To determine where the secant of [tex]\(\theta\)[/tex] is undefined, we need to understand the relationship between secant and cosine. The secant function is the reciprocal of the cosine function:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \][/tex]
Therefore, [tex]\(\sec \theta\)[/tex] will be undefined wherever [tex]\(\cos \theta = 0\)[/tex], since division by zero is undefined.
Now, let's look at each given value of [tex]\(\theta\)[/tex] to determine where [tex]\(\cos \theta = 0\)[/tex]:
1. For [tex]\(\theta = \pi\)[/tex]:
[tex]\[ \cos(\pi) = -1 \][/tex]
Since [tex]\(\cos(\pi) \neq 0\)[/tex], [tex]\(\sec(\pi)\)[/tex] is defined.
2. For [tex]\(\theta = 0\)[/tex]:
[tex]\[ \cos(0) = 1 \][/tex]
Since [tex]\(\cos(0) \neq 0\)[/tex], [tex]\(\sec(0)\)[/tex] is defined.
3. For [tex]\(\theta = \frac{3\pi}{2}\)[/tex]:
[tex]\[ \cos\left(\frac{3\pi}{2}\right) = 0 \][/tex]
Since [tex]\(\cos\left(\frac{3\pi}{2}\right) = 0\)[/tex], [tex]\(\sec\left(\frac{3\pi}{2}\right)\)[/tex] is undefined.
4. For [tex]\(\theta = \frac{\pi}{2}\)[/tex]:
[tex]\[ \cos\left(\frac{\pi}{2}\right) = 0 \][/tex]
Since [tex]\(\cos\left(\frac{\pi}{2}\right) = 0\)[/tex], [tex]\(\sec\left(\frac{\pi}{2}\right)\)[/tex] is undefined.
Based on these evaluations, [tex]\(\sec(\theta)\)[/tex] is undefined for:
[tex]\[ \theta = \frac{3\pi}{2} \quad \text{and} \quad \theta = \frac{\pi}{2} \][/tex]
Therefore, the correct answers are:
C. [tex]\(\frac{3\pi}{2}\)[/tex]
D. [tex]\(\frac{\pi}{2}\)[/tex]
