To determine when [tex]\(\tan \theta\)[/tex] is undefined on the unit circle within the interval [tex]\(0 < \theta \leq 2\pi\)[/tex], we need to consider the definition of the tangent function in terms of sine and cosine:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
The tangent function is undefined wherever the denominator [tex]\(\cos \theta\)[/tex] is zero, because division by zero is undefined. Therefore, we need to find the values of [tex]\(\theta\)[/tex] where [tex]\(\cos \theta = 0\)[/tex].
The cosine function is zero at specific points within one full cycle of the unit circle. These points occur at:
[tex]\[
\theta = \frac{\pi}{2} \quad \text{and} \quad \theta = \frac{3\pi}{2}
\][/tex]
Thus, the values of [tex]\(\theta\)[/tex] at which [tex]\(\tan \theta\)[/tex] is undefined are:
[tex]\[
\theta = \frac{\pi}{2} \quad \text{and} \quad \theta = \frac{3\pi}{2}
\][/tex]
Hence, the correct choice is:
[tex]\[
\theta = \frac{\pi}{2} \text{ and } \theta = \frac{3\pi}{2}
\][/tex]
