Arctan Calculator
Find the inverse tangent of any real number. Enter a value for x to compute θ = arctan(x).
What does the arctan calculator do?
This calculator computes the inverse tangent function, often written as arctan(x) or tan-1(x). If tangent takes an angle and returns a ratio, arctan does the reverse: it takes a ratio and returns an angle.
In right-triangle terms, if you know the ratio opposite / adjacent = x, then arctan(x) gives you the angle that matches that ratio.
How arctan works
Core idea
The tangent function is periodic, so many angles can have the same tangent value. To make arctan single-valued, we use its principal range:
- Radians: from -π/2 to π/2 (not including endpoints)
- Degrees: from -90° to 90° (not including endpoints)
That means your calculator always returns the principal angle in that interval.
Formula used
The calculator uses JavaScript’s built-in numeric function:
- Radians:
Math.atan(x) - Degrees:
Math.atan(x) × 180 / π
Radians vs. degrees
Engineers and scientists frequently use radians because they simplify many calculus and physics formulas. Degrees are often more intuitive for everyday interpretation.
To help both use cases, this tool outputs both radians and degrees from the same input.
Quick examples
- arctan(1) = π/4 radians = 45°
- arctan(0) = 0 radians = 0°
- arctan(-1) = -π/4 radians = -45°
- arctan(10) ≈ 84.289°
As input magnitude gets very large, arctan approaches ±90° but never reaches it for finite x.
Where arctan is used
- Trigonometry: recovering angles from side ratios
- Physics: vector direction and slope angles
- Engineering: control systems and signal phase calculations
- Computer graphics: camera orientation and geometric transforms
- Data analysis: interpreting slopes and directional relationships
Common mistakes to avoid
1) Confusing tan and arctan
tan(angle) gives a ratio, while arctan(ratio) gives an angle. They are inverse operations, but not the same input/output type.
2) Ignoring angle units
A value like 0.785398 is radians, which equals 45°. Always check which unit you are reading.
3) Expecting outputs beyond principal range
Even though many angles have the same tangent, arctan returns only the principal angle in the range (-π/2, π/2).
FAQ
Can arctan return exactly 90°?
No. For finite input, arctan approaches 90° but does not equal it.
Is tan-1(x) the same as 1/tan(x)?
No. tan-1(x) means inverse tangent (arctan), not reciprocal tangent. The reciprocal of tangent is cotangent.
Can I enter expressions like pi/4?
Yes. This calculator accepts simple arithmetic expressions and the constant pi (or π).