Precalculus (6th Edition) Blitzer

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

Chapter 6 - Section 6.5 - Complex Numbers in Polar Form; DeMoivre's Theorem - Exercise Set - Page 767: 14

Answer

$$z=\sqrt{8} \left ( \cos \frac{7\pi }{4} +i \sin \frac{7\pi }{4} \right ).$$
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Work Step by Step

We plot the complex number $z=2-2i$ the same way we plot $(2,-2)$ in the rectangular coordinate system. Please note that the complex number $z=a+bi$ is written in polar form as$$z=r(\cos \theta +i \sin \theta ),$$where $a=r\cos \theta$, $b=r\sin \theta$, $r=\sqrt{a^2+b^2}$, and $\tan \theta = \frac{b}{a}$, $0 \le \theta \lt 2\pi $. So, for the complex number $z=2-2i$, we have$$r=\sqrt{2^2+(-2)^2}=\sqrt{8}, \\ \tan \theta =\frac{-2}{2}=-1.$$Since $\cos \theta =\frac{a}{r} = \frac{2}{\sqrt{8}} \gt 0$, the argument, $\theta$, must lie in quadrant $IV$. So$$\theta = \frac{3\pi }{4} +\pi = \frac{7\pi }{4}.$$Thus, the complex number has the polar form$$z=\sqrt{8} \left ( \cos \frac{7\pi }{4} +i \sin \frac{7\pi }{4} \right ).$$
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