830 CHAPTER 8 Applications of Trigonometry Find each product. Write answers in rectangular form. See Example 6. 73. 331cos 60° + i sin60°24321cos 90° + i sin90°24 74. 341cos 30° + i sin30°24351cos 120° + i sin120°24 75. 341cos 60° + i sin60°24361cos 330° + i sin330°24 76. 381cos 300° + i sin300°24351cos 120° + i sin120°24 77. 321cos 135° + i sin135°24321cos 225° + i sin225°24 78. 381cos 210° + i sin210°24321cos 330° + i sin330°24 79. A 23 cis 45°B A 23 cis 225°B 80. A 26 cis 120°B C 26 cis1-30°2D 81. 15 cis 90°213 cis 45°2 82. 13 cis 300°217 cis 270°2 Find each quotient. Write answers in rectangular form. In Exercises 89–94, first convert the numerator and the denominator to trigonometric form. See Example 7. 83. 41cos 150° + i sin150°2 21cos 120° + i sin120°2 84. 241cos 150° + i sin150°2 21cos 30° + i sin30°2 85. 101cos 50° + i sin50°2 51cos 230° + i sin230°2 86. 121cos 23° + i sin23°2 61cos 293° + i sin293°2 87. 3 cis 305° 9 cis 65° 88. 16 cis 310° 8 cis 70° 89. 823 + i 90. 2i -1 - i23 91. -i 1 + i 92. 1 2 - 2i 93. 226 - 2i22 22 - i26 94. -322 + 3i26 26 + i22 Use a calculator to find the following. Write answers in rectangular form, expressing real and imaginary parts to four decimal places. 95. 32.51cos 35° + i sin35°2433.01cos 50° + i sin50°24 96. 34.61cos 12° + i sin12°2432.01cos 13° + i sin13°24 97. 112 cis 18.5°213 cis 12.5°2 98. 14 cis 19.25°217 cis 41.75°2 99. 45Acos 2p 3 + i sin 2p 3 B 22.5Acos 3p 5 + i sin 3p 5 B 100. 30Acos 2p 5 + i sin 2p 5 B 10Acos p 7 + i sin p 7 B 101. c 2 cis 5p 9 d 2 102. c 24.3 cis 7p 12 d 2 Work each problem. 103. Note that 1r cis u22 = 1r cis u21r cis u2 = r2 cis1u + u2 = r2 cis 2u. Explain how we can square a complex number in trigonometric form. (In the next section, we will develop this idea more fully.) 104. Without actually performing the operations, state why the following products are the same. 321cos 45° + i sin45°24 # 351cos 90° + i sin90°24 and C 23cos1-315°2 + i sin1-315°24D # C 53cos1-270°2 + i sin1-270°24D 105. Show that 1 z = 1 r 1cos u - i sinu2, where z = r1cos u + i sinu2.
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