Question

In a $△ABC,2s=$ perimeter and $R$ circum-radius . then $\frac{s}{R}$ is equal to-

• A. $\sin A+\sin B+\sin C$
• B. $\cos A+\cos B+\cos C$
• C. $\sin \frac{A}{2}+\sin \frac{B}{2}+\sin \frac{C}{2}$
• D. $Noneofthese$

Answer 1

The correct option is A $\sin A+\sin B+\sin C$

U\sin g \sin e rule we can say that,

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$

Therefore,

$\Rightarrow a=2R\sin A$

$\Rightarrow b=2R\sin B$

$\Rightarrow c=2R\sin C$

we know that in $△ABC,a+b+c=2s$ ....$\left(1\right)$

Substituting the values of $a,b,c$ in $\left(1\right)$ we get,

$\Rightarrow 2R\sin A+2R\sin B+2R\sin C=2s$

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

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

$\Rightarrow \sin A+\sin B+\sin C=\frac{s}{R}$

Therefore, $\Rightarrow \frac{s}{R}=\sin A+\sin B+\sin C$