Chapter 4.5, Problem 56E

Solution

To identify: The correct option for the statement “The graph of $y=\csc x$ has the same set of asymptotes as the graph of $y=$ ”

Option A) $\sin x.$

Option B) $\tan x.$

Option C) $\cot x.$

Option D) $\sec x.$

Option E) $\csc 2x.$

Option(C) is correct option for the statement “The graph of $y=\csc x$ has the same set of asymptotes as the graph of $y=$ ”.

Given information:

The statement “The graph of $y=\csc x$ has the same set of asymptotes as the graph of $y=$ ”.

Explanation:

Consider the function $y=\csc x$ ,

Now find the range of $y$ .

  $\csc x=\frac{1}{\sin x}$

As the function $\sin x$ has zeroes on the integer multiple of $\pi$ , so $y=\csc x$ is not defined at $n\pi$ .

The domain of $y=\csc x$ is all the real number except integer multiples of $\pi$ .

The vertical asymptotes of $y=\csc x$ are $n\pi$ .

Now consider $y=\cot x$ ,

  $\cot x=\frac{\cos x}{\sin x}$

As the function $\sin x$ has zeroes on the integer multiple of $\pi$ , so $y=\cot x$ is not defined at $n\pi$ .

The domain of $y=\cot x$ is all the real number except integer multiples of $\pi$ .

The vertical asymptotes of $y=\cot x$ are $n\pi$ .

Thus, Option (C) is correct.