To determine where the secant function, [tex]\(\sec \theta\)[/tex], is undefined, we need to look at the definition of [tex]\(\sec \theta\)[/tex]. The secant function is defined as:
[tex]\[
\sec \theta = \frac{1}{\cos \theta}
\][/tex]
This function is undefined whenever the denominator, [tex]\(\cos \theta\)[/tex], is equal to 0, because division by zero is undefined. Hence, we need to find the values of [tex]\(\theta\)[/tex] where [tex]\(\cos \theta = 0\)[/tex].
Let's examine each of the given angles:
A. [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]\(\sec \left(\frac{\pi}{2}\right) = \frac{1}{0}\)[/tex], which is undefined.
B. [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]\(\sec \left(\frac{3\pi}{2}\right) = \frac{1}{0}\)[/tex], which is also undefined.
C. [tex]\(\theta = 0\)[/tex]
[tex]\[
\cos(0) = 1
\][/tex]
Since [tex]\(\cos(0) = 1\)[/tex], [tex]\(\sec(0) = \frac{1}{1} = 1\)[/tex], which is defined.
D. [tex]\(\theta = \pi\)[/tex]
[tex]\[
\cos(\pi) = -1
\][/tex]
Since [tex]\(\cos(\pi) = -1\)[/tex], [tex]\(\sec(\pi) = \frac{1}{-1} = -1\)[/tex], which is also defined.
After analyzing these angles, we conclude that [tex]\(\sec \theta\)[/tex] is undefined for:
- A. [tex]\(\frac{\pi}{2}\)[/tex]
- B. [tex]\(\frac{3 \pi}{2}\)[/tex]
Thus, the correct answers are:
A. [tex]\(\frac{\pi}{2}\)[/tex]
B. [tex]\(\frac{3 \pi}{2}\)[/tex]
