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: 40

Answer

$[{cis - 22.5^{\circ},cis -112.5^{\circ},cis -202.5^{\circ}},cis -292.5^{\circ}]$

Work Step by Step

Given: $x^{4}+i=0$ or $x^{4}=-i$ $-i$ can be written in trigonometric form as: $-i=0-i=(\cos 90^{\circ}-\sin 90^{\circ})$ Absolute value of fourth root is given as $\sqrt[4] 1=1$ Now, the arguments can be given as: $k=0,1,2,3$ Roots: $(\cos 22.5^{\circ}+\sin 22.5^{\circ})$, $(\cos 112.50^{\circ}+\sin 112.50^{\circ})$, $(\cos 202.5^{\circ}+\sin 202.5^{\circ})$, $(\cos 292.5^{\circ}+\sin 292.5^{\circ})$, Solution set of the equation can be written as: $[{cis - 22.5^{\circ},cis -112.5^{\circ},cis -202.5^{\circ}},cis -292.5^{\circ}]$
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