To determine for which value of [tex]\(\theta\)[/tex] the cosecant function, [tex]\(\csc(\theta)\)[/tex], is undefined, let's first recall the definition of cosecant. The cosecant of an angle [tex]\(\theta\)[/tex] is defined as:
[tex]\[
\csc(\theta) = \frac{1}{\sin(\theta)}
\][/tex]
This means that [tex]\(\csc(\theta)\)[/tex] is undefined whenever [tex]\(\sin(\theta) = 0\)[/tex], since division by zero is undefined.
Next, we need to identify for which values of [tex]\(\theta\)[/tex] the sine function equals zero. The sine function, [tex]\(\sin(\theta)\)[/tex], equals zero at integer multiples of [tex]\(\pi\)[/tex]:
[tex]\[
\sin(\theta) = 0 \quad \text{if} \quad \theta = k\pi \quad \text{for} \quad k \in \mathbb{Z}
\][/tex]
Now, let's check each given option to see if [tex]\(\sin(\theta)\)[/tex] equals zero:
1. [tex]\(\theta = 0\)[/tex]:
[tex]\[
\sin(0) = 0
\][/tex]
Therefore, [tex]\(\csc(0)\)[/tex] is undefined.
2. [tex]\(\theta = \frac{\pi}{2}\)[/tex]:
[tex]\[
\sin\left(\frac{\pi}{2}\right) = 1
\][/tex]
Therefore, [tex]\(\csc\left(\frac{\pi}{2}\right) = \frac{1}{1} = 1\)[/tex], so it is defined.
3. [tex]\(\theta = \frac{3\pi}{2}\)[/tex]:
[tex]\[
\sin\left(\frac{3\pi}{2}\right) = -1
\][/tex]
Therefore, [tex]\(\csc\left(\frac{3\pi}{2}\right) = \frac{1}{-1} = -1\)[/tex], so it is defined.
4. [tex]\(\theta = \frac{7\pi}{2}\)[/tex]:
[tex]\[
\sin\left(\frac{7\pi}{2}\right) = 1
\][/tex]
Therefore, [tex]\(\csc\left(\frac{7\pi}{2}\right) = \frac{1}{1} = 1\)[/tex], so it is defined.
Based on our evaluations, the only value of [tex]\(\theta\)[/tex] for which [tex]\(\csc(\theta)\)[/tex] is undefined is:
[tex]\[
\theta = 0
\][/tex]
