The inverse tangent answers one question: which angle has this tangent? It is written in three common ways, arctan(x), atan(x) and tan⁻¹(x), and all three mean the same thing.
Tangent first
In a right triangle, the tangent of an angle θ is the length of the side opposite θ divided by the side adjacent to it:
tan θ = opposite ÷ adjacent
So if you know the angle, tangent gives you a ratio. The inverse tangent goes the other way: if you know the ratio, arctan gives you the angle.
θ = arctan(opposite ÷ adjacent)
A first example
Suppose the opposite side is 1 and the adjacent side is 1. The ratio is 1, and
- tan 45° = 1, so
- arctan(1) = 45°, which is π/4 radians.
Why arctan only returns angles between −90° and 90°
The tangent function repeats every 180°. For example, tan 45° and tan 225° are both 1. A function can only return one answer, so arctan is defined to return the principal value, the one angle in the open interval (−90°, 90°), or (−π/2, π/2) in radians.
| x | arctan x (degrees) | arctan x (radians) |
|---|---|---|
| −√3 | −60° | −π/3 |
| −1 | −45° | −π/4 |
| 0 | 0° | 0 |
| 1/√3 | 30° | π/6 |
| 1 | 45° | π/4 |
| √3 | 60° | π/3 |
As x grows very large, arctan(x) gets closer and closer to 90° but never reaches it. That is why the graph of arctan has two horizontal asymptotes, at y = π/2 and y = −π/2.
tan⁻¹ is not 1/tan
The notation tan⁻¹(x) is easy to misread. It means the inverse function, not the reciprocal. The reciprocal of tangent is the cotangent:
- tan⁻¹(1) = 45°
- 1 / tan(45°) = cot(45°) = 1
They happen to agree at 45°, but not elsewhere. For example tan⁻¹(2) ≈ 63.43°, while 1/tan(2°) ≈ 28.64.
Key facts at a glance
- Domain: every real number.
- Range: −π/2 < arctan x < π/2.
- Odd function: arctan(−x) = −arctan(x).
- Derivative: d/dx arctan x = 1 / (1 + x²).
- Complement identity: arctan(x) + arctan(1/x) = π/2 for x > 0.
Where you will meet arctan
- Turning a slope into an angle (roads, ramps, roofs).
- Finding a direction from x and y components in physics and games.
- Solving right triangles in surveying and construction.
- Signal processing and complex numbers, where arctan gives the phase.
When you need the full direction of a point in any quadrant, use atan2(y, x) instead of plain arctan. Our article on atan vs atan2 explains why.
Try it yourself: open the inverse tangent calculator to check any value from this article with full step-by-step working.