To determine where the tangent function [tex]\(\tan(\theta)\)[/tex] is undefined, we need to consider the trigonometric properties of the tangent function.
The tangent of an angle [tex]\(\theta\)[/tex] is defined as:
[tex]\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\][/tex]
Since the denominator [tex]\(\cos(\theta)\)[/tex] cannot be zero (as division by zero is undefined), [tex]\(\tan(\theta)\)[/tex] will be undefined wherever [tex]\(\cos(\theta) = 0\)[/tex].
The values of [tex]\(\theta\)[/tex] where [tex]\(\cos(\theta) = 0\)[/tex] are at odd multiples of [tex]\(\frac{\pi}{2}\)[/tex]. These values are:
[tex]\[
\theta = \left(\frac{\pi}{2} + k\pi\right) \quad \text{for integers } k
\][/tex]
Now, let's check each of the given options:
A. [tex]\(\frac{3\pi}{2}\)[/tex]
[tex]\[
\cos\left(\frac{3\pi}{2}\right) = 0
\][/tex]
Since [tex]\(\cos\left(\frac{3\pi}{2}\right) = 0\)[/tex], [tex]\(\tan\left(\frac{3\pi}{2}\right)\)[/tex] is undefined.
B. [tex]\(\frac{\pi}{2}\)[/tex]
[tex]\[
\cos\left(\frac{\pi}{2}\right) = 0
\][/tex]
Since [tex]\(\cos\left(\frac{\pi}{2}\right) = 0\)[/tex], [tex]\(\tan\left(\frac{\pi}{2}\right)\)[/tex] is undefined.
C. [tex]\(\pi\)[/tex]
[tex]\[
\cos(\pi) = -1
\][/tex]
Since [tex]\(\cos(\pi) \neq 0\)[/tex], [tex]\(\tan(\pi)\)[/tex] is defined.
D. [tex]\(0\)[/tex]
[tex]\[
\cos(0) = 1
\][/tex]
Since [tex]\(\cos(0) \neq 0\)[/tex], [tex]\(\tan(0)\)[/tex] is defined.
Therefore, [tex]\(\tan(\theta)\)[/tex] is undefined for:
- A. [tex]\(\frac{3\pi}{2}\)[/tex]
- B. [tex]\(\frac{\pi}{2}\)[/tex]
The correct choices are:
- A. [tex]\(\frac{3\pi}{2}\)[/tex]
- B. [tex]\(\frac{\pi}{2}\)[/tex]
