To determine when [tex]\(\tan \theta\)[/tex] is undefined on the unit circle for [tex]\(0 < \theta \leq 2\pi\)[/tex], it's important to understand the definition of the tangent function in terms of the sine and cosine functions. The tangent function is given by:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
The tangent function will be undefined whenever the denominator is zero, which occurs when [tex]\(\cos \theta = 0\)[/tex].
On the unit circle, the cosine of an angle equals zero at specific points. Let's identify these points within the interval [tex]\(0 < \theta \leq 2\pi\)[/tex]:
1. The cosine of [tex]\(\theta\)[/tex] is zero at [tex]\(\theta = \frac{\pi}{2}\)[/tex] because [tex]\(\cos \left(\frac{\pi}{2}\right) = 0\)[/tex].
2. Similarly, the cosine of [tex]\(\theta\)[/tex] is also zero at [tex]\(\theta = \frac{3\pi}{2}\)[/tex] because [tex]\(\cos \left(\frac{3\pi}{2}\right) = 0\)[/tex].
Hence, [tex]\(\tan \theta\)[/tex] is undefined at these points. Therefore, the values of [tex]\(\theta\)[/tex] 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]
Thus, the correct option is:
[tex]\[
\theta = \frac{\pi}{2} \quad \text{and} \quad \theta = \frac{3\pi}{2}
\][/tex]
