IDNLearn.com makes it easy to find accurate answers to your questions. Our platform offers detailed and accurate responses from experts, helping you navigate any topic with confidence.

Which are vertical asymptotes for [tex]\( y = \csc x \)[/tex]?

A. [tex]\(-\pi, \frac{\pi}{3}, \frac{5\pi}{3}\)[/tex]
B. [tex]\(-\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}\)[/tex]
C. [tex]\(-\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}\)[/tex]
D. [tex]\(-\pi, 0, 2\pi\)[/tex]


Sagot :

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]