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: 36


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 $-20+22.4i$.
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