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: 7

Answer

$27i$

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 $[ r cis\theta]^{n}=[ r^{n} (cis \theta)]$ $[ 3 (\cos 30^{\circ}+i \sin30^{\circ})]^{3}=[ 3^{3} (\cos 3\times30^{\circ}+i \sin 3\times30^{\circ})]$ $=[ 27 (\cos 90^{\circ}+i \sin 90^{\circ})]$ $=[ 27 (0+i.1]$ $=27i$
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