To determine which of the following values is an asymptote of [tex]\( y = \csc(x) \)[/tex], we must consider the behavior of [tex]\( y = \csc(x) \)[/tex], which is defined as [tex]\( \csc(x) = \frac{1}{\sin(x)} \)[/tex]. For [tex]\( y = \csc(x) \)[/tex] to have a vertical asymptote, the denominator [tex]\(\sin(x)\)[/tex] must be equal to zero since the cosecant function is undefined when [tex]\(\sin(x) = 0\)[/tex].
The sine function, [tex]\(\sin(x)\)[/tex], equals zero at integer multiples of [tex]\(\pi\)[/tex]:
[tex]\[ x = k\pi \quad \text{where } k \text{ is an integer} \][/tex]
Now, let's check each given value to see if [tex]\(\sin(x) = 0\)[/tex]:
1. [tex]\( x = -\pi \)[/tex]
[tex]\[ \sin(-\pi) = 0 \][/tex]
Since [tex]\(\sin(-\pi) = 0\)[/tex], there is a vertical asymptote at [tex]\( x = -\pi \)[/tex].
2. [tex]\( x = -\frac{\pi}{3} \)[/tex]
[tex]\[ \sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \][/tex]
Since [tex]\(\sin\left(-\frac{\pi}{3}\right) \neq 0\)[/tex], there is no vertical asymptote at [tex]\( x = -\frac{\pi}{3} \)[/tex].
3. [tex]\( x = \frac{\pi}{4} \)[/tex]
[tex]\[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \][/tex]
Since [tex]\(\sin\left(\frac{\pi}{4}\right) \neq 0\)[/tex], there is no vertical asymptote at [tex]\( x = \frac{\pi}{4} \)[/tex].
4. [tex]\( x = \frac{\pi}{2} \)[/tex]
[tex]\[ \sin\left(\frac{\pi}{2}\right) = 1 \][/tex]
Since [tex]\(\sin\left(\frac{\pi}{2}\right) \neq 0\)[/tex], there is no vertical asymptote at [tex]\( x = \frac{\pi}{2} \)[/tex].
Based on our analysis, the only value among the options given that corresponds to a vertical asymptote for [tex]\( y = \csc(x) \)[/tex] is:
[tex]\[ x = -\pi \][/tex]
