---
tag: 065XW2V6C3
kind: definition
name: Arctangent
msc: [33B10]
defines: 0CE768XNVP
uses: [06T0HS14PP, 07BY47HG5S]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    locator: Principal values
---

## Definition

The arctangent is the function $\arctan \colon \mathbb{R} \to (-\pi/2, \pi/2)$ such that, for each real $x$, $\arctan x$ is the unique $y \in (-\pi/2, \pi/2)$ with $\tan y = x$.

## Remarks

The [[tangent|06T0HS14PP]] is strictly increasing and continuous on $(-\pi/2, \pi/2)$, with range $\mathbb{R}$. So $\arctan$ is well defined and [[continuous|07BY47HG5S]].
For example, $\arctan 0 = 0$ and $\arctan 1 = \pi/4$.
Also $\arctan x \to \pi/2$ as $x \to \infty$, and $\arctan x \to -\pi/2$ as $x \to -\infty$. Mathlib defines it as `Real.arctan`.

## Explanations

### L1 human

The arctangent of $x$ is the angle between $-\pi/2$ and $\pi/2$ whose tangent is $x$.
It is the angle of a line with slope $x$.