834 CHAPTER 8 Applications of Trigonometry nth RootTheorem If n is any positive integer, r is a positive real number, and u is in degrees, then the nonzero complex number r 1cos u + i sin u2 has exactly n distinct nth roots, given by the following. n! r 1cos A +i sin A2 or n! r cis A, where A = U +360° # k n , or A = U n + 360° # k n , k = 0, 1, 2, . . . , n - 1. If u is in radians, then A = U +2Pk n , or A = U n + 2Pk n , k = 0, 1, 2, . . . , n - 1. EXAMPLE 2 Finding Complex Roots Find the two square roots of 4i. Write the roots in rectangular form. SOLUTION First write 4i in trigonometric form. 4 acos p 2 + i sin p 2b Trigonometric form (using radian measure) Here r = 4 and u = p 2 . The square roots have absolute value 24 = 2 and arguments as follows. a = p 2 2 + 2pk 2 = p 4 + pk Because there are two square roots, let k = 0 and 1. If k = 0, then a = p 4 + p # 0 = p 4 . If k = 1, then a = p 4 + p # 1 = 5p 4 . Using these values for a, the square roots are 2 cis p 4 and 2 cis 5p 4 , which can be written in rectangular form as 2 2 + i22 and -22 - i22. S Now Try Exercise 23(a). Be careful simplifying here. This screen confirms the results of Example 2. This screen shows how a calculator finds r and u for the number in Example 3. EXAMPLE 3 Finding Complex Roots Find all fourth roots of -8 + 8i23. Write the roots in rectangular form. SOLUTION -8 + 8i23 = 16 cis 120° Write in trigonometric form. Here r = 16 and u = 120°. The fourth roots of this number have absolute value 2 4 16 = 2 and arguments as follows. a = 120° 4 + 360° # k 4 = 30° + 90° # k
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