Trigonometry 7th Edition

Published by Cengage Learning
ISBN 10: 1111826854
ISBN 13: 978-1-11182-685-7

Chapter 8 - Section 8.3 - Products and Quotients in Trigonometric Form - 8.3 Problem Set - Page 439: 30

Answer

$32i$

Work Step by Step

Using de Moivre's theorem ($z^{n}=r^{n} cis \,n\theta$ where $z=r\, cis\,\theta$ and $n$ is an integer), we get $(\sqrt {2}\,cis\, \frac{\pi}{4})^{10}=(\sqrt {2})^{10}(cis\, 10\cdot\frac{\pi}{4})$ $=32(\cos \frac{5\pi}{2}+i\sin\frac{5\pi}{2})$ $=32(\cos \frac{\pi}{2}+i\sin\frac{\pi}{2})$ ($\frac{\pi}{2}$ is coterminal with $\frac{5\pi}{2}$) In standard form, our result is $=32(0+i\cdot1)$ $=32i$
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