On the unit circle, for $0<\theta <\pi$, when is $tan\left(\theta \right)$ undefined?

• A. When $\theta =\frac{\pi }{2}$
• B. When $\theta =\pi$
• C. When $\theta =\frac{\pi }{4}$
• D. When $\theta =0$

Answer: (A)

Verified step by step guidance

1. Recall that on the unit circle, the tangent of an angle \( \theta \) is defined as the ratio of the sine to the cosine: \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
2. Understand that tangent is undefined when the denominator of this ratio, \( \cos(\theta) \), is zero because division by zero is undefined.
3. Identify the values of \( \theta \) between 0 and \( \pi \) where \( \cos(\theta) = 0 \). On the unit circle, \( \cos(\theta) = 0 \) at \( \theta = \frac{\pi}{2} \).
4. Note that at \( \theta = \pi \), \( \cos(\pi) = -1 \), which is not zero, so tangent is defined there.
5. Conclude that within the interval \( 0 < \theta < \pi \), \( \tan(\theta) \) is undefined only when \( \theta = \frac{\pi}{2} \).