Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 8 - Complex Numbers, Polar Equations, and Parametric Equations - Section 8.4 De Moivre's Theorem: Powers and Roots of Complex Numbers - 8.4 Exercises - Page 376: 4

Answer

$8$

Work Step by Step

De Moivre’s Theorem states that when $ r (\cos\theta+i \sin\theta)$ is a complex number, and if $n$ is any real number, then the following relationship holds. $[ r (\cos\theta+i \sin\theta)]^{n}=[ r^{n} (\cos n\theta+i \sin n\theta)]$ In compact form, this is written $[ r cis\theta]^{n}=[ r^{n} (cis \theta)]$ $[2 (\cos 120^{\circ}+i \sin 120^{\circ})]^{3}=[ (8\cos 3\times 120^{\circ}+i \sin 3\times 120^{\circ})]$ $=[ 8(\cos 360^{\circ}+i \sin 360^{\circ})]$ $=8(1+i.0)$ $=8$
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