University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 3 - Section 3.9 - Inverse Trigonometric Functions - Exercises - Page 181: 17


$\displaystyle \frac{\pi}{2}$

Work Step by Step

$y=\sec^{-1}x$ is the number in $[0, \pi/2) \cup(\pi/2, \pi$] for which $\sec y=x.$ (In terms of cosine, $\quad \displaystyle \cos y=\frac{1}{x}$) We want an angle (in radians) from $[0, \pi/2$) $\cup(\pi/2, \pi$] such that its cosine approaches 0 (so that its secant grows to $\infty$). This angle is $\displaystyle \frac{\pi}{2}$. Alternatively (if in doubt), we can reach the same conclusion by observing the graph of $y=\sec x$ (also written as $\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{e}\mathrm{c} x$) when $ x\rightarrow\infty$. See below.
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