Question

The perimeter of a triangle ABC is six times the arithmetic mean of the sines of its angles. Given that side a is 1, angle A can be written in the form of $\frac{\pi }{x}$. Find the value of $x$, where $x$ is an integer.

Answer 1

Given that $a+b+c=6\times \frac{\sin A+\sin B+\sin C}{3}$

using sine rule

$2R(\sin A+\sin B+\sin C)=2(\sin A+\sin B+\sin C)\Rightarrow R=1$

Now, $\frac{a}{\sin A}=2R$
$\sin A=\frac{1}{2}(\because a=1)\Rightarrow A={30}^{\circ }$ or $\frac{\pi }{6}.$

by comparing with $\frac{\pi }{x}\Rightarrow x=6$