which of the following is an asymptote of y = csc(x)?\no x=-π\no x=-π/3\no x=π/4\no x=π/2

Explanation:

Step1: Recall the definition of cosecant

The cosecant function is defined as $\csc(x)=\frac{1}{\sin(x)}$. An asymptote occurs where $\sin(x) = 0$ because division by zero is undefined.

Step2: Find the zeros of sine function

The zeros of the sine - function $\sin(x)$ occur at $x = n\pi$, where $n\in\mathbb{Z}$ (the set of all integers).

Step3: Check the options

When $n=- 1$, $x=- \pi$ and $\sin(-\pi)=0$. So, $\csc(-\pi)=\frac{1}{\sin(-\pi)}$ is undefined. While for $x =-\frac{\pi}{3}$, $\sin(-\frac{\pi}{3})=-\frac{\sqrt{3}}{2}\neq0$; for $x=\frac{\pi}{4}$, $\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\neq0$; for $x = \frac{\pi}{2}$, $\sin(\frac{\pi}{2}) = 1\neq0$.

Answer:

$x =-\pi$