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Inverse Tangent (Arctan) Calculator

Calculate the inverse tangent (arctan) of any real number. Returns the angle in degrees, radians, or gradians instantly.

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Inputs

Value (x)

Output Unit

Angle (θ)

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Angle (θ)—°

The formula

How the
result is
computed.

What Is the Inverse Tangent (Arctan) Function?

The inverse tangent function, written as arctan(x) or tan^-1(x), answers the question: which angle has a tangent equal to x? While the standard tangent function maps an angle to a ratio, the arctangent reverses that process — it maps a ratio back to an angle. This tan inverse calculator automates that computation for any real number input.

The Core Formula

The fundamental relationship is expressed as:

θ = arctan(x) = tan^-1(x)

Where x is any real number (the ratio of the opposite side to the adjacent side in a right triangle), and θ is the resulting angle. Unlike the tangent function, which is undefined at ±90°, arctangent accepts every real number from −∞ to +∞ and always returns a value strictly within the open interval (−90°, 90°), or equivalently (−π/2, π/2) radians. This bounded output is the principal value of the inverse tangent.

Understanding the Output Units

Degrees

Degrees are the most intuitive unit for everyday applications. A full circle contains 360°, a right angle is 90°, and arctangent results fall between −90° and 90°. For example, arctan(1) = 45° because tan(45°) = 1.

Radians

Radians are the standard unit in calculus, physics, and engineering. One radian equals approximately 57.296°. The arctangent range in radians is (−π/2, π/2), roughly (−1.5708, 1.5708). Calculus formulas for derivatives and integrals require radians; the derivative of arctan(x) is 1/(1 + x²), valid only when x is in radians.

Gradians

Gradians (also called gon or grad) divide a right angle into 100 units, making a full circle 400 gradians. Surveyors and civil engineers in some countries prefer this system. In gradians, arctan(1) = 50 grad.

Key Variables Explained

Value (x): The tangent ratio — any real number. Positive values yield angles in the first quadrant (0° to 90°); negative values yield angles in the fourth quadrant (−90° to 0°); arctan(0) = 0°.
Output Unit: The angular unit for the result — degrees, radians, or gradians. The numerical answer changes with the unit, but the geometric angle stays the same.

Practical Step-by-Step Example

Suppose a ramp rises 3 meters vertically over a horizontal distance of 4 meters. The slope ratio is 3/4 = 0.75. To find the ramp's angle:

Set x = 0.75
Calculate θ = arctan(0.75)
Result: approximately 36.87° (or 0.6435 radians)

This angle tells engineers whether the ramp meets accessibility standards (typically ≤ 4.76° for ADA compliance) or safety codes for steeper industrial ramps.

Real-World Applications

Navigation and Bearing

GPS and mapping systems use arctan to convert Cartesian coordinate differences (Δy, Δx) into compass bearings. The two-argument variant, atan2(y, x), extends arctan to cover the full 360° circle by accounting for the signs of both components.

Physics and Engineering

Phase angles in AC circuits, projectile launch angles, and the angle of incidence in optics all rely on inverse tangent calculations. When a force of 50 N acts horizontally and 50 N vertically, the resultant direction is arctan(50/50) = 45°.

Computer Graphics

3D rendering engines use arctan (via atan2) to compute the angle between vectors, enabling correct rotation matrices, camera orientation, and lighting calculations.

Mathematical Derivation and Series Expansion

The arctangent function can be expressed as an infinite series (Gregory-Leibniz series) for |x| ≤ 1:

arctan(x) = x − x³/3 + x⁵/5 − x⁷/7 + …

This series converges slowly for values near ±1 but demonstrates why π/4 = arctan(1) = 1 − 1/3 + 1/5 − 1/7 + …, a historically significant identity. Modern calculators use optimized CORDIC algorithms rather than this series for faster convergence.

Sources and Further Reading

The mathematical foundations of inverse trigonometric functions are documented by Xavier University of Louisiana — Math 1030 and the step-by-step application to calculator-based problem solving is covered in detail by Paul's Online Math Notes: Solving Trig Equations with Calculators. Additional context on the function's role in calculus is available at UNL Calculus — Trigonometric Functions.

Reference

Frequently asked questions

What does tan inverse (arctan) actually calculate?

Tan inverse, or arctan(x), finds the angle whose tangent equals x. For instance, arctan(1) = 45° because tan(45°) = 1. The function accepts any real number as input and always returns an angle strictly between −90° and 90° (or −π/2 and π/2 radians), which is called the principal value.

Why is the arctan output always between −90° and 90°?

The tangent function is not one-to-one over its full domain — many different angles can share the same tangent value. To define a true inverse, mathematicians restrict the output to the principal value range of (−90°, 90°). This guarantees a unique, unambiguous answer for every real-number input, making the function mathematically well-defined and consistent across all calculators and software.

How do I convert an arctan result from radians to degrees?

Multiply the radian value by 180 and then divide by π (approximately 3.14159). For example, arctan(1) = π/4 radians. Multiplying π/4 by 180/π gives exactly 45°. Conversely, to go from degrees to radians, multiply by π/180. This calculator handles the conversion automatically when the output unit is changed.

What is the difference between arctan and atan2?

Standard arctan(x) takes a single ratio and returns an angle in (−90°, 90°), losing information about which quadrant the original vector lies in. The two-argument function atan2(y, x) takes both the vertical and horizontal components separately and returns an angle over the full (−180°, 180°] range, preserving quadrant information. Navigation, robotics, and game engines almost always use atan2 for this reason.

Can arctan be used to find angles in non-right triangles?

Directly, no — arctan is derived from right-triangle definitions where the ratio opposite/adjacent is well-defined. For oblique (non-right) triangles, the standard approach is to first apply the Law of Sines or the Law of Cosines to find a missing side or another angle, then use that result within a right-triangle framework or apply arcsin/arccos as appropriate to solve for remaining angles.

What is the derivative of arctan(x) and why does it matter?

The derivative of arctan(x) with respect to x is 1/(1 + x²). This result is important in calculus because it shows the function is always increasing (the derivative is always positive) and because it leads to a widely used integration formula: the integral of 1/(1 + x²) dx equals arctan(x) + C. This integral appears frequently in engineering, probability distributions, and signal processing.