Question

A positive acute angle is divided into two parts whose tangents are $\frac{1}{2}$ and $\frac{1}{3}$. Then, the angle is

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{5}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{6}$

Answer 1

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

According to the question,
$\theta ={\tan }^{-1}\frac{1}{2}+{\tan }^{-1}\frac{1}{3}$
$={\tan }^{-1}\left(\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2}\cdot \frac{1}{3}}\right)$
$={\tan }^{-1}\left(\frac{\frac{5}{6}}{\frac{5}{6}}\right)={\tan }^{-1}\left(1\right)$
$=\frac{\pi }{4}$