The angle θ,

Let angle θ increase twice as fast as its sine.

It is given that,

$\frac{d \theta}{d t}=2 \frac{d}{d t}(\sin \theta)$

$\Rightarrow \frac{d \theta}{d t}=2 \cos \theta \frac{d \theta}{d t}$

$\Rightarrow 2 \cos \theta=1$

$\Rightarrow \cos \theta=\frac{1}{2}=\cos \frac{\pi}{3}$

$\Rightarrow \theta=\frac{\pi}{3}$              $\left(0<\theta<\frac{\pi}{2}\right)$

Thus, the angle $\theta$ is $\frac{\pi}{3}$.

The angle $\theta, 0<\theta<\frac{\pi}{2}$, which increases twice as fast as its sine, is $\frac{\pi}{3}$