Inverse Tangent
TrigonometryInverse 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.
Example
A ladder leans against a wall: $6$ feet up and $4$ feet out. The angle the ladder makes with the ground $= \arctan(6/4) = \arctan(1.5) \approx 56.3^\circ$.
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.
Definition
$\arctan(x)$ is the inverse of tan restricted to $(-\pi/2, \pi/2)$. 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.
Example
A ramp rises $3$ m over a horizontal run of $10$ m. The angle of inclination $= \arctan(3/10) = \arctan(0.3) \approx 16.7^\circ$.
Key Insight
As $x$ approaches infinity, $\arctan(x)$ approaches $\pi/2$ ($90^\circ$) but never reaches it, because no finite number is the tangent of $90^\circ$. This asymptotic behavior has applications in signal processing.
Definition
$\arctan: \mathbb{R} \to (-\pi/2, \pi/2)$ with $d/dx[\arctan(x)] = 1/(1 + x^2)$. The Gregory-Leibniz series: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + \ldots = \arctan(1)$, via $\arctan(x) = x - x^3/3 + x^5/5 - \ldots$ for $|x| \le 1$.
Example
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. This is essential in computer graphics and robotics.
Key Insight
The integral of $1/(1 + x^2) = \arctan(x) + C$. This arises in partial fractions: any rational function with irreducible quadratic denominators integrates to arctan expressions, connecting algebra to trigonometry.