## Precalculus (6th Edition) Blitzer

Published by Pearson

# Chapter 6 - Section 6.5 - Complex Numbers in Polar Form; DeMoivre's Theorem - Exercise Set - Page 767: 36

#### Answer

The rectangular form of the given complex number is $-20+22.4i$.

#### 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=30\left( \cos 2.3+i\sin 2.3 \right) \\ & z=x+iy \\ \end{align} Simplify it further to get, \begin{align} & 30\left( \cos 2.3+i\sin 2.3 \right)=30\cos 2.3+30i\sin 2.3 \\ & =\left( 20\times -0.66 \right)+i\left( 20\times 0.74 \right) \\ & =-20+22.4i \\ & 30\left( \cos 2.3+i\sin 2.3 \right)=-20+22.4i \end{align} The rectangular form of the complex number is $-20+22.4i$.

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.