Arctan Calculator

Compute the arctangent of any real number — the inverse of the tangent function. Get results in both radians and degrees, visualize the principal branch of y = arctan(x), and explore fundamental properties, real‑world applications, and historical context.

Arctan(x) returns the angle θ ∈ (−π/2, π/2) such that tan(θ) = x.
arctan(-100) ≈ -89.43° arctan(-10) ≈ -84.29° arctan(-√3) = -60° arctan(-1) = -45° arctan(-1/√3) = -30° arctan(0) = 0° arctan(1/√3) = 30° arctan(1) = 45° arctan(√3) = 60° arctan(10) ≈ 84.29° arctan(100) ≈ 89.43°
Privacy-first computation: All calculations happen locally in your browser. No data is transmitted or stored.
Arctan Result & Properties
arctan(1) =
0.785398 radians
45.000000 degrees
Principal value range
θ ∈ (−π/2, π/2)   (−90° < θ < 90°)
✔ valid input (any real number)
Inverse property: tan(arctan(x)) = x   for all real x;   arctan(tan(θ)) = θ only if θ ∈ (−π/2, π/2).
Graph of y = arctan(x) (principal branch)
y = arctan(x)
Current point (x₀, arctan(x₀))
Asymptotes y = ±π/2

Domain: all real numbers   |   Range: (-π/2, π/2) (≈ -1.5708 to 1.5708 rad)

Understanding Arctan (Inverse Tangent): definition, derivation & significance

The arctangent function, denoted arctan(x) or tan⁻¹(x), is the inverse of the tangent function restricted to the interval (−π/2, π/2). For any real number x, arctan(x) returns the unique angle θ in that open interval whose tangent equals x. This function is essential for converting slopes to angles, solving right triangles, and appears extensively in calculus, physics, and engineering.

For all real x,   θ = arctan(x)   ⇔   tan(θ) = x   and   θ ∈ (−π/2, π/2).

The derivative: d/dx arctan(x) = 1/(1+x²). This simple rational derivative makes arctan a favorite in integral calculus (e.g., ∫ dx/(1+x²) = arctan(x) + C).

Historical & mathematical roots

The arctangent function has a rich history, notably in the development of infinite series for π. In 1671, James Gregory discovered the series arctan(x) = x − x³/3 + x⁵/5 − x⁷/7 + … (for |x| ≤ 1), and later Leibniz used it to derive the famous Leibniz formula for π/4. The notation "arctan" became standard in the 18th century. Today, arctan is fundamental in coordinate transformations (cartesian to polar), signal processing, and in the atan2 function which handles quadrant ambiguity.

Key properties and identities

  • Symmetry: arctan(−x) = − arctan(x) — odd function.
  • Limits: lim_{x→+∞} arctan(x) = π/2, lim_{x→−∞} arctan(x) = −π/2 (horizontal asymptotes).
  • Relationship with arccot: arctan(x) + arccot(x) = π/2 for all real x.
  • Addition formula: arctan(u) + arctan(v) = arctan((u+v)/(1−uv)) + kπ, where k is adjusted to place result in (−π/2, π/2).
  • Derivative and integral: ∫ arctan(x) dx = x arctan(x) − ½ ln(1+x²) + C.
  • Series expansion: arctan(x) = x − x³/3 + x⁵/5 − x⁷/7 + … for |x| ≤ 1.

Practical applications across disciplines

Coordinate Systems & Navigation

Given Cartesian coordinates (x, y), the angle to the positive x-axis is atan2(y, x) = arctan(y/x) with quadrant adjustment. Used in robotics, GPS, and computer graphics to compute headings and orientations.

Signal Processing & Control Theory

Arctan is used in phase detection, frequency demodulation, and in the computation of the argument of a complex number. The arctan function also appears in PID controller tuning and system identification.

Statistics & Machine Learning

The arctan function is used as an activation function in neural networks (e.g., arctan activation) and in robust regression (M-estimators). It also appears in the Cauchy distribution's cumulative distribution function.

Step-by-step calculation method

  1. Accept any real number input x (no domain restrictions).
  2. Use the built‑in Math.atan function (IEEE 754 compliant) returning value in radians between -π/2 and π/2.
  3. Convert to degrees by multiplying by 180/π.
  4. Validate with known identities: arctan(0) = 0, arctan(1) = π/4, arctan(∞) → π/2.

Our calculator implements double‑precision floating‑point arithmetic, ensuring accuracy to 15 decimal digits. The interactive graph shows the exact point (x₀, arctan(x₀)) on the principal branch, and illustrates the horizontal asymptotes at y = ±π/2.

Common mistakes and misconceptions

  • Confusing arctan with cot: arctan(x) is not the reciprocal of tangent; cot(x) = 1/tan(x) is different.
  • Quadrant confusion: arctan(y/x) alone loses quadrant information; use atan2(y, x) for correct angle in all four quadrants.
  • Principal value nuance: arctan(tan(150°)) ≠ 150°; it returns -30° because the result must lie within (-90°, 90°).
  • Large inputs: For very large |x|, arctan(x) approaches ±π/2 but never exceeds it.

Frequently Asked Questions

They are identical notations. tan⁻¹(x) represents the inverse function, not the multiplicative inverse. For reciprocal, use cot(x).

On (−π/2, π/2), tangent is strictly increasing and covers all real numbers exactly once, making it the natural choice for the principal branch. This range also aligns with the typical slope-to-angle conversion.

atan2(y, x) returns the angle from the positive x-axis to the point (x, y), taking into account the quadrant of (x, y). It returns values in (−π, π]. arctan(y/x) loses the sign of x and may give an angle off by π. Many programming languages provide atan2 for this reason.

Very accurate. Math.atan(1e12) returns π/2 minus a tiny value (~1e-12), within machine precision. The calculator will show the result correctly as ≈ 90° but never exceeding 90°.

If a line has slope m, then the angle θ it makes with the positive x-axis is θ = arctan(m), assuming the angle is in (−π/2, π/2). For vertical lines, the angle is ±π/2, which corresponds to infinite slope.
Authoritative references: Wolfram MathWorld: Inverse Tangent | Khan Academy: Intro to arctan | Abramowitz, M. and Stegun, I.A. "Handbook of Mathematical Functions" (1972).

Expert verification: This tool implements the standard IEEE-754 arctangent function. All content has been reviewed by getzenquery tech team. Last update: May 2026.