Arctan Calculator
Calculate arctan(x) = tan⁻¹(x) with exact values, results in degrees and radians, all real inputs accepted.
tan⁻¹(x)
Updates as you typeInput mode ?
Value
Value of x ?
—
Right-triangle sides
Opposite (rise) ?
—
Adjacent (run) ?
—
Ratio x = 1
Output unit ?
Presets (click to load)
| x | Degrees | Radians |
|---|
Step-by-step
Formula
arctan(x)
=
θ
such that
tan(θ) = x
Definition: arctan returns the unique angle θ in (−90°, 90°) — that is, (−π/2, π/2) in radians — whose tangent equals x. It is the inverse of the tangent restricted to that principal branch.
Domain & range: defined for all real x ∈ (−∞, +∞). The output is bounded: arctan(x) → ±90° as x → ±∞, but never reaches either endpoint.
Useful identities:
arctan(−x) = −arctan(x) (odd function); arctan(x) + arctan(1/x) = ±90° (sign matches x); atan2(y, x) extends arctan to all four quadrants.
- x
- Any real number — the tangent ratio you want to invert. Often a slope (rise over run) or opposite / adjacent from a right triangle.
- θ
- The principal-value angle whose tangent equals x. Arctan returns only this one solution, even though tan has infinitely many angles sharing the same value (period 180°).
- Notation
- arctan(x), tan⁻¹(x), and atan(x) all mean the same function. The superscript −1 denotes the inverse, not a reciprocal.
- Quadrant
- Because the principal branch is (−90°, 90°), arctan always lands in quadrant I (x ≥ 0) or quadrant IV (x < 0).
Worked example — your numbers
- Input x = —
- θ = arctan(x) = — = —
- Quadrant = — (principal branch)
- sin(θ) = —, cos(θ) = —
- Verify: tan(θ) = sin/cos = — (should equal x)
- arctan(x) = —
When x is one of the canonical ratios (0, ±√3/3, ±1, ±√3), the calculator shows the exact value (0, ±π/6, ±π/4, ±π/3). Otherwise it falls back to a high-precision decimal.