Inverse Tangent (Arctangent) $y = \arctan x$

$\arctan x$ (the inverse tangent, or arctangent) is the inverse of the tangent function $\tan$. Since $\tan$ has period $\pi$, we restrict it to the interval $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ on which it is increasing and define the inverse there (the principal value) as $\arctan$.

Definition

$y = \arctan x$ is the value $y$ satisfying $\tan y = x$ with $-\dfrac{\pi}{2} < y < \dfrac{\pi}{2}$.

Domain and range

Because $\tan$ takes every real value on this interval, the domain of $\arctan$ is all real numbers $(-\infty, \infty)$. The range is the open interval $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$.

Symmetry and monotonicity

$\arctan$ is an odd function with $\arctan(-x) = -\arctan x$, so its graph is symmetric about the origin. Its derivative

is always positive, so the function is strictly increasing. Because the denominator is never $0$, this derivative is smooth for all real $x$.

Asymptotes and limits

As $x \to +\infty$, $\arctan x \to \dfrac{\pi}{2}$, and as $x \to -\infty$, $\arctan x \to -\dfrac{\pi}{2}$. Hence it has the horizontal asymptotes $y = \dfrac{\pi}{2}$ and $y = -\dfrac{\pi}{2}$, giving a gentle S-shaped curve. Its values never cross these two lines.

Notable points

The graph passes through the origin $(0, 0)$, which is an inflection point, and near the origin $\arctan x \approx x$. For example $\arctan 1 = \dfrac{\pi}{4}$, $\arctan \dfrac{1}{\sqrt{3}} = \dfrac{\pi}{6}$, and $\arctan \sqrt{3} = \dfrac{\pi}{3}$.

Relations and applications

$\arctan$ is used to compute $\pi$: from $\arctan 1 = \dfrac{\pi}{4}$ one derives the Leibniz series and others. It underlies $\operatorname{atan2}$, which recovers the polar angle of a point $(x, y)$, and is widely used in signal processing, control engineering, and computer graphics.