Tan Inverse Calculator

Enter a value x to find θ = tan⁻¹(x), the angle whose tangent equals x, in degrees, radians, and gradians.

Quick Facts

Formula
θ = tan⁻¹(x)
The angle whose tangent equals x; the inverse of tan(θ) = opposite / adjacent.
Principal range
-90° < θ < 90° (-π/2 to π/2 rad)
tan⁻¹ always returns a value in this range, regardless of how large x is.
Known value
tan⁻¹(1) = 45°
Since tan(45°) = 1 in an isosceles right triangle.
Derivative
d/dx[tan⁻¹(x)] = 1 / (1 + x²)
Used in calculus when tan⁻¹(x) appears inside another function.

Your Results

Calculated
Angle (Degrees)
-
θ = tan⁻¹(x), principal value
Angle (Radians)
-
θ × π/180
Angle (Gradians)
-
θ × 200/180
Check: tan(θ)
-
Should equal your input x

Ready

Enter a value for x, then press Calculate.

Formula and Method for the Tan Inverse Calculator

The inverse tangent, written tan⁻¹(x) or arctan(x), is the angle θ whose tangent equals x. If tan(θ) = x, then θ = tan⁻¹(x). Since tan(θ) = opposite / adjacent in a right triangle, tan⁻¹(x) answers the question "what angle produces this ratio of opposite to adjacent side?" This calculator takes any real number x — positive, negative, zero, or very large — and returns the principal angle θ in degrees, radians, and gradians, plus a check that recomputes tan(θ) so you can confirm it matches your input.

How the calculation works

Enter x, the ratio you want the angle for. The calculator computes θ = tan⁻¹(x) in radians using the arctangent function, then converts that radian value to degrees (θ° = θ_rad × 180/π) and gradians (θ_grad = θ_rad × 200/π). Because tangent repeats every 180°, tan⁻¹ always returns its principal value, strictly between -90° and 90° (-π/2 and π/2 radians) — it never returns 90° or -90° exactly, since tan(θ) is undefined at those points. The tool also shows tan(θ) recomputed from the result, which should match your original x (subject to rounding) as a sanity check.

arctan(x) vs. atan2(y, x)

tan⁻¹(x) only sees the ratio, so it cannot tell whether the original opposite and adjacent sides were both positive or both negative — 1/1 and (-1)/(-1) both give x = 1 and the same 45° answer, even though those correspond to different quadrants. If you need the full angle for a coordinate pair (x, y), including which quadrant it lies in across the whole -180° to 180° range, use the two-argument function atan2(y, x) instead, which examines the sign of each side separately.

Common applications

  • Finding an unknown angle in a right triangle when the opposite and adjacent side lengths (or their ratio) are known.
  • Computing the angle of a slope, ramp, or roof pitch from its rise-over-run ratio.
  • Determining the direction of a 2D vector or line segment in physics, engineering, and computer graphics.
  • Evaluating derivatives and integrals involving tan⁻¹(x) in calculus, where d/dx[tan⁻¹(x)] = 1/(1 + x²).

Frequently Asked Questions

What is the difference between arctan(x) and atan2(y, x)?
tan⁻¹(x), or arctan(x), takes a single ratio and always returns a principal angle between -90° and 90°. It cannot tell (1, 1) apart from (-1, -1) because both give the same ratio. atan2(y, x) takes the numerator and denominator separately and uses their individual signs to return the correct angle across the full -180° to 180° range.
What is the range of the inverse tangent function?
The principal value of tan⁻¹(x) is always between -90° and 90° (exclusive), or -π/2 and π/2 radians. This is because the tangent function repeats every 180°, so arctan returns only one representative angle from that repeating pattern.
How do I convert the arctan result between degrees and radians?
Multiply radians by 180/π to get degrees, or multiply degrees by π/180 to get radians. For example, tan⁻¹(1) = π/4 radians = 0.7854 rad = 45°.
What is tan⁻¹(1) and why?
tan⁻¹(1) = 45° (π/4 radians), because tan(45°) = 1. A 45° angle forms an isosceles right triangle where the opposite and adjacent sides are equal, so their ratio is exactly 1.

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