3.Trigonometrical Ratios, Functions and Identities
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Let $A, B$ and $C$ are the angles of a plain triangle and $\tan \frac{A}{2} = \frac{1}{3},\,\,\tan \frac{B}{2} = \frac{2}{3}$. Then $\tan \frac{C}{2}$ is equal to

A

$7/9$

B

$2/9$

C

$1/3$

D

$2/3$

Solution

(a) $A + B + C = \pi $ 

$\therefore \,\,\,\tan \left( {\frac{{A + B}}{2}} \right) = \tan \left( {\frac{\pi }{2} – \frac{C}{2}} \right)$ 

==> $\frac{{\tan \frac{A}{2} + \tan \frac{B}{2}}}{{1 – \tan \frac{A}{2}.\tan \frac{B}{2}}} = \cot \frac{C}{2} $

$\Rightarrow \frac{{\frac{1}{3} + \frac{2}{3}}}{{1 – \frac{1}{3}.\frac{2}{3}}} = \frac{9}{7} = \cot \frac{C}{2}$ 

$\therefore  \,\,  \tan \frac{C}{2} = \frac{7}{9}$.

Standard 11
Mathematics

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