Question
If \( 0 \leq \theta<2 \pi \), for what angles, \( \theta \), if any, is \( \tan \theta \) undefined?
Ask by Haynes Kirk. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \tan \theta \) is undefined at \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \) within the interval \( 0 \leq \theta < 2\pi \).
Solution
The tangent function, \( \tan \theta \), is defined as:
\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]
For \( \tan \theta \) to be undefined, the denominator must be zero:
\[
\cos \theta = 0
\]
Within the interval \( 0 \leq \theta < 2\pi \), the cosine function equals zero at:
\[
\theta = \frac{\pi}{2} \quad \text{and} \quad \theta = \frac{3\pi}{2}
\]
At these angles, the tangent function is undefined because it would require division by zero.
**Answer:**
All angles equal to π⁄2 or to 3 π⁄2 radians. In symbols, θ = π/2 and θ = 3π/2.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
The tangent function, \( \tan \theta \), is undefined wherever the cosine function, \( \cos \theta \), is zero since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). The cosine function is zero at the angles where \( \theta \) corresponds to the odd multiples of \( \frac{\pi}{2} \). Within the interval \( 0 \leq \theta < 2 \pi \), the specific angles at which \( \tan \theta \) is undefined are: 1. \( \theta = \frac{\pi}{2} \) (90 degrees) 2. \( \theta = \frac{3\pi}{2} \) (270 degrees) Thus, \( \tan \theta \) is undefined at \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \).
