Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 8 - Application of Trigonometry - 8.6 De Moivre's Theorem; Powers and Roots of Complex Numbers - 8.6 Exercises: 43

Answer

$[{cis45^{\circ},cis 135^{\circ},cis 225^{\circ},cis 315^{\circ}}]$

Work Step by Step

Given: $x^{4}+1=0$ or $x^{4}=-1$ $-1$ can be written in trigonometric form as: $-1=1(\cos 180^{\circ}+\sin 180^{\circ})$ $x=\cos (180^{\circ}+2k\pi)+i \sin (180^{\circ}+2k\pi)$ $x=[(\cos (180^{\circ}+2k\pi)+i \sin (180^{\circ}+2k\pi))]^{\frac{1}{4}}$ Apply De-Moivre's Theorem $x=[(\cos \frac{(180^{\circ}+2k\pi)}{4}+i \sin\frac{(180^{\circ}+2k\pi)}{4})]$ Now, the arguments can be written as: $\frac{180^{\circ}+2k\pi}{4} $ for $k=0,1,2,3$ Arguments are: $45^{\circ}$ for $k=0$ $135^{\circ}$ for $k=1$ $225^{\circ}$ for $k=2$ $315^{\circ}$ for $k=3$ Solution set of the equation can be written as: $[{cis45^{\circ},cis 135^{\circ},cis 225^{\circ},cis 315^{\circ}}]$
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