Question

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

• A. ${30}^o$
• B. ${45}^o$
• C. ${60}^o$
• D. ${75}^o$

Answer 1

Answer: (B)

${45}^o$

In a $△$ Sum of 3 angle $={180}^0$
Let third angle $={\tan }^{-1}\theta$
${\tan }^{-1}3+{\tan }^{-1}2+{\tan }^{-1}\theta =\pi$
${\tan }^{-1}a+{\tan }^{-1}b={\tan }^{-1}\frac{(a+b)}{(1-ab)}$
So ${\tan }^{-1}(\frac{3+2}{1-3.2})+{\tan }^{-1}\theta =\pi$
${\tan }^{-1}(\frac{5}{-5})+{\tan }^{-1}\theta =\pi$
${\tan }^{-1}(-1)+{\tan }^{-1}\theta =\pi$
We know that ${\tan }^{-1}(-1)={135}^0$or$\frac{3\pi }{4}$
So,
=>$\frac{3\pi }{4}+{\tan }^{-1}(\theta )=\pi$
=>${\tan }^{-1}(\theta )=\pi -\frac{3\pi }{4}=\frac{\pi }{4}$
So, third angle $={\tan }^{-1}(\theta )={45}^0$ or $\frac{\pi }{4}$