Question
For which value of \( \theta \) is \( \cot (\theta) \) undefined?
Ask by Vaughn King. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\(\cot(\theta)\) is undefined when \(\theta = n\pi\) for any integer \(n\).
Solution
1. Write the definition of the cotangent function:
\[
\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
\]
2. The expression \(\cot(\theta)\) is undefined when the denominator is zero. That is, when:
\[
\sin(\theta) = 0
\]
3. The sine function equals zero for angles given by:
\[
\theta = n\pi \quad \text{where } n \text{ is an integer}
\]
Thus, \(\cot(\theta)\) is undefined for:
\[
\theta = n\pi \quad \text{for any integer } n.
\]
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Simplify this solution Mind Expander
\( \cot (\theta) \) is undefined whenever the sine function is zero because \( \cot (\theta) = \frac{\cos(\theta)}{\sin(\theta)} \). This occurs at angles where \( \theta \) is an integer multiple of \( \pi \) (or 180 degrees), specifically at \( \theta = n\pi \), where \( n \) is any integer. So, for angles like 0, \( \pi \), \( 2\pi \), etc., \( \cot (\theta) \) is undefined. Additionally, it's also essential to remember that these specific points correspond to the values of \( \theta \) where the unit circle intersects the x-axis, where the sine value is zero, leading to the cotangent being undefined.
