In a triangle, if $b=5, c=6, an \frac{A}{2}=\frac{1}{\sqrt{2}}$, then $a=$

Question

In a triangle, if $b=5, c=6, 	an \frac{A}{2}=\frac{1}{\sqrt{2}}$, then $a=$

Aicharya · Accepted Answer

$\because \cos A=\frac{1-	an ^2 \frac{A}{2}}{1+	an ^2 \frac{A}{2}}$ $=\frac{1-\left(\frac{1}{\sqrt{2}}ight)^2}{1+\left(\frac{1}{\sqrt{2}}ight)^2}=\frac{1}{3}$ $\cos A=\frac{b^2+c^2-a^2}{2 b c}$ $\frac{1}{3}=\frac{(5)^2+(6)^2-a^2}{2(5)(6)}$ $\begin{aligned} & 20=25+36-a^2 \ & a^2=41 \ & a=\sqrt{41}\end{aligned}$

In a triangle, if $b=5, c=6, \tan \frac{A}{2}=\frac{1}{\sqrt{2}}$, then $a=$

Solution