Question

If $ {{\tan }^{-1}}\,2 $ and $ {{\tan }^{-1}}\,3 $ are two angles of a triangle, then the third angle is

• A. $ \frac{\pi }{2} $
• B. $ \frac{\pi }{3} $
• C. $ \frac{\pi }{4} $
• D. $ \frac{\pi }{6} $

Answer 1

Answer: (C)

Given two angles are
$ {{\tan }^{-1}}2 $ and $ {{\tan }^{-1}}3. $ Let third angle be $ \theta , $
then, $ {{\tan }^{-1}}2+{{\tan }^{-1}}3+\theta ={{180}^{o}} $
$ \Rightarrow $ $ {{\tan }^{-1}}\left( \frac{2+3}{1-2\times 3} \right)={{180}^{o}}-\theta $
$ \Rightarrow $ $ \frac{5}{-5}=\tan ({{180}^{o}}-\theta )=-\tan \theta $
$ \Rightarrow $ $ \tan \theta =1=\tan \frac{\pi }{4} $
$ \Rightarrow $ $ \theta =\frac{\pi }{4} $