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: 34


The rectangular form of the given complex number is $-7i$.

Work Step by Step

Consider any complex number, given by $z=x+iy$; for a complex number in rectangular form, $z=x+iy$ …… (1) The polar form is given by, $z=r\left( \cos \theta +i\sin \theta \right)$ …… (2) Here, $x=r\cos \theta \ \text{ and }\ y=r\sin \theta $. Divide the value of y by x, to get, $\tan \theta =\frac{y}{x}$ Also, the value of r is called as moduli of the complex number, given by, $r=\sqrt{{{x}^{2}}+{{y}^{2}}}$ For any complex number in polar form, $z=r\left( \cos \theta +i\sin \theta \right)$, the rectangular form is, Using (1) and (2), $\begin{align} & z=7\left( \cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2} \right) \\ & z=x+iy \\ \end{align}$ Simplify it further to get, $\begin{align} & 7\left( \cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2} \right)=7\cos \frac{3\pi }{2}+7i\sin \frac{3\pi }{2} \\ & =\left( 7\times 0 \right)+i\left( 7\times -1 \right) \\ & =-7i \\ & 7\left( \cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2} \right)=-7i \end{align}$ The rectangular form of the complex number is $-7i$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.