Question

on the unit circle, where ( 0 < \theta leq 2pi ), when is ( \tan \theta ) undefined?

( \theta = pi ) and ( \theta = 2pi )
( sin \theta = cos \theta )
( \theta = \frac{pi}{2} ) and ( \theta = \frac{3pi}{2} )
( sin \theta = \frac{1}{cos \theta} )

Explanation:

Step1: Recall tanθ definition

$\tan\theta = \frac{\sin\theta}{\cos\theta}$, undefined when $\cos\theta=0$.

Step2: Find θ where cosθ=0

On unit circle, $\cos\theta=0$ at $\theta=\frac{\pi}{2}$ and $\theta=\frac{3\pi}{2}$ (0<θ≤2π).

Step3: Match with options

Check options: third option matches the valid θ values.

Answer:

$\theta=\frac{\pi}{2}$ and $\theta=\frac{3\pi}{2}$