Calculate the inverse tangent (arctan) of any real number. Get the angle in degrees, radians, and gradians. Unlike arcsin and arccos, arctan accepts all real numbers as input.
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Sources & Methodology
⏱ Last reviewed: April 2026
How to Calculate the Inverse Tangent
The inverse tangent function (arctan, also written tan⁻¹ or atan) returns the angle whose tangent equals a given value. Unlike arcsin and arccos which require input in [-1, 1], arctan accepts any real number as input because tangent itself spans all real values. The principal value range is −90° to +90° (exclusive), approaching but never reaching those limits as x approaches ±∞.
The Arctan Function: Domain and Range
arctan(x) accepts all real numbers as input (domain = all reals). The output range is strictly between −90° and +90° (open interval). Key values: arctan(0) = 0°, arctan(1) = 45°, arctan(−1) = −45°, arctan(∞) approaches 90°, arctan(−∞) approaches −90°. The function is odd: arctan(−x) = −arctan(x).
Arctan vs. Arcsin and Arccos — Key Difference
All three inverse trig functions return angles, but they differ in input domain and output range. arcsin and arccos require input in [-1, 1]. arctan accepts any real number. arcsin range: [-90°, 90°]. arccos range: [0°, 180°]. arctan range: (-90°, 90°) — open interval, never reaching exactly ±90°. Arctan is the most commonly used in programming and engineering.
atan2: The Two-Argument Arctangent
atan2(y, x) is a variant that takes two arguments (y, x) and returns the angle in the correct quadrant for all four quadrants, ranging from −180° to +180°. It solves the ambiguity of arctan when both coordinates are needed: atan2(1, −1) = 135° (second quadrant), while arctan(1/−1) = arctan(−1) = −45° (fourth quadrant). atan2 is essential in programming for converting Cartesian to polar coordinates.
Applications of Arctan
Arctan appears in: 1) Slope-to-angle conversion: angle = arctan(rise/run). 2) Vector angle from components: angle = arctan(y/x) or atan2(y,x). 3) Camera/optics: field of view = 2 x arctan(sensor_size / (2 x focal_length)). 4) Electronics: phase angle of RC circuits = arctan(1/(omega x R x C)). 5) Approximations: pi = 4 x arctan(1) = 4 x 45°.
Inverse Tangent Common Values
| x value | arctan(x) degrees | arctan(x) radians | Notes |
|---|---|---|---|
| -∞ | -90° (limit) | -π/2 (limit) | approaches but never reaches |
| -1.732 | -60° | -π/3 (-1.0472) | tan(-60°) = -sqrt(3) |
| -1 | -45° | -π/4 (-0.7854) | tan(-45°) = -1 |
| -0.577 | -30° | -π/6 (-0.5236) | tan(-30°) = -1/sqrt(3) |
| 0 | 0° | 0 | tan(0°) = 0 |
| 0.577 | 30° | π/6 (0.5236) | tan(30°) = 1/sqrt(3) |
| 1 | 45° | π/4 (0.7854) | tan(45°) = 1 |
| 1.732 | 60° | π/3 (1.0472) | tan(60°) = sqrt(3) |
| +∞ | 90° (limit) | π/2 (limit) | approaches but never reaches |
atan2(y, x) instead of atan(y/x) when working with 2D coordinates. atan2 handles all four quadrants correctly, avoids division by zero when x=0, and returns the correct quadrant angle in the range (-180°, 180°]. JavaScript: Math.atan2(y, x). Python: math.atan2(y, x).