Free Arctan Calculator (Inverse Tangent)

The arctan calculator (also known as the inverse tangent calculator) is a free online tool designed to compute the angle that corresponds to a given tangent value. Whether you need to find the arctan of 1, 0, or any real number, this arctangent calculator outputs both radians and degrees instantly. It also includes a detailed explanation of the arctan function, its graph, common values, relationships with other trigonometric functions, its derivative and integral.

What is Arctan?

Arctangent, denoted as $\arctan(x)$ or $\tan^{-1}(x)$, is the inverse of the tangent function. While the tangent function gives the ratio of opposite to adjacent sides in a right triangle for a given angle, arctan does the reverse: it returns the angle whose tangent equals the input value. However, because the tangent function is periodic, it is not one-to-one over its entire domain. To define a proper inverse function, we restrict the domain of the tangent to $(-\pi/2, \pi/2)$ radians (or $-90$ to $90$ degrees). This restricted interval is called the principal range of the arctan function, and its output is the principal value. The domain of the arctan function is all real numbers $\mathbb{R}$, meaning you can input any real number and get a valid angle.

It’s important to distinguish arctan from the cotangent function. The notation $\tan^{-1}(x)$ can be misinterpreted as $(\tan(x))^{-1} = 1/\tan(x)$, which is the cotangent. However, $\arctan(x)$ is the inverse trigonometric function, not the multiplicative inverse. To avoid confusion, this calculator uses the $\arctan(x)$ notation, though the $\tan^{-1}(x)$ form is also common. Remember: $\cot(x) = 1/\tan(x)$, while $\arctan(x)$ is the angle whose tangent is $x$.

Arctan Graph and Common Values

The graph of arctan is obtained by reflecting the restricted tangent graph (from $-\pi/2$ to $\pi/2$) along the line $y = x$. As $x$ approaches $-\infty$, $\arctan(x)$ approaches $-\pi/2$ ($-90$ degrees), and as $x$ approaches $\infty$, it approaches $\pi/2$ ($90$ degrees). The following table lists some frequently used values:

$x$	$\arctan(x)$ (rad)	$\arctan(x)$ (deg)
$-\infty$	$-\pi/2$	$-90$
$-3$	$-1.2490$	$-71.565$
$-2$	$-1.1071$	$-63.435$
$-\sqrt{3}$	$-\pi/3$	$-60$
$-1$	$-\pi/4$	$-45$
$-\frac{\sqrt{3}}{3}$	$-\pi/6$	$-30$
$0$	$0$	$0$
$\frac{\sqrt{3}}{3}$	$\pi/6$	$30$
$1$	$\pi/4$	$45$
$\sqrt{3}$	$\pi/3$	$60$
$2$	$1.1071$	$63.435$
$3$	$1.2490$	$71.565$
$\infty$	$\pi/2$	$90$

Relationships with Trigonometric Functions

From a right triangle with legs $1$ and $x$, the following relationships hold:

• $\sin(\arctan(x)) = \dfrac{x}{\sqrt{1+x^2}}$
• $\cos(\arctan(x)) = \dfrac{1}{\sqrt{1+x^2}}$
• $\tan(\arctan(x)) = x$

Other useful identities include:

• $\arctan(x) + \arctan\!\left(\frac{1}{x}\right) = \dfrac{\pi}{2}$ for $x > 0$, and $-\dfrac{\pi}{2}$ for $x < 0$.
• $\arctan(-x) = -\arctan(x)$ (odd function property).
• $\arcsin(x) = \arctan\!\left(\dfrac{x}{\sqrt{1-x^2}}\right)$, for $|x| < 1$.
• $\arctan(x) = \dfrac{\pi}{2} - \operatorname{arccot}(x)$.

The identity for $\arctan(x)+\arctan(1/x)$ can be visualized in a right triangle: if the legs are $1$ and $x$, the two acute angles are $\arctan(x)$ and $\arctan(1/x)$, and together they sum to $\pi/2$ (or $90$ degrees).

Integral and Derivative of Arctan

The derivative of the arctan function is:

$$\frac{d}{dx} \arctan(x) = \frac{1}{1+x^2}.$$

The indefinite integral (antiderivative) is:

$$\int \arctan(x) \, dx = x \arctan(x) - \frac{1}{2} \ln|1+x^2| + C,$$

where $C$ is the constant of integration.

How to Use the Arctan Calculator

This arctan calculator is straightforward. Enter any real number (positive, negative, or zero) into the input field. The calculator instantly returns the arctan of that number in both radians and degrees. For example, typing 1 gives $\pi/4$ radians or $45$ degrees. You can also use the calculator in reverse: supply an angle to obtain its tangent value. This feature makes it useful for solving right‑triangle problems, working with trigonometric equations, or verifying identities.