836 CHAPTER 8 Applications of Trigonometry The solution set of the equation can be written as 5cis 0°, cis 72°, cis 144°, cis 216°, cis 288°6. The first of these roots is the real number 1. The others cannot easily be expressed in rectangular form but can be approximated using a calculator. The tips of the arrows representing the five fifth roots all lie on a unit circle and are equally spaced around it every 72°, as shown in Figure 55. y x –1 1 1 –1 1(cos 08 + i sin 08) 1(cos 1448 + i sin 1448) 1(cos 728 + i sin 728) 1(cos 2168 + i sin 2168) 1(cos 2888 + i sin 2888) 0 728 Figure 55 S Now Try Exercise 41. 8.6 Exercises CONCEPT PREVIEWFill in the blanks to correctly complete each problem. 1. If z = 31cos 30° + i sin30°2, it follows that z3 = _____1cos_____ + i sin_____2 = _____1_____ + i _____2 = _____ + _____i, or simply_____ . 2. If we are given z = 161cos 80° + i sin80°2, then any fourth root of z has r = _____, and the fourth root with least positive argument has u = _____. 3. 3cos 6° + i sin6°430 = cos_____ + i sin_____ = _____ + _____i 4. Based on the result of Exercise 3, cos 6° + i sin6° is a(n) root of . CONCEPT PREVIEWAnswer each question. 5. How many real tenth roots of 1 exist? 6. How many nonreal complex tenth roots of 1 exist? Find each power. Write answers in rectangular form. See Example 1. 7. 341cos 30° + i sin30°243 8. 321cos 135° + i sin135°244 9. 1cos 45° + i sin45°28 10. 321cos 120° + i sin120°243 11. 33 cis 100°43 12. 33 cis 40°43 13. A 23 + i B 5 14. A2 - 2i23B 4 15. A222 - 2i22B 6 16. ¢ 22 2 - 22 2 i≤ 8 17. 1-2 - 2i25 18. 1-1 + i27
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