\(\frac{x}{\sin 2 \theta}=\frac{r}{\sin (90-\theta)}\)

\(\Rightarrow x=2 r \sin \theta\)

\(\therefore \frac{d x}{d t}=2 r \cos \theta \times \frac{d \theta}{d t}\)

\(\frac{d \theta}{d t}=\frac{d x / d t}{2 r \cos \theta}=\frac{v_0}{2 r \cos 60^{\circ}}=\frac{v_0}{r}\)