## Precalculus (10th Edition)

Published by Pearson

# Chapter 9 - Polar Coordinates; Vectors - 9.3 The Complex Plane; De Moivre's Theorem - 9.3 Assess Your Understanding - Page 594: 39

#### Answer

$zw=4(\cos({\frac{9\pi}{40})}+i\sin{(\frac{9\pi}{40}})$ and $\frac{z}{w}=(\cos({\frac{\pi}{40})}+i\sin{\frac{\pi}{40}})$.

#### Work Step by Step

We know that if $z=a(\cos{\alpha}+i\sin{\alpha})$ and $w=b(\cos{\beta}+i\sin{\beta})$, then $zw=ab(\cos({\alpha+\beta)}+i\sin{(\alpha+\beta})$ and $\frac{z}{w}=\frac{a}{b}(\cos({\alpha-\beta)}+i\sin{(\alpha-\beta})$. Hence here: $zw=(2)(12(\cos({\frac{\pi}{8}+\frac{\pi}{10})}+i\sin{(\frac{\pi}{8}+\frac{\pi}{10}})\\zw=4(\cos({\frac{9\pi}{40})}+i\sin{(\frac{9\pi}{40}})$ and $\frac{z}{w}=\frac{2}{2}(\cos({\frac{\pi}{8}-\frac{\pi}{10})}+i\sin{\frac{\pi}{8}-\frac{\pi}{10}})\\\frac{z}{w}=(\cos({\frac{\pi}{40})}+i\sin{\frac{\pi}{40}})$.

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