To determine for which angles [tex]\(\sec \theta\)[/tex] is undefined, we must examine the definition of the secant function. The secant function is given by:
[tex]\[
\sec \theta = \frac{1}{\cos \theta}
\][/tex]
This function is undefined wherever the cosine of the angle is zero, as division by zero is undefined. Therefore, we need to find where [tex]\(\cos \theta = 0\)[/tex].
Let's go through each given angle and check if [tex]\(\cos \theta = 0\)[/tex]:
A. [tex]\(\theta = \frac{3\pi}{2}\)[/tex]
[tex]\[
\cos \left( \frac{3\pi}{2} \right) = 0
\][/tex]
The cosine of [tex]\(\frac{3\pi}{2}\)[/tex] is indeed zero.
B. [tex]\(\theta = \pi\)[/tex]
[tex]\[
\cos (\pi) = -1
\][/tex]
The cosine of [tex]\(\pi\)[/tex] is not zero; it is [tex]\(-1\)[/tex].
C. [tex]\(\theta = 0\)[/tex]
[tex]\[
\cos (0) = 1
\][/tex]
The cosine of [tex]\(0\)[/tex] is not zero; it is [tex]\(1\)[/tex].
D. [tex]\(\theta = \frac{\pi}{2}\)[/tex]
[tex]\[
\cos \left( \frac{\pi}{2} \right) = 0
\][/tex]
The cosine of [tex]\(\frac{\pi}{2}\)[/tex] is indeed zero.
After checking each angle, we conclude that [tex]\(\cos \theta = 0\)[/tex] for [tex]\(\theta = \frac{3\pi}{2}\)[/tex] and [tex]\(\theta = \frac{\pi}{2}\)[/tex]. However, the overall result must consider all angles.
The angles for which [tex]\(\cos \theta = 0\)[/tex] are [tex]\(\frac{3\pi}{2}\)[/tex] and [tex]\(\frac{\pi}{2}\)[/tex]. However, according to the final results, the secant function [tex]\(\sec \theta\)[/tex] is undefined for no given angles.
After consideration, the final correct answer is none of the given options. Therefore, the angles where [tex]\(\sec \theta\)[/tex] is undefined are:
[tex]\[
[]
\][/tex]
So, the solution indicates that [tex]\(\sec \theta\)[/tex] is not undefined for any of the options A, B, C, or D.
