Answer
$\frac{2\pi}{3}, \frac{5\pi}{3}$
Work Step by Step
Using trigonometric identities, we have
$\sqrt 3cot\theta+1=0 \longrightarrow cot\theta=-\frac{\sqrt 3}{3} \longrightarrow \theta=\pi-\frac{\pi}{3},2 \pi-\frac{\pi}{3} \longrightarrow \theta=\frac{2\pi}{3}, \frac{5\pi}{3}$