To determine which of the given options correspond to an asymptote of the function [tex]\( y = \csc(x) \)[/tex], we need to recall some key properties of the cosecant function. The cosecant function is defined as the reciprocal of the sine function:
[tex]\[ \csc(x) = \frac{1}{\sin(x)} \][/tex]
Vertical asymptotes occur in the cosecant function at points where [tex]\(\sin(x) = 0\)[/tex], because the cosecant function becomes undefined at these points, causing it to approach infinity.
Therefore, we need to identify the values in the given options where [tex]\(\sin(x) = 0\)[/tex].
1. Option [tex]\( x = -\pi \)[/tex]:
[tex]\[\sin(-\pi) = 0\][/tex]
This is true because [tex]\(\sin(\theta) = 0\)[/tex] at integer multiples of [tex]\(\pi\)[/tex]. Thus, [tex]\( x = -\pi \)[/tex] is a vertical asymptote of [tex]\( y = \csc(x) \)[/tex].
2. Option [tex]\( x = -\frac{\pi}{3} \)[/tex]:
[tex]\[\sin(-\frac{\pi}{3}) = -\frac{\sqrt{3}}{2} \neq 0\][/tex]
Therefore, [tex]\( x = -\frac{\pi}{3} \)[/tex] is not a vertical asymptote.
3. Option [tex]\( x = \frac{\pi}{4} \)[/tex]:
[tex]\[\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \neq 0\][/tex]
Therefore, [tex]\( x = \frac{\pi}{4} \)[/tex] is not a vertical asymptote.
4. Option [tex]\( x = \frac{\pi}{2} \)[/tex]:
[tex]\[\sin(\frac{\pi}{2}) = 1 \neq 0\][/tex]
Therefore, [tex]\( x = \frac{\pi}{2} \)[/tex] is not a vertical asymptote.
Based on this analysis, the only value from the given options where [tex]\(\sin(x) = 0\)[/tex] and hence where [tex]\( y = \csc(x) \)[/tex] has a vertical asymptote is:
[tex]\[ x = -\pi \][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{-\pi}
\][/tex]
