Precalculus (6th Edition)

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

Chapter 7 - Trigonometric Identities and Equations - 7.6 Trigonometric Equations - 7.6 Exercises - Page 720: 22


$\frac{\pi}{4}$, $\frac{3\pi}{4}$, $\frac{7\pi}{6}$, $\frac{11\pi}{6}$

Work Step by Step

$(\csc x+2)(\csc x-\sqrt{2})=0$ $\csc x+2=0$ or $\csc x-\sqrt{2}=0$ If $\csc x+2=0$: $\csc x=-2$ $\frac{1}{\sin x}=-2$ $\sin x=-\frac{1}{2}$ The only solutions in $[0, 2\pi)$ are $\frac{7\pi}{6}$ and $\frac{11\pi}{6}$. If $\csc x-\sqrt{2}=0$: $\csc x=\sqrt{2}$ $\frac{1}{\sin x}=\sqrt{2}$ $\sin x=\frac{1}{\sqrt{2}}$ $\sin x=\frac{\sqrt{2}}{2}$ The only solutions in $[0, 2\pi)$ are $\frac{\pi}{4}$ and $\frac{3\pi}{4}$.
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