# How do you evaluate arctan(1/2)+arctan(1/3)?

Jul 21, 2016

By using the formula for the tangent of a sum, we find that the given sum is just $\setminus \frac{\pi}{4}$.

#### Explanation:

Let $\tan \setminus {\theta}_{1} = \frac{1}{2} , \tan \setminus {\theta}_{2} = \frac{1}{3}$. From the formula for the tangent of a sum:$\tan \left(\setminus {\theta}_{1} + \setminus {\theta}_{2}\right) = \frac{\tan \setminus {\theta}_{1} + \tan \setminus {\theta}_{2}}{1 - \tan \setminus {\theta}_{1} \tan \setminus {\theta}_{2}}$
$= \frac{\frac{1}{2} + \frac{1}{3}}{1 - \left(\frac{1}{2}\right) \left(\frac{1}{3}\right)} = 1$

Since each individual term ${\theta}_{1} , \setminus {\theta}_{2}$ is between $0$ and $\setminus \frac{\pi}{4}$, their sum must be between $0$ and $\setminus \frac{\pi}{2}$. So then:

$\setminus {\theta}_{1} + \setminus {\theta}_{2} = \arctan \left(1\right) = \setminus \frac{\pi}{4}$.