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: 73



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: $\begin{align} & {{\left[ \frac{1}{2}\left( \cos \frac{\pi }{14}+i\sin \frac{\pi }{14} \right) \right]}^{7}}={{\left( \frac{1}{2} \right)}^{7}}\left[ \cos 7\left( \frac{\pi }{14} \right)+i\sin 7\left( \frac{\pi }{14} \right) \right] \\ & =\frac{1}{128}\left( \cos \frac{\pi }{2}+i\sin \frac{\pi }{2} \right) \\ & =\frac{1}{128}\left\{ 0+i\left( 1 \right) \right\} \\ & =\frac{1}{128}i \end{align}$ Hence, ${{\left[ \frac{1}{2}\left( \cos \frac{\pi }{14}+i\sin \frac{\pi }{14} \right) \right]}^{7}}=\frac{1}{128}i$.
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