Question

1. multiple choice the graph of $y = \csc x$ has the same set of asymptotes as the graph of $y =$ (a) $\sin x$. (b) $\tan x$. (c) $\cot x$. (d) $\sec x$. (e) $\csc 2x$.

Explanation:

Step1: Recall the asymptotes of trigonometric functions

• For \( y = \csc x=\frac{1}{\sin x} \), the vertical asymptotes occur where \( \sin x = 0 \), i.e., \( x = n\pi, n\in\mathbb{Z} \).
• For \( y=\sin x \), it has no vertical asymptotes (it's a continuous function with range \([-1,1]\)).
• For \( y = \tan x=\frac{\sin x}{\cos x} \), vertical asymptotes occur where \( \cos x = 0 \), i.e., \( x=\frac{\pi}{2}+n\pi, n\in\mathbb{Z} \).
• For \( y = \cot x=\frac{\cos x}{\sin x} \), vertical asymptotes occur where \( \sin x = 0 \), i.e., \( x = n\pi, n\in\mathbb{Z} \).
• For \( y=\sec x=\frac{1}{\cos x} \), vertical asymptotes occur where \( \cos x = 0 \), i.e., \( x=\frac{\pi}{2}+n\pi, n\in\mathbb{Z} \).
• For \( y = \csc 2x=\frac{1}{\sin 2x} \), vertical asymptotes occur where \( \sin 2x = 0 \), i.e., \( 2x=n\pi\Rightarrow x=\frac{n\pi}{2}, n\in\mathbb{Z} \).

Step2: Compare the asymptotes

We see that \( y = \csc x \) and \( y=\cot x \) both have vertical asymptotes at \( x = n\pi, n\in\mathbb{Z} \).

Answer:

C. \( \cot x \)