Question

In $△ABC$, if $a=13,b=14$ and $c=15$, then $\sin \left(\frac{A}{2}\right)=?$

• A. $\frac{1}{5}$
• B. $\sqrt{\frac{1}{5}}$
• C. $\frac{4}{5}$
• D. $\frac{2}{5}$

Answer 1

The correct option is B $\sqrt{\frac{1}{5}}$

Using cosine rule, $\cos A=\frac{b^2+c^2-a^2}{2bc}=\frac{{14}^2+{15}^2-{13}^2}{2\cdot 14\cdot 15}=\frac{{14}^2+(15+13)(15-13)}{2\cdot 14\cdot 15}=\frac{{14}^2+14\cdot 4}{2\cdot 14\cdot 15}=\frac{18}{30}=\frac{3}{5}$

Now using double angle formula, $\cos A=1-2{\sin }^2\frac{A}{2}$

$\Rightarrow 2{\sin }^2\frac{A}{2}=1-\cos A=1-\frac{3}{5}=\frac{2}{5}$

$\Rightarrow {\sin }^2\frac{A}{2}=\frac{1}{5}\therefore \sin \frac{A}{2}=\sqrt{\frac{1}{5}}$