which of the following is an asymptote of $y = csc(x)$?\n$x = - pi$\n$x = - \frac{pi}{3}$\n$x =…
which of the following is an asymptote of $y = csc(x)$?\n$x = - pi$\n$x = - \frac{pi}{3}$\n$x = \frac{pi}{4}$\n$x = \frac{pi}{2}$
Answer
Explanation:
Step1: Recall the formula for (y = \csc(x))
We know that (\csc(x)=\frac{1}{\sin(x)}). The vertical asymptotes of (y = \csc(x)) occur where (\sin(x)=0).
Step2: Find the values of (x) for which (\sin(x) = 0)
The general solution of the equation (\sin(x)=0) is (x = k\pi), where (k\in\mathbb{Z}) ( (k = 0,\pm1,\pm2,\cdots) ).
Step3: Check each option
- For (x=-\pi), since (\sin(-\pi)=0), then (y = \csc(-\pi)=\frac{1}{\sin(-\pi)}) is undefined.
- For (x =-\frac{\pi}{3}), (\sin\left(-\frac{\pi}{3}\right)=-\frac{\sqrt{3}}{2}\neq0).
- For (x=\frac{\pi}{4}), (\sin\left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}\neq0).
- For (x=\frac{\pi}{2}), (\sin\left(\frac{\pi}{2}\right) = 1\neq0).
Answer:
(x =-\pi)