Question

The perimeter of a $△ABC$ is $6$ times the arithmetic mean of the sines of its angles.

If the side $a$ is $1$, then the angle $A$ is?

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

Answer 1

The correct option is A $\frac{\pi }{6}$

We have, $a+b+c=\frac{6(\sin A+\sin B+\sin C)}{3}$

$\Rightarrow k(\sin A+\sin B+\sin C)=2(\sin A+\sin B+\sin C)$

where $k=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

$\Rightarrow k=2[\because \sin A+\sin B+\sin C\ne 0]$

$\Rightarrow \frac{a}{\sin A}=2\Rightarrow \sin A=\frac{1}{2}\Rightarrow A=\frac{\pi }{6}$