Question

In a $△ABC,a=5,b=4$ and $\tan \frac{C}{2}=\sqrt{\frac{7}{9}}$, then $c=?$

• A. $6$
• B. $3$
• C. $2$
• D. $5$

Answer 1

The correct option is A $6$

Since,

$\cos 2\theta =\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }\Rightarrow \cos C=\frac{1-{\tan }^2\frac{C}{2}}{1+{\tan }^2\frac{C}{2}}=\frac{1-\frac{7}{9}}{1+\frac{7}{9}}=\frac{1}{8}$

Using cosine rule,

$c^2=a^2+b^2-2ab\cos C=25+16-2\times 4\times 5\times \frac{1}{8}=36c=\sqrt{36}=6$

Therefore, Answer is $A$