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3.Trigonometrical Ratios, Functions and Identities
medium
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
