Q. 6 - Precalculus Enhanced with Graphing Utilities

Question

True or False The graphs of

$y=\tan x, y=cotx, y=secx, and y=cscx$

 each have infinitely many vertical asymptotes.

Step-by-Step Solution

Answer

It is true that the graphs of $y=\tan x, y=cotx, y=secx, and y=csx$ each have infinitely many vertical asymptotes.

Step 1. Given Information

The given functions are $y=\tan x, y=cotx, y=secx, and y=csx.$

We have to state whether it is true or false that each function have infinitely many vertical asymptotes.

Step 2. Finding whether the statement is true or false

The function $y=\tan x$ has asymptotes at $\pm \frac{n\mathrm{\pi }}{2}$. Hence it has infinite asymptotes.

The function $y=cotx$ has asymptotes at  $\pm n\mathrm{\pi }.$ Hence it has infinite asymptotes.

The function $y=secx$ has asymptotes at $\pm \frac{n\mathrm{\pi }}{2}.$ Hence it has infinite asymptotes.

The function $y=cscx$ has asymptotes at $\pm n\mathrm{\pi }.$ Hence it has infinite asymptotes.

Thus, the statement is true.