Question

Say true or false:

If $A,B$ and $C$ are interior angles of a triangle $ABC$, then $\sin \left(\frac{B+C}{2}\right)=\cos \frac{A}{2}$.

• A. True
• B. False

Answer 1

The correct option is A True

$A+B+C={180}^{\circ }$ ...Angle sum property of triangle
$\Rightarrow B+C={180}^{\circ }-A$
$\Rightarrow \frac{B+C}{2}=\frac{{180}^{\circ }-A}{2}\Rightarrow \frac{B+C}{2}={90}^{\circ }-\frac{A}{2}$
$\Rightarrow \sin \left(\frac{B+C}{2}\right)=\sin ({90}^{\circ }-\frac{A}{2})=\cos \frac{A}{2}[\because \sin ({90}^{\circ }-\theta )=\cos \theta ]$