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)

$\frac{\pi }{4}$

Answer 2

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