829 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients 30. 2 + 6i, -2i 31. 1 2 + 2 3 i, 2 3 + 1 2 i 32. - 1 5 + 2 7 i, 3 7 - 3 4 i Write each complex number in rectangular form. See Example 2. 33. 21cos 45° + i sin45°2 34. 41cos 60° + i sin60°2 35. 101cos 90° + i sin90°2 36. 81cos 270° +i sin270°2 37. 41cos 240° + i sin240°2 38. 21cos 330° + i sin330°2 39. 3 cis 150° 40. 2 cis 30° 41. 5 cis 300° 42. 6 cis 135° 43. 22 cis 225° 44. 23 cis 315° 45. 41cos1-30°2 + i sin1-30°22 46. 221cos1-60°2 + i sin1-60°22 Write each complex number in trigonometric form r1cos u + i sinu2, with u in the interval 30°, 360°2. See Example 3. 47. -3 - 3i23 48. 1 + i23 49. 23 - i 50. 423 + 4i 51. -5 - 5i 52. -2 + 2i 53. 2 + 2i 54. 4 + 4i 55. 5i 56. -2i 57. -4 58. 7 Write each complex number in its alternative form, using a calculator. Approximate answers in rectangular form to four decimal places and angles in degrees to two decimal places as necessary. See Example 4. Rectangular Form Trigonometric Form 59. 2 + 3i 60. cos 35° + i sin35° 61. 31cos 250° + i sin250°2 62. -4 + i 63. 12i 64. 3 cis 180° 65. 3 + 5i 66. cis 110.5° Concept Check The complex number z, where z = x + yi, can be graphed in the plane as 1x, y2. Describe the graphs of all complex numbers z satisfying the given conditions. 67. The absolute value of z is 1. 68. The imaginary part of z is 1. 69. The real part of z is 1. 70. The real and imaginary parts of z are equal. Julia Set Refer to Example 5. 71. Is z = -0.2i in the Julia set? 72. The graph of the Julia set in Figure 53 appears to be symmetric with respect to both the x-axis and the y-axis. Complete the following to show that this is true. (a) Show that complex conjugates have the same absolute value. (b) Compute z1 2 - 1 and z 2 2 - 1, where z 1 = a + bi and z2 = a - bi. (c) Discuss why if 1a, b2 is in the Julia set, then so is 1a, -b2. This reveals that the graph of the Julia set must be symmetric with respect to the x-axis. (d) Using a similar argument, show that the Julia set must also be symmetric with respect to the y-axis.
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