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 769: 95


See the explanation below.

Work Step by Step

The polar form of a complex number is yet another form to represent a complex number, other than rectangular form. A complex number $z=a+ib$ can be represented in polar form as: $z=r\left( \cos \theta +i\sin \theta \right)$ Here $a=r\cos \theta $, $b=r\sin \theta $, $r=\sqrt{{{a}^{2}}+{{b}^{2}}}$ and $\tan \theta =\frac{b}{a}$ The values $r$ and $\theta $ are called the modulus and argument of a complex number $z$ respectively, with $0\le \theta \le 2\pi $. Example: The above explanation can be justified with the help of an example. The provided complex number $z=5\left( \cos 30{}^\circ +i\sin 30{}^\circ \right)$ is in simple polar form.The modulus and argument of a complex number is $5$ and $30{}^\circ $ respectively. Simplify the complex number, $\begin{align} & z=5\left( \cos 30{}^\circ +i\sin 30{}^\circ \right) \\ & =5\left( \frac{\sqrt{3}}{2}+i\frac{1}{2} \right) \\ & =\frac{5\sqrt{3}}{2}+i\frac{1}{2} \end{align}$ The real part of a complex number is $\frac{5\sqrt{3}}{2}$ and the imaginary part of a complex number is $\frac{1}{2}$. The above expression of a complex number in polar form is converted into rectangular form.
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.