To determine where [tex]\(\tan \theta\)[/tex] is undefined on the unit circle for [tex]\(0 \leq \theta \leq 2\pi\)[/tex], we need to understand the definition of the tangent function in terms of sine and cosine:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
For the tangent function to be undefined, the denominator (the cosine of the angle) must be zero. Thus, we need to find the values of [tex]\(\theta\)[/tex] for which [tex]\(\cos \theta = 0\)[/tex].
On the unit circle, the cosine of an angle is the x-coordinate of the corresponding point. The cosine function is zero at two specific angles within one full rotation [tex]\([0, 2\pi]\)[/tex]:
1. [tex]\(\theta = \frac{\pi}{2}\)[/tex]
2. [tex]\(\theta = \frac{3\pi}{2}\)[/tex]
At these angles, the sine function [tex]\(\sin \theta\)[/tex] is either [tex]\(1\)[/tex] or [tex]\(-1\)[/tex] respectively, but the cosine function [tex]\(\cos \theta\)[/tex] is zero, making the tangent function [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex] undefined.
Therefore, 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]
So, the correct answer is [tex]\(\theta=\frac{\pi}{2}\)[/tex] and [tex]\(\theta=\frac{3\pi}{2}\)[/tex].
