To determine where [tex]\(\tan \theta\)[/tex] is undefined on the unit circle for [tex]\(0 < \theta \leq 2\pi\)[/tex], let's analyze the behavior of the tangent function.
The tangent of [tex]\(\theta\)[/tex] is defined as:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
For [tex]\(\tan \theta\)[/tex] to be undefined, the denominator of this fraction, [tex]\(\cos \theta\)[/tex], must be zero. Therefore, we need to determine where [tex]\(\cos \theta = 0\)[/tex] within the interval [tex]\(0 < \theta \leq 2\pi\)[/tex].
On the unit circle, [tex]\(\cos \theta = 0\)[/tex] at two specific points:
1. [tex]\(\theta = \frac{\pi}{2}\)[/tex]
2. [tex]\(\theta = \frac{3\pi}{2}\)[/tex]
At these angles, the cosine function is zero, which makes the tangent function undefined because division by zero is not possible.
To summarize, the locations where [tex]\(\tan \theta\)[/tex] is undefined on the unit circle for [tex]\(0 < \theta \leq 2\pi\)[/tex] are:
[tex]\[
\theta = \frac{\pi}{2} \quad \text{and} \quad \theta = \frac{3\pi}{2}
\][/tex]
Hence, the correct answer from the given choices is:
[tex]\[
\theta = \frac{\pi}{2} \quad \text{and} \quad \theta = \frac{3\pi}{2}
\][/tex]
