# How do you evaluate arctan(1)?

Aug 13, 2016

$= {45}^{\circ}$
$= \frac{\pi}{4}$

#### Explanation:

$\arctan \left(1\right)$
$= {\tan}^{-} 1 \left(1\right)$
$= {45}^{\circ}$
$= \frac{\pi}{4}$

Aug 13, 2016

$\arctan \left(1\right) = \frac{\pi}{4}$

#### Explanation:

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$\theta = \arctan \left(1\right)$ is the angle $\theta \in \left(- \frac{\pi}{2} , \frac{\pi}{2}\right)$ satisfying $\tan \left(\theta\right) = 1$

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Note that the triangle formed by bisecting a unit square diagonally is a right angled triangle with sides $1$, $1$, $\sqrt{2}$ and angles $\frac{\pi}{4}$, $\frac{\pi}{4}$ and $\frac{\pi}{2}$.

So we find:

$\tan \left(\frac{\pi}{4}\right) = \text{opposite"/"adjacent} = \frac{1}{1} = 1$

So $\theta = \frac{\pi}{4}$ satisfies $\tan \left(\theta\right) = 1$ and is in the required range.

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So:

$\arctan \left(1\right) = \frac{\pi}{4}$