Inverse Tangent
Inverse tangent (arctan) is the function that returns the angle whose tangent equals a given value.
Formula
\arctan(x) = \theta \text{ such that } \tan(\theta) = x
Definition
Inverse tangent (also called arctan or $\tan^{-1}$) is the reverse of tangent: given any number, it tells you what angle has that tangent value. Formally, $\arctan(x)$ is the inverse of tan restricted to $(-\pi/2, \pi/2)$, so for any real $x$, $\arctan(x)$ is the unique angle $\theta$ in $(-90^\circ, 90^\circ)$ with $\tan(\theta) = x$; unlike arcsin and arccos, arctan accepts all real numbers as input. It maps $\mathbb{R} \to (-\pi/2, \pi/2)$ with derivative $d/dx[\arctan(x)] = 1/(1 + x^2)$.
Example
A ladder leans against a wall $6$ feet up and $4$ feet out; the angle it makes with the ground $= \arctan(6/4) = \arctan(1.5) \approx 56.3^\circ$. A ramp rising $3$ m over a horizontal run of $10$ m has an angle of inclination $= \arctan(0.3) \approx 16.7^\circ$. The two-argument form $\text{atan2}(y, x)$ gives the full angle in $(-\pi, \pi]$ for a point $(x, y)$ by handling all four quadrants correctly, unlike $\arctan(y/x)$ which loses quadrant information, essential in computer graphics and robotics.
Key Insight
Unlike sine and cosine, which only take inputs from $-1$ to $1$, tangent can be any number, so arctan accepts any real number as input; as $x$ approaches infinity, $\arctan(x)$ approaches $\pi/2$ but never reaches it, since no finite number is the tangent of $90^\circ$. The Gregory-Leibniz series $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + \ldots = \arctan(1)$ and the integral of $1/(1 + x^2) = \arctan(x) + C$ connect this asymptotic behavior to partial-fraction integration and signal processing.