To solve the equation [tex]\(\cot \theta = 0\)[/tex], let's go through the steps in detail:
### Understanding [tex]\(\cot \theta\)[/tex]
1. Definition of [tex]\(\cot \theta\)[/tex]:
[tex]\(\cot \theta\)[/tex] is defined as the reciprocal of the tangent function: [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex].
2. When is [tex]\(\cot \theta = 0\)[/tex]?:
Since [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex], the cotangent function equals zero when the tangent function is undefined (which happens when the tangent function approaches infinity).
### Finding [tex]\(\theta\)[/tex]
3. Behavior of [tex]\(\tan \theta\)[/tex]:
The tangent function, [tex]\(\tan \theta\)[/tex], approaches infinity at odd multiples of [tex]\(\frac{\pi}{2}\)[/tex]. This occurs because the tangent function has vertical asymptotes at these points.
4. Key Values for [tex]\(\theta\)[/tex]:
Therefore, [tex]\(\tan \theta\)[/tex] is undefined at:
[tex]\[
\theta = \frac{\pi}{2} + k\pi
\][/tex]
where [tex]\(k\)[/tex] is any integer. These are the points where [tex]\(\cot \theta = 0\)[/tex].
### Conclusion:
5. General Solution:
The solutions to the equation [tex]\(\cot \theta = 0\)[/tex] are given by:
[tex]\[
\theta = \frac{\pi}{2} + k\pi \quad \text{where } k \text{ is any integer}
\][/tex]
In conclusion, the set of all possible solutions to the equation [tex]\(\cot \theta = 0\)[/tex] can be expressed as:
[tex]\[
\theta = \frac{\pi}{2} + k\pi \quad \text{where } k \text{ is any integer}
\][/tex]
This means that [tex]\(\theta\)[/tex] can be [tex]\(\frac{\pi}{2}\)[/tex], [tex]\(\frac{3\pi}{2}\)[/tex], [tex]\(\frac{5\pi}{2}\)[/tex], [tex]\(-\frac{\pi}{2}\)[/tex], and so on.
