Arctan Calculator - Inverse Tangent Function
Calculate the inverse tangent (arctan) of any real number and get the angle in degrees or radians instantly.
Enter any real number, choose degrees or radians, and compute the angle whose tangent equals that value.
Arctan Calculator - Inverse Tangent Function
Calculate the inverse tangent (arctan) of any real number and get the angle in degrees or radians instantly.
Domain: all real numbers (-∞ to +∞)
About the arctan calculator
The arctangent function, written arctan(x) or tan⁻¹(x), is the inverse of the tangent function. Given any real number x, arctan(x) returns the angle θ whose tangent equals x. Unlike arcsine and arccosine, arctangent is defined for all real numbers — there is no domain restriction — because the tangent function takes all real values as x ranges from -90° to 90°.
The output of arctan is in the open interval (-90°, 90°) for degrees, or (-π/2, π/2) for radians. Note that these endpoints are excluded: arctan never equals exactly ±90° because the tangent function approaches infinity as the angle approaches these limits, meaning no finite x gives a 90° result. As x → +∞, arctan(x) → 90°, and as x → -∞, arctan(x) → -90°.
Arctan is the most frequently used of the inverse trigonometric functions, appearing in an enormous range of applications. In geometry, if you know the lengths of the opposite and adjacent sides of a right triangle, the angle is arctan(opposite/adjacent). For example, a slope of 1 (rise equals run) corresponds to an angle of arctan(1) = 45°. A slope of 2 gives arctan(2) ≈ 63.43°.
In computer graphics and game development, arctan is used to find the angle of a 2D vector from its x and y components. The atan2(y, x) function (available in virtually every programming language) extends arctan to handle all four quadrants by checking the signs of both components, returning values in the full range (-180°, 180°].
In signal processing and control theory, arctan appears in phase calculations: the phase of a complex number a + bi is arctan(b/a). In navigation, the bearing between two geographic coordinates is computed using arctan of the differences in longitude and latitude. In physics, the angle of a pendulum or of a refracted ray involves arctan in its exact form.
Key values: arctan(0) = 0°, arctan(1) = 45°, arctan(-1) = -45°, arctan(√3) = 60°, arctan(1/√3) = 30°. These arise from the standard triangles and are worth memorising. The function is odd — arctan(-x) = -arctan(x) — so the negative values are simply the negatives of the corresponding positive angles.
The relationship between arctan and π is famous in mathematics. The Leibniz formula for π states: π/4 = 1 - 1/3 + 1/5 - 1/7 + ... = arctan(1). More efficient formulas like Machin's formula π/4 = 4·arctan(1/5) - arctan(1/239) were historically used to compute many digits of π by hand.
Arctan examples
Classic input values covering the key points on the arctangent curve.
| Input | arctan(x) | Explanation |
|---|---|---|
| arctan(0) | 0° = 0 rad | The tangent of 0° is 0, so arctan(0) = 0. The origin of the arctan curve. |
| arctan(1) | 45° ≈ 0.7854 rad | The tangent of 45° is 1. A slope of 1 (equal rise and run) corresponds to a 45° angle. |
| arctan(1.7321) | 60° ≈ 1.0472 rad | arctan(√3) = 60°. Arises from the 30-60-90 triangle where the tangent of 60° equals √3 ≈ 1.7321. |
| arctan(-1) | -45° ≈ -0.7854 rad | Negative input gives a negative angle. arctan(-1) = -45°, illustrating the odd-function symmetry. |
How to use the arctan calculator
- Type any real number in the Input Value (x) field. Unlike arcsin and arccos, arctan accepts all real numbers with no domain restriction.
- Select the Output Unit: click Degrees to receive the result in degrees, or Radians for the result in radians.
- Click Calculate Arctan to compute the inverse tangent. The result is in the range (-90°, 90°) for degrees.
- The result panel shows arctan(x) = result, together with the underlying formula for reference.
- Click Reset to clear the input and compute a new value.
Arctan calculator FAQ
What is the domain of arctan?
The domain of arctan is all real numbers (-∞, +∞). Because the tangent function takes every real value as its argument ranges over (-90°, 90°), there is no input restriction. You can compute arctan of very large numbers (arctan(1000) ≈ 89.943°) or very negative numbers (arctan(-1000) ≈ -89.943°).
Why does arctan never quite reach ±90°?
The tangent function approaches ±∞ as the angle approaches ±90°. Working backwards, to get arctan(x) = 90°, you would need x = tan(90°) = ∞. Since infinity is not a finite real number, arctan never reaches exactly ±90°. It approaches these limits asymptotically as x grows large in magnitude.
What is the difference between arctan and atan2?
arctan(y/x) only returns angles in (-90°, 90°) and cannot distinguish opposite quadrants — the vectors (1,1) and (-1,-1) both give a ratio of 1. The atan2(y, x) function takes both y and x separately and returns angles in the full range (-180°, 180°], correctly placing the result in the right quadrant. Use atan2 for vector angles in programming.
How is arctan used to find the slope angle of a line?
The slope m of a line is rise/run = tan(θ), so the inclination angle is θ = arctan(m). For a line with slope 0.5, the angle is arctan(0.5) ≈ 26.57°. A vertical line has an undefined slope (infinite tangent) and an inclination of 90°, which arctan approaches but never reaches for finite input.
Is arctan odd or even?
Arctan is an odd function: arctan(-x) = -arctan(x). This mirrors the fact that the tangent function is also odd: tan(-θ) = -tan(θ). Practically, this means if you know arctan(2) ≈ 63.43°, you immediately know arctan(-2) ≈ -63.43° without any extra computation.
What is the derivative of arctan?
The derivative of arctan(x) with respect to x is 1/(1 + x²). This relatively simple derivative makes arctan useful in integral calculus: the integral of 1/(1 + x²) dx equals arctan(x) + C. This integral appears frequently in problems involving rational functions and in deriving the series expansion for π.