Question

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 the opposite angle is

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is C $\frac{\pi }{3}$

Let $ABC$ be a variable triangle inscribed in a circle of radius $R$.

Let the rate of change of a side is $R$ times the rate of change of the opposite angle.

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

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

$\Rightarrow a=2R\sin A$

Differentiate both sides with respect to $t$, we get

$\frac{da}{dt}=2R\cos A\frac{dA}{dt}$

$\Rightarrow R\frac{dA}{dt}=2R\cos A\frac{dA}{dt}$

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

$\Rightarrow A={\cos }^{-1}\left(\frac{1}{2}\right)$

$\Rightarrow A=\frac{\pi }{3}$

Hence, the opposite angle is $\frac{\pi }{3}$.