Chapter 6.5, Problem 15AYU

In Problems 7-16, if necessary, refer to the graphs of the functions to answer each question.

For What numbers $x,-2\pi \le x\le \text{2}\pi$, does the graph of $y=\tan x$ have vertical asymptotes?

To determine

To find: What numbers does in the given function $y$ have vertical asymptotes?

Answer to Problem 15AYU

Solution:

$x=\frac{-3\pi }{2},x=\frac{\pi }{2},x=-\frac{\pi }{2},x=\frac{3\pi }{2}$

Explanation of Solution

Given:

$y=\tan x$

Calculation:

Since $\tan x=\frac{\sin x}{\cos x}$ then tangent has a vertical asymptote wherever it is undefined.

This occurs precisely where $\cos x=0$, and for the interval$-2\pi \le x\le 2\pi$.

There are four values that satisfy this, namely

$x=\frac{-3\pi }{2},x=\frac{\pi }{2},x=-\frac{\pi }{2},x=\frac{3\pi }{2}$

Therefore,

$x=\frac{-3\pi }{2},x=\frac{\pi }{2},x=-\frac{\pi }{2},x=\frac{3\pi }{2}$