What this calculator does
Arctan, also written tan⁻¹, is the inverse of the tangent function: instead of starting from an angle and finding its tangent, arctan starts from a ratio and finds the angle that produces it. If tan(θ) = x, then arctan(x) = θ.
This calculator takes any value, positive, negative, small or large, and returns the angle whose tangent equals it, in both degrees and radians, along with a check that recalculating the tangent of that angle returns the original value.
The formula
Arctan always returns an angle strictly between -90° and 90° (or -π/2 and π/2 radians), because tangent itself repeats every 180° and arctan picks the one result in that principal range. The angle is converted between radians (the unit trigonometric functions use internally) and degrees (the more familiar unit) so both are shown.
| Term | Meaning |
|---|---|
| Arctan (tan⁻¹) | The inverse tangent function: arctan(x) is the angle whose tangent equals x. |
| Principal value | The single angle between -90° and 90° that arctan returns, chosen because tangent is not one-to-one across a full circle. |
| Radian | The angle unit in which one full circle equals 2π; degrees and radians describe the same angle on different scales. |
The inputs explained
| Field | What to enter |
|---|---|
| Value (the tangent ratio) | The ratio you want the inverse tangent of. This can be any real number, including negative values. |
When to use it
Finding an angle from a slope or gradient
A slope expressed as rise over run is exactly a tangent ratio, so arctan of that ratio gives the angle of incline directly, useful for ramps, roofs and roads.
Working backwards in a right triangle
If you know the lengths of the opposite and adjacent sides of a right triangle but not the angle between them, dividing one by the other and taking arctan recovers that angle.
Converting a bearing or direction ratio to an angle
Navigation and vector problems often produce a ratio of two components (such as northing over easting); arctan turns that ratio back into an angle measured from a reference direction.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Arctan of common ratios, in degrees and radians
A range of input values and the angle each produces.
| Value (x) | Angle (degrees) | Angle (radians) |
|---|---|---|
| 0.0 | 0° | 0 |
| 0.5 | 26.565° | 0.463648 |
| 1.0 | 45.000° | 0.785398 |
| 2.0 | 63.435° | 1.1071 |
| 5.0 | 78.690° | 1.3734 |
| 10.0 | 84.289° | 1.4711 |
Questions
What is the difference between arctan and 1/tan?
They are unrelated despite the similar notation. Arctan (tan⁻¹) is the inverse function that returns an angle from a ratio; 1/tan (which equals cotangent) is a different ratio entirely, calculated from an angle, not the other way around.
Why does arctan only return values between -90° and 90°?
Tangent repeats every 180°, so infinitely many angles share the same tangent value. Arctan is defined to return just one of them, the one in the range -90° to 90°, so it behaves as a proper function with a single output.
Can arctan take any input value?
Yes. Unlike arcsin and arccos, which are only defined for inputs between -1 and 1, arctan accepts any real number, because tangent itself can produce any value as its angle approaches, without reaching, 90°.
How does arctan relate to a right-angled triangle?
If you know the lengths of the two legs (opposite and adjacent) of a right triangle, arctan of opposite ÷ adjacent gives the angle between the adjacent leg and the hypotenuse.
To solve a full right-angled triangle from one known angle and one known side, see the trigonometry calculator.