832 CHAPTER 8 Applications of Trigonometry Powers of Complex Numbers (De Moivre’s Theorem) Because raising a number to a positive integer power is a repeated application of the product rule, it would seem likely that a theorem for finding powers of complex numbers exists. Consider the following. 3r 1cos u + i sin u242 = 3r 1cos u + i sin u243r 1cos u + i sin u24 a2 = a # a = r # r3cos1u + u2 + i sin1u + u24 Product theorem = r21cos 2u + i sin 2u2 Multiply and add. In the same way, 3r 1cos u + i sin u243 is equivalent to r31cos 3u + i sin 3u2. These results suggest the following theorem for positive integer values of n. Although the theorem is stated and can be proved for all n, we use it only for positive integer values of n and their reciprocals. 8.6 De Moivre’s Theorem; Powers and Roots of Complex Numbers ■ Powers of Complex Numbers (De Moivre’s Theorem) ■ Roots of Complex Numbers Abraham De Moivre (1667 – 1754) Named after this French expatriate friend of Isaac Newton, De Moivre’s theorem relates complex numbers and trigonometry. De Moivre’sTheorem If r 1cos u + i sin u2 is a complex number, and if n is any real number, then the following holds true. 3 r 1cos U +i sin U2 4 n =rn1cos nU +i sin nU2 In compact form, this is written 3 r cis U4 n =rn1cis nU2. EXAMPLE 1 Finding a Power of a Complex Number Find A1 + i23 B 8 and write the answer in rectangular form. SOLUTION Using earlier methods, write 1 + i23 in trigonometric form. 21cos 60° + i sin 60°2 Trigonometric form of 1 + i 23 Now, apply De Moivre’s theorem. A1 + i23 B 8 = 321cos 60° + i sin 60°248 Trigonometric form = 283cos18 # 60°2 + i sin18 # 60°24 De Moivre’s theorem = 2561cos 480° + i sin 480°2 Apply the exponent and multiply. = 2561cos 120° + i sin 120°2 480° and 120° are coterminal. = 256 ¢- 1 2 + i 23 2 ≤ cos 120° = - 1 2 ; sin 120° = 23 2 = -128 + 128i23 Rectangular form S Now Try Exercise 13.
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