Question

In $\Delta ABC$ if $b=24,c=20$ and $\sin \left(\frac{A}{2}\right)=\frac{1}{\sqrt{8}}$, find $a$.

Answer 1

$\sin \left(\frac{A}{2}\right)=\frac{1}{\sqrt{8}}$

From triangle $\cos \left(\frac{A}{2}\right)=\frac{\sqrt{7}}{\sqrt{8}}$

$\sin \left(A\right)=2\sin \left(\frac{A}{2}\right)\cos \left(\frac{A}{2}\right)$

$=2\times \frac{1}{\sqrt{8}}\times \frac{\sqrt{7}}{\sqrt{8}}=\frac{\sqrt{7}}{4}$

${\cos }^{2}A+{\sin }^{2}A=1$
${\cos }^{2}A=1-{\sin }^{2}A=1-\frac{7}{16}=\frac{9}{16}$

$\cos A=\frac{3}{4}$

By cosine rule $\cos A=\frac{b^2+c^2-a^2}{2bc}$

Substituting $\cos A,b,c$

$\frac{3}{4}=\frac{(24{)}^2+(20{)}^2-a^2}{2\left(24\right)\left(20\right)}$

$a^2=256a=16$