Explanation Given information: Need to find all the complex solutions of z 7 = ( 1 − i ) 2 Formula used: 1) Demoivre’s Theorem: Let z = r ( cos θ + i sin θ ) be a complex number in polar form. If n is a positive integer, then z to the n t h power, z n is z n = [ r ( cos θ + i sin θ ) ] n = r n ( cos n θ + i sin n θ ) w h e r e n = 0 , ± 1 , ± 2 , ... 2) Polar form of z = x + i y is z = r ( cos θ + i sin θ ) Where r = x 2 + y 2 a n d θ = − tan − 1 | y x | 3) sin x and cos x are periodic function with period 2 n π so sin x = sin ( 2 n π + x ) , cos x = cos ( 2 n π + x ) Calculation: z 7 = ( 1 − i ) 2 [Given] z 7 = ( 1 ) 2 − 2 ( 1 ) ( i ) + ( i ) 2 z 7 = 1 − 2 i − 1 z 7 = − 2 i ------eq(1) In above equation x = 0 and y = -2 so r = x 2 + y 2 = 0 + ( − 2 ) 2 = 2 a n d θ = − tan − 1 | y x | = − tan − 1 | − 2 0 | = − tan − 1 ( ∞ ) = − π 2 Polar form of above equation (1) is z 7 = 2 [ cos ( − π 2 ) + i sin ( − π 2 ) ] z 7 = 2 ( cos π 2 − i sin π 2 ) ------eq(2) [Since cos ( − θ ) = cos θ a n d sin ( − θ ) = sin θ ] Since sinx and cosx are periodic function, so we can write equation (2) z 7 = 2 [ cos ( 2 n π + π 2 ) − i sin ( 2 n π + π 2 ) ] Now take the seventh root of both sides z = 2 1 7 [ cos ( 2 n π + π 2 ) − i sin ( 2 n π + π 2 ) ] 1 7 z = 2 1 7 [ cos ( 2 n π + π 2 7 ) − i sin ( 2 n π + π 2 7 ) ] [Using Demoivre’s Theorem] z = 2 1 7 [ cos ( 4 n π + π 14 ) − i sin ( 4 n π + π 14 ) ]

Precalculus: A Unit Circle Approach, Books a la Carte Edition plus MyMathLab with Pearson eText -- Access Card Package (2nd Edition)

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ISBN: 9780321869401

Author: J. S. Ratti

Publisher: PEARSON

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Chapter 6.7, Problem 92E

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Chapter 6 Solutions

Precalculus: A Unit Circle Approach, Books a la Carte Edition plus MyMathLab with Pearson eText -- Access Card Package (2nd Edition)

Ch. 6.1 - If is an acute angle and sin=223 , then cos=_____.Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - In Exercises 9—14, find the exact values for the...Ch. 6.1 - In Exercises 9-14, find the exact values for the...Ch. 6.1 - In Exercises 9—14, find the exact values for the...Ch. 6.1 - In Exercises 9—14, find the exact values for the...

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To determine all complex solution.

Solution Summary: The author calculates the complex solution of z7=(1-i)2.

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To determine all complex solution.