Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 6 - Section 6.7 - The Dot Product - Exercise Set - Page 799: 72



Work Step by Step

DeMoivre’s Theorem: For any complex number $z=r\left( \cos \theta +i\sin \theta \right)$, if n is a positive integer $\left( z>0 \right)$ then, $\begin{align} & {{z}^{n}}={{\left[ r\left( \cos \theta +i\sin \theta \right) \right]}^{n}} \\ & ={{r}^{n}}\left( \cos n\theta +i\sin n\theta \right) \end{align}$ So, the given complex number can be written as: $\begin{align} & {{\left[ 4\left( \cos 50{}^\circ +i\sin 50{}^\circ \right) \right]}^{3}}={{4}^{3}}\left[ \cos 3\left( 50{}^\circ \right)+i\sin 3\left( 50{}^\circ \right) \right] \\ & =64\left( \cos 150{}^\circ +i\sin 150{}^\circ \right) \\ & =64\left( -\frac{\sqrt{3}}{2}+i\frac{1}{2} \right) \\ & =-32\sqrt{3}+32i \end{align}$ Hence, ${{\left[ 4\left( \cos 50{}^\circ +i\sin 50{}^\circ \right) \right]}^{3}}=-32\sqrt{3}+32i$.
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