Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively, then $\sin \frac{A}{2}$ is?

• A. less then $\frac{b+c}{a}$
• B. less than or equal to $\frac{a}{b+c}$
• C. greater than $\frac{2a}{a+b+c}$
• D. none of these

Answer 1

The correct option is B less than or equal to $\frac{a}{b+c}$

$\frac{a}{b+c}=\frac{\sin A}{\sin B+\sin C}=\frac{2\sin \frac{A}{2}\cos \frac{A}{2}}{2\sin \frac{B+C}{2}\cos \frac{B-C}{2}}=\frac{\sin \frac{A}{2}}{\cos \frac{B-C}{2}}$

$\sin \frac{A}{2}=\frac{a}{b+c}\cos \frac{B-C}{2}$

$\cos \frac{B-C}{2}\le 1$

$\Rightarrow \sin \frac{A}{2}\le \frac{a}{b+c}$

So option $B$ is correct.