SECTION 9.3 The Complex Plane; De Moivre’s Theorem 635 10. True or False The polar form of a nonzero complex number is unique. 11. Multiple Choice If = + z x yi is a complex number, then the magnitude of z is: (a) + x y 2 2 (b) + x y (c) + x y 2 2 (d) + x y 12. Multiple Choice If = θ z r ei 1 1 1 and = θ z r ei 2 2 2 are complex numbers, then ≠ z z z , 0, 1 2 2 equals: (a) θ θ ( ) − r r ei 1 2 1 2 (b) θ θ ( ) ⋅ r r ei 1 2 1 2 (c) θ θ ( ) + r r ei 1 2 1 2 (d) θ θ ( ) r r ei 1 2 1 2 Skill Building In Problems 13–24, plot each complex number in the complex plane and write it in polar form and in exponential form. 13. +i 1 14. − +i 1 15. −i 3 16. − i 1 3 17. − i3 18. −2 19. − i 4 4 20. + i 9 3 9 21. − i 3 4 22. + i 2 3 23. − + i 2 3 24. −i 5 In Problems 25–36, write each complex number in rectangular form. 25. π π ( ) +i 2 cos 2 3 sin 2 3 26. π π ( ) +i 3 cos 7 6 sin 7 6 27. π ⋅ e4 i 7 4 28. π ⋅ e2 i 5 6 29. π π ( ) +i 3 cos 3 2 sin 3 2 30. π π ( ) +i 4 cos 2 sin 2 31. π e7 i 32. π⋅ e3 i 2 33. π π ( ) +i 0.2 cos 5 9 sin 5 9 34. π π ( ) +i 0.4 cos 10 9 sin 10 9 35. π⋅ e2 i 18 36. π⋅ e3 i 10 In Problems 37–44, find zw and z w . Write each answer in polar form and in exponential form. 37. π π ( ) = + z i 2 cos 2 9 sin 2 9 π π ( ) = + w i 4 cos 9 sin 9 38. π π = + z i cos 2 3 sin 2 3 π π = + w i cos 5 9 sin 5 9 39. = π ⋅ z e3 i 13 18 = π ⋅ w e4 i 3 2 40. = π ⋅ z e2 i 4 9 = π ⋅ w e6 i 10 9 41. π π ( ) = + z i 2 cos 8 sin 8 π π ( ) = + w i 2 cos 10 sin 10 42. π π ( ) = + z i 4 cos 3 8 sin 3 8 π π ( ) = + w i 2 cos 9 16 sin 9 16 43. = + z i 2 2 = − w i 3 44. = − z i 1 = − w i 1 3 In Problems 45–56, write each expression in exponential form, polar form, and rectangular form. 45. π π ( ) + ⎡ ⎣ ⎢ ⎤ ⎥ ⎦ i 4 cos 2 9 sin 2 9 3 46. π π ( ) + ⎡ ⎣ ⎢ ⎤ ⎥ ⎦ i 3 cos 4 9 sin 4 9 3 47. π π ( ) + ⎡ ⎣ ⎢ ⎤ ⎥ ⎦ i 2 cos 10 sin 10 5 48. π π ( ) + ⎡ ⎣ ⎢ ⎤ ⎥ ⎦ i 2 cos 5 16 sin 5 16 4 49. π π ( ) + ⎡ ⎣ ⎢ ⎤ ⎥ ⎦ i 3 cos 18 sin 18 6 50. π π ( ) + ⎡ ⎣ ⎢ ⎤ ⎥ ⎦ i 1 2 cos 2 5 sin 2 5 5 51. [ ] π ⋅ e5 i 3 16 4 52. [ ] π ⋅ e3 i 5 18 6 53. ( ) −i 1 5 54. ( ) −i 3 6 55. ( ) −i 2 6 56. ( ) − i 1 5 8 In Problems 57–64, find all the complex roots. Write your answers in exponential form and in polar form. 57. The complex cube roots of +i 1 58. The complex fourth roots of −i 3 59. The complex fourth roots of − i 4 4 3 60. The complex cube roots of − − i 8 8 61. The complex fourth roots of − i 16 62. The complex cube roots of −8 63. The complex fifth roots of i 64. The complex fifth roots of −i
RkJQdWJsaXNoZXIy NjM5ODQ=