Precalculus (6th Edition)

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

Chapter 7 - Trigonometric Identities and Equations - 7.5 Inverse Circular Functions - 7.5 Exercises - Page 708: 29


$y=-\displaystyle \frac{\pi}{6}$

Work Step by Step

$y=\csc^{-1}x$ Domain: $(-\infty, -1]\cup[1, \infty)$ Range: $[-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$ Quadrants (unit circle): I and IV --------- $y$ is the number from $[-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$ such that $\csc y=-2\quad (\displaystyle \sin y=-\frac{1}{2})$ Since $\displaystyle \sin\frac{\pi}{6}=\frac{1}{2}$, $\displaystyle \sin(-\frac{\pi}{6})=-\frac{1}{2}$, that is, $\csc (-\displaystyle \frac{\pi}{6})=-2$, and, $-\displaystyle \frac{\pi}{6}\in [-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$, so, $y=-\displaystyle \frac{\pi}{6}$
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