Question

The perimeter of a $\Delta$ABC is $3$ times the arithmetic mean of the sine of its angles.

If the side $b=1$, then the angle $B$ is equal to?

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. None of these

Answer 1

The correct option is C $\frac{\pi }{2}$

Applying sine rule we get
$\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$

Now $b=1$

Hence $a=\frac{sinA}{sinB}$

$c=\frac{sinC}{sinB}$

Now $a+b+c=sinA+sinB+sinC$

Or $\frac{sinA}{sinB}+1+\frac{sinC}{sinB}=sinA+sinB+sinC$

$\frac{1}{sinB}(sinA+sinB+sinC)=sinA+sinB+sinC$

Or $sinB=1$

Or $B=si{n}^{-1}\left(1\right)$
Or $B=\frac{\pi }{2}$.