel b r cos

9 Chapter 7 Group Activit 571 TABLE 1 The Planets Ma. distance from sun Min. distance from sun Planet Eccentricit (millions of miles) (millions of miles) Mercur , ,028.6 Venus , ,066.8 Earth , ,091.4 Mars ,155 0,129 Jupiter ,507 0,461 Saturn ,938 0,838 Uranus ,860 1,670 Neptune ,820 2,760 Pluto ,550 2,760 (B) Show that Earth s orbit is given approimatel b r (C) Show that Mars orbit is given approimatel b r cos cos Problem 6: Plotting the Orbits for Mercur, Earth, and Mars. Plot all three orbits (Mercur, Earth, and Mars) from the equations in parts A through C in the same viewing window of a graphing utilit. Choose the window dimensions so that Mars orbit fills up most of the window. Problem 7: Finding Distances and Angles Related to Orbits. Figure 3 represents a schematic drawing showing Earth at two locations during its orbit. Find the straight-line distance between the position at A and the position at B to 3 significant digits. Find the measures of the angles BAO and ABO in degree measure to one decimal place. The Earth s orbit crosses the polar ais at aphelion (the greatest distance from the sun). FIGURE 3 Earth s orbit. B A Sun