To determine which set of points corresponds to the vertical asymptotes for the function [tex]\( y = \csc(x) \)[/tex], we need to understand under what conditions the cosecant function, [tex]\( \csc(x) = \frac{1}{\sin(x)} \)[/tex], is undefined. This occurs where [tex]\( \sin(x) \)[/tex] is zero because division by zero is undefined.
Mathematically, [tex]\(\sin(x) = 0\)[/tex] at:
[tex]\[ x = n\pi \quad \text{where} \quad n \in \mathbb{Z} \][/tex]
We'll check each set of points to see if all values in each set are such that [tex]\(\sin(x) = 0\)[/tex].
1. [tex]\(-\pi, \frac{\pi}{3}, \frac{5\pi}{3}\)[/tex]
- [tex]\(\sin(-\pi) = 0\)[/tex]
- [tex]\(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \neq 0\)[/tex]
- [tex]\(\sin\left(\frac{5\pi}{3}\right) = -\frac{\sqrt{3}}{2} \neq 0\)[/tex]
Since [tex]\(\sin\left(\frac{\pi}{3}\right) \neq 0\)[/tex] and [tex]\(\sin\left(\frac{5\pi}{3}\right) \neq 0\)[/tex], this set does not contain only vertical asymptotes.
2. [tex]\(-\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}\)[/tex]
- [tex]\(\sin\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \neq 0\)[/tex]
- [tex]\(\sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \neq 0\)[/tex]
- [tex]\(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \neq 0\)[/tex]
Since none of these values of [tex]\(x\)[/tex] yield [tex]\(\sin(x) = 0\)[/tex], this set does not contain vertical asymptotes.
3. [tex]\(-\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}\)[/tex]
- [tex]\(\sin\left(-\frac{\pi}{2}\right) = -1 \neq 0\)[/tex]
- [tex]\(\sin\left(\frac{\pi}{2}\right) = 1 \neq 0\)[/tex]
- [tex]\(\sin\left(\frac{3\pi}{2}\right) = -1 \neq 0\)[/tex]
Since none of these values of [tex]\(x\)[/tex] yield [tex]\(\sin(x) = 0\)[/tex], this set does not contain vertical asymptotes.
4. [tex]\(-\pi, 0, 2\pi\)[/tex]
- [tex]\(\sin(-\pi) = 0\)[/tex]
- [tex]\(\sin(0) = 0\)[/tex]
- [tex]\(\sin(2\pi) = 0\)[/tex]
Since all these values of [tex]\(x\)[/tex] yield [tex]\(\sin(x) = 0\)[/tex], this set does indeed contain vertical asymptotes.
Therefore, the correct option that lists the vertical asymptotes for [tex]\( y = \csc(x) \)[/tex] is:
[tex]\[ -\pi, 0, 2\pi \][/tex]
