The Arctangent Function

Definition The restricted tangent branch is the tangent function on the open interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Tangent is undefined at the endpoints $\pm\frac{\pi}{2}$, where its graph has the vertical asymptotes $x = \pm\frac{\pi}{2}$. On this interval tangent is increasing and takes every real value exactly once, so the branch is one-to-one with range $(-\infty, \infty)$.

Definition For every real number $u$, the arctangent of $u$ is the unique angle $\theta$ in radians with

It is written $\theta = \arctan u$ or $\theta = \tan^{-1} u$, where $\tan^{-1}$ means the inverse function and not the reciprocal $\frac{1}{\tan u} = \cot u$.

Definition The value $\arctan u$ is the principal angle with tangent $u$. It is a signed angle, negative when $u < 0$, zero when $u = 0$ and positive when $u > 0$. Every other angle with tangent $u$ is $\arctan u + k\pi$ for some integer $k$. The inverse identities are

Definition The domain of arctangent is the set of all real numbers, $(-\infty, \infty)$, because the restricted tangent branch takes every real value. Unlike arcsine and arccosine, arctangent needs no condition on its input. A composition $\arctan(g(x))$ is defined exactly where $g$ is defined.

Definition The range of arctangent is the open interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. The endpoints $\pm\frac{\pi}{2}$ are never attained, since $\tan \theta$ is undefined at $\theta = \pm\frac{\pi}{2}$ and no real $u$ can have them as its arctangent. On a graph these heights are marked by dashed lines, never by dots.

Definition The graph of $y = \arctan x$ is the set of points $(b, a)$ for which $(a, b)$ lies on the graph of the restricted tangent branch. With equal scales on both axes it is the reflection of that branch across the line $y = x$. Swapping coordinates turns the vertical asymptotes $x = \pm\frac{\pi}{2}$ of the branch into the horizontal lines $y = \pm\frac{\pi}{2}$ of the inverse, and the open ends of the interval stay open.

Definition A horizontal line $y = L$ is a horizontal asymptote of the graph of $f$ if $f(x)$ stays as close to $L$ as you like once $x$ is large enough, or once $x$ is negative enough. This is written $f(x) \to L$ as $x \to \infty$, or as $x \to -\infty$, and calculus makes it precise as a limit at infinity. The graph of $y = \arctan x$ has two horizontal asymptotes,

The reason is visible on the tangent branch. Because tangent is increasing there, for any small $\varepsilon > 0$ the angle $\arctan x$ lies above $\frac{\pi}{2} - \varepsilon$ exactly when $x > \tan\left(\frac{\pi}{2} - \varepsilon\right)$. So every input beyond that number gives an output within $\varepsilon$ of $\frac{\pi}{2}$, and oddness gives the same picture near $-\frac{\pi}{2}$ on the left.

Definition Arctangent is odd, which means

This holds because $\tan(-\theta) = -\tan \theta$ and $-\theta$ lies in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ whenever $\theta$ does. The graph is symmetric about the origin.