Explanation Given information: 3 − i Concept Involved: We use de Moivre’s Theorem to find powers of complex numbers written in trigonometric form. If z = r ( cos θ + i sin θ ) = r c i s θ is a complex number in trigonometric form and n is an integer, then z n = [ r ( cos θ + i sin θ ) ] n = r n ( cos n θ + i sin n θ ) = r n c i s n θ To find the nth power of a complex number, find the nth power of moduli and multiply the arguments by nth power Modulus: The absolute value or modulus of the complex number z = x + y i is the distance from the origin to the point ( x , y ) in the complex plane. If this distance is denoted by r , then r = | z | = | x + y i | = x 2 + y 2 Argument: The argument of the complex number z = x + y i , denoted a r g ( z ) , is the smallest positive angle θ from the positive real axis to the graph of z. Trigonometric Form: If z = x + y i is a complex number in standard form, then the trigonometric form of z is given by: z = r ( cos θ + i sin θ ) = r c i s θ Where r is the modulus of z and θ is the argument of z. We can convert back and forth between standard form and trigonometric form using the relationships that follow: For z = x + y i = r ( cos θ + i sin θ ) = r c i s θ r = x 2 + y 2 and θ is such that cos θ = x r , sin θ = y r , a n d tan θ = y x The sign of trigonometric ratio in different quadrants Calculation: TABLE 1 Steps Description z = 3 − i Complex number in Standard Form x + y i ⇒ 3 − i x = 3 y = − 1 Identifying x and y by comparison r = x 2 + y 2 = ( 3 ) 2 + ( − 1 ) 2 = 3 + 1 = 2 Identifying x and y cos θ = x r = 3 2 sin θ = y r = − 1 2 Identifying cos θ and sin θ

8th Edition

ISBN: 9781305652224

Author: Charles P. McKeague, Mark D. Turner

Publisher: Cengage Learning

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Chapter 8.3, Problem 41PS

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The reciprocal of the given expression using de Moivre’s Theorem

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Chapter 8.3, Problem 40PS

Chapter 8.3, Problem 42PS

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Chapter 8 Solutions

Trigonometry (MindTap Course List)

Ch. 8.1 - For Questions 1 through 10, fill in the blank with...Ch. 8.1 - Prob. 2PS Ch. 8.1 - Prob. 3PS Ch. 8.1 - Prob. 4PS Ch. 8.1 - Prob. 5PS Ch. 8.1 - Prob. 6PS Ch. 8.1 - Prob. 7PS Ch. 8.1 - Prob. 8PS Ch. 8.1 - Prob. 9PS Ch. 8.1 - Prob. 10PS

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The reciprocal of the given expression using de Moivre’s Theorem

Solution Summary: The author explains how to find the reciprocal of the given expression using de Moivre’s Theorem.

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To find: The reciprocal of the given expression using de Moivre’s Theorem

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Chapter 8 Solutions

To find:

The reciprocal of the given expression using de Moivre’s Theorem