## Calculus 8th Edition

Published by Cengage

# Appendix G - Complex Numbers - G Exercises: 29

#### Answer

$zw=4[cos(\frac{\pi}{2})+isin(\frac{\pi}{2})]$, $\frac{z}{w}=cos(-\pi/6)+isin(-\pi/6)]$, $\frac{1}{z}=\frac{1}{2}[cos(-\frac{\pi}{6})+isin(-\frac{\pi}{6})]$

#### Work Step by Step

$z=2[cos\frac{\pi}{6}+isin\frac{\pi}{6}]$ and $w=2[cos\frac{\pi}{3}+isin\frac{\pi}{3}]$ Calculate $zw$, $\frac{z}{w}$ and $\frac{1}{z}$ $zw=2[cos\frac{\pi}{6}+isin\frac{\pi}{6}]\times 2[cos\frac{\pi}{3}+isin\frac{\pi}{3}]$ $zw=4[cos(\frac{\pi}{6}+\frac{\pi}{3})+i(sin(\frac{\pi}{6}+\frac{\pi}{3})]$ Hence, $zw=4[cos(\frac{\pi}{2})+isin(\frac{\pi}{2})]$ $\frac{z}{w}=\frac{2[cos\frac{\pi}{6}+isin\frac{\pi}{6}]}{2[cos\frac{\pi}{3}+isin\frac{\pi}{3}]}$ $\frac{z}{w}=\frac{2}{2} [cos(\pi /6-\pi/3)+isin(\pi/6-\pi/3)]$ Hence, $\frac{z}{w}=cos(-\pi/6)+isin(-\pi/6)]$ $\frac{1}{z}=\frac{1}{2[cos\frac{\pi}{6}+isin\frac{\pi}{6}]}$ $\frac{1}{z}=\frac{1(cos 0+isin0)}{2[cos\frac{\pi} {6}+isin\frac{\pi}{6}]}$ Hence, $\frac{1}{z}=\frac{1}{2}[cos(-\frac{\pi}{6})+isin(-\frac{\pi}{6})]$

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