Free Arcus Tangent Calculator

What Is Arcus Tangent (Inverse Tangent)?

The arcus tangent — also written as arctan or tan⁻¹ — is the inverse of the tangent function. Given a real number $x$, it returns the angle $\alpha$ (in radians) whose tangent equals $x$, provided $\alpha$ lies in the principal interval $(-\pi/2, \pi/2)$. Formally, if $\tan \alpha = x$, then $\alpha = \arctan(x)$. This restriction is necessary because the standard tangent function is periodic and many‑to‑one; only a one‑to‑one branch can be inverted.

Domain and Range of Arctan

Because the tangent function can produce any real output, the domain of the arcus tangent is all real numbers: $-\infty < x < \infty$. Its range is confined to the open interval $(-\pi/2, \pi/2)$ (or $-90^\circ$ to $90^\circ$). On a graph, the arctan curve passes through the origin, increases monotonically, and approaches horizontal asymptotes at $y = \pm \pi/2$ as $x \to \pm \infty$. This shape reflects the chosen principal branch of the tangent function.

How to Use This Free Arcus Tangent Calculator Online

With this Inverse Tangent Calculator, you can instantly compute the arctan of any real input. Enter the tangent value (a real number) into the input field, and the Arctan Calculator displays the corresponding angle in radians (a degree setting is often available). Because the domain covers every real number, all valid numeric inputs return a result. Moreover, this Tan Inverse Calculator can also work in reverse: if you supply an angle, it functions as a standard tangent calculator, giving you the flexibility to switch between direct and inverse operations. There is no download or registration needed — it is a Free Arcus Tangent Calculator Online ready to use anywhere.

The notation $\arctan$ and $\tan^{-1}$ are used interchangeably, but remember that $\tan^{-1}$ does not mean $\frac{1}{\tan x}$; it denotes the functional inverse.

Common Arctan Values at a Glance

The table below lists frequently encountered arctan results (rounded where appropriate):

$x$ (tangent)	$\arctan(x)$ (radians)	$\arctan(x)$ (degrees)
0	0	0°
1	$\pi/4$	45°
$\sqrt{3}$	$\pi/3$	60°
$-\sqrt{3}$	$-\pi/3$	–60°
$\to \infty$	$\to \pi/2$	$\to 90^\circ$
$\to -\infty$	$\to -\pi/2$	$\to –90^\circ$

Practical Applications of Arctan

Arctan is widely used in science and engineering:

• Slope to angle conversion: Given a rise/run ratio, arctan yields the angle of inclination.
• Polar coordinates: Convert rectangular components $(x, y)$ to an angle with $\arctan(y/x)$.
• Phase angles: In AC circuits, the phase shift between voltage and current is found via arctan of the reactive over the real impedance.
• Navigation: Determine bearings from rectangular position data.

Having a dedicated Arctan Calculator at hand removes the need for manual table lookup or iterative solving.

In summary, the arcus tangent function maps any real number to an angle in $(-\pi/2, \pi/2)$. The online calculator provided here makes that operation instantaneous, whether you need the inverse tangent of a simple value like 1 or a more complex irrational number.