Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 41

Estimate the equations by graphing all the vertical asymptotes of the curve $y = \tan (2 \sin x) \quad -\pi \leq x \leq \pi$. Then find the exact equation of these asymptotes.



To find the exact equation of the asymptotes, we know that the tangent function has vertical asymptotes at $\displaystyle x = \frac{\pi}{2} + n \pi$, where $n$ is an integer and $n$ $\pi$ for its multiple cycles. So,



$
\begin{equation}
\begin{aligned}

2 \sin x & = \frac{\pi}{2} + n \pi\\

\sin x =& \frac{\pi}{4} + \frac{n \pi}{2} ; \quad \text{ for } -\pi < x < \pi, \\

\sin x =& \frac{\pi}{4}\\

x =& \sin ^ {-1} \left[\frac{\pi }{4}\right] = 0.9033

\end{aligned}
\end{equation}
$


The other value of $x$ is obtained from taking the supplementary angle

$\displaystyle x = \pi - \sin^{-1} \left[\frac{\pi}{4}\right]$

$x = 2.2383$

Therefore, the value of the asymptotes are exactly $x = \pm 0.9033 \text{ and } x = \pm 2.2383$