Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 8 - Application of Trigonometry - 8.6 De Moivre's Theorem; Powers and Roots of Complex Numbers - 8.6 Exercises: 8

Answer

$-16$

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 $[ r cis\theta]^{n}=[ r^{n} (cis \theta)]$ $[ 2 (\cos 135^{\circ}+i \sin135^{\circ})]^{4}=[ 2^{4} (\cos 4\times 135^{\circ}+i \sin 4\times 135^{\circ})]$ $=[ 16 (\cos 540^{\circ}+i \sin 540^{\circ})]$ $=[ 16 (-1+i.0]$ $=-16$
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