Answer
$0,\frac{2\pi}{3}$
Work Step by Step
1. $\sqrt 3sin\theta+cos\theta=1 \Longrightarrow \frac{\sqrt 3}{2}sin\theta+\frac{1}{2}cos\theta=\frac{1}{2} \Longrightarrow cos\frac{\pi}{6}sin\theta+sin\frac{\pi}{6}cos\theta=\frac{1}{2} \Longrightarrow sin(\theta+\frac{\pi}{6})=\frac{1}{2}$
2. $\theta+\frac{\pi}{6}=2k\pi+\frac{\pi}{6}$ or $\theta+\frac{\pi}{6}=2k\pi+\frac{5\pi}{6}$
3. $\theta=2k\pi$ or $\theta=2k\pi+\frac{2\pi}{3}$
4. Within $[0,2\pi)$, we have $\theta=0,\frac{2\pi}{3}$
