To determine when the tangent function [tex]\(\tan \theta\)[/tex] is undefined, we need to look for the values of [tex]\(\theta\)[/tex] where the denominator of the tangent function [tex]\(\cos \theta\)[/tex] is equal to zero. Recall that [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex], and a fraction is undefined whenever its denominator is zero.
1. The unit circle and cosine function:
- The cosine function [tex]\(\cos \theta\)[/tex] is zero at [tex]\(\theta = \frac{\pi}{2}\)[/tex] and [tex]\(\theta = \frac{3\pi}{2}\)[/tex], within the interval [tex]\(0 < \theta \leq 2\pi\)[/tex].
2. Values of [tex]\(\theta\)[/tex] where [tex]\(\cos \theta = 0\)[/tex]:
- At [tex]\(\theta = \frac{\pi}{2}\)[/tex], the cosine function [tex]\(\cos \theta\)[/tex] is zero.
- At [tex]\(\theta = \frac{3\pi}{2}\)[/tex], the cosine function [tex]\(\cos \theta\)[/tex] is zero.
Therefore, [tex]\(\tan \theta\)[/tex] is undefined exactly at [tex]\(\theta = \frac{\pi}{2}\)[/tex] and [tex]\(\theta = \frac{3\pi}{2}\)[/tex] because these are the points where [tex]\(\cos \theta = 0\)[/tex].
So, the correct answer is:
[tex]\[
\theta = \frac{\pi}{2} \text{ and } \theta = \frac{3\pi}{2}
\][/tex]
