Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 6 - Inverse Circular Functions and Trigonometric Equations - Section 6.1 Inverse Circular Functions - 6.1 Exercises - Page 264: 29


The value of $y$ in need to find is $$y=-\frac{\pi}{6}$$

Work Step by Step

$$y=\csc^{-1} (-2)$$ First, we see that the domain of inverse cosecant function is $(-\infty,\infty)$. Therefore, in fact when we deal with inverse cosecant function, we do not need to do this checking step. The range of inverse cotangent function is $[-\frac{\pi}{2},0)\hspace{0.2cm}U\hspace{0.2cm}(0,\frac{\pi}{2}]$. In other words, $y\in[-\frac{\pi}{2},0)\hspace{0.2cm}U\hspace{0.2cm}(0,\frac{\pi}{2}]$. We can rewrite $y=\csc^{-1}(-2)$ into $\csc y=(-2)$ We know that $$\csc\frac{\pi}{6}=2$$ which means $$-\csc\frac{\pi}{6}=-2$$ $$\cot(-\frac{\pi}{6})=-2$$ And $-\frac{\pi}{6}$ belongs to the range $[-\frac{\pi}{2},0)\hspace{0.2cm}U\hspace{0.2cm}(0,\frac{\pi}{2}]$. Therefore, the exact value of $y$ here is $$y=-\frac{\pi}{6}$$
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