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, ${\tan }^{-1}2+{\tan }^{-1}3+\theta ={180}^o$
$\Rightarrow$ ${\tan }^{-1}\left(\frac{2+3}{1-2\times 3}\right)={180}^o-\theta$
$\Rightarrow$ $\frac{5}{-5}=\tan ({180}^o-\theta )=-\tan \theta$
$\Rightarrow$ $\tan \theta =1=\tan \frac{\pi }{4}$
$\Rightarrow$ $\theta =\frac{\pi }{4}$