Question

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

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

Answer 1

The correct option is A $\frac{\pi }{4}$

Given two angles are ${\tan }^{-1}\left(2\right)$ and ${\tan }^{-1}\left(3\right)$. Now, $\left(2\right)\left(3\right)>1$
$\Rightarrow {\tan }^{-1}\left(2\right)+{\tan }^{-1}\left(3\right)=\pi +{\tan }^{-1}\frac{2+3}{1-2\times 3}$
$=\pi +{\tan }^{-1}(-1)=\pi -\frac{\pi }{4}=\frac{3\pi }{4}$
Hence the third angle is $\pi -\frac{3\pi }{4}=\frac{\pi }{4}$