A variable triangle is inscribed in a circle of radius R. If the rate of change of a side is R times the rate of change of the opposite angle, then that angle, is

$\pi / 3$

Let the side be BC = a and A be the opposite angle.

Then,

$R =\frac{a}{2 \sin A}$

$\Rightarrow a =2 R \sin A$

$\Rightarrow \frac{d a}{d t} =2 R \cos A \frac{d A}{d t}$

$\Rightarrow R \frac{d A}{d t} =2 R \cos A . \frac{d A}{d t}$          [∵ $\frac{d a}{d t}=R \frac{d A}{d t}$ (given)]

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