Question

lf the perimeter of a triangle is six times the arithmetic mean of the sine angles and the side '$a$' is unity, then $A=$

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

Answer 1

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

Given $2s=\frac{6(\sin A+\sin B+\sin C)}{3}$ .....(1)
We have $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=2R$
Given $a=1$
$\Rightarrow \sin A=2aR,\sin B=2bR,\sin c=2cR$
Put these values in (1), we get
$(1+b+c)=2(2R+2bR+2cR)$
$\Rightarrow (1+b+c)=4R(1+b+c)$
$\Rightarrow R=\frac{1}{4}$
So, $\sin A=\frac{1}{2}$
$\Rightarrow A=\frac{\pi }{6}$