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: 32


The value of $y$ here is $$y=\frac{3\pi}{4}$$

Work Step by Step

$$y=\sec^{-1}(-\sqrt 2)$$ First, we see that the domain of inverse secant function is $(-\infty,\infty)$. Therefore, in fact when we deal with inverse secant function, we do not need to do this checking step. The range of inverse secant function is $[0,\frac{\pi}{2})\hspace{0.2cm}U\hspace{0.2cm}(\frac{\pi}{2},\pi]$. In other words, $y\in[0,\frac{\pi}{2})\hspace{0.2cm}U\hspace{0.2cm}(\frac{\pi}{2},\pi]$. We can rewrite $y=\sec^{-1}(-\sqrt 2)$ into $\sec y=-\sqrt2$ We know that $$\sec\frac{3\pi}{4}=\frac{1}{\cos\frac{3\pi}{4}}=\frac{1}{-\frac{\sqrt 2}{2}}=-\frac{2}{\sqrt2}=-\sqrt2$$ And $\frac{3\pi}{4}$ belongs to the range $[0,\frac{\pi}{2})\hspace{0.2cm}U\hspace{0.2cm}(\frac{\pi}{2},\pi]$. Therefore, the exact value of $y$ here is $$y=\frac{3\pi}{4}$$
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