MCQ
If in a triangle $ABC$, $A = {\tan ^{ - 1}}2$ and $B = {\tan ^{ - 1}}3,$ then angle $C$ is equal to
- A$\pi /2$
- B$\pi /3$
- ✓$\pi /4$
- DNone of these
✓
Answer
Correct option: C.
$\pi /4$
c
(c) Given that $\angle A = {\tan ^{ - 1}}2,\;\angle B = {\tan ^{ - 1}}3$
We know that $\angle A + \angle B + \angle C = \pi $
$ \Rightarrow {\tan ^{ - 1}}2 + {\tan ^{ - 1}}3 + \angle C = \pi $
$ \Rightarrow {\tan ^{ - 1}}\left( {\frac{{2 + 3}}{{1 - 2 \times 3}}} \right) + \angle C = \pi $$ \Rightarrow {\tan ^{ - 1}}( - 1) + \angle C = \pi $
$ \Rightarrow \frac{{3\pi }}{4} + \angle C = \pi \Rightarrow \angle C = \frac{\pi }{4}$.
(c) Given that $\angle A = {\tan ^{ - 1}}2,\;\angle B = {\tan ^{ - 1}}3$
We know that $\angle A + \angle B + \angle C = \pi $
$ \Rightarrow {\tan ^{ - 1}}2 + {\tan ^{ - 1}}3 + \angle C = \pi $
$ \Rightarrow {\tan ^{ - 1}}\left( {\frac{{2 + 3}}{{1 - 2 \times 3}}} \right) + \angle C = \pi $$ \Rightarrow {\tan ^{ - 1}}( - 1) + \angle C = \pi $
$ \Rightarrow \frac{{3\pi }}{4} + \angle C = \pi \Rightarrow \angle C = \frac{\pi }{4}$.
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