Arctan Calculator (Inverse Tangent)

Calculate the inverse tangent (arctan) of a value. Supports both atan(x) and atan2(y, x) with results in degrees and radians.

Select Mode & Enter Value

Result

arctan(1)
45°
degrees
Degrees45°
Radians0.7854 rad
Exact Valueπ/4
Tangent Verificationtan(45°) = 1
Range(-90°, 90°)

Common Arctan Values

xatan(x) Degreesatan(x) RadiansExact

Understanding the Inverse Tangent Function

The inverse tangent (also called arctangent or atan) is a fundamental trigonometric function that returns the angle whose tangent equals a given value. If tan(θ) = x, then arctan(x) = θ.

Mathematical Definition

The arctangent function is defined as the inverse of the tangent function restricted to the interval (-π/2, π/2). For any real number x:

θ = arctan(x), where θ ∈ (-90°, 90°) or (-π/2, π/2)

Unlike the tangent function which has a period of π and is not one-to-one, the arctangent is a true function that maps every real number to exactly one angle in its range.

atan(x) vs atan2(y, x)

There are two common variants of the inverse tangent function:

  • atan(x) — The standard inverse tangent. Takes a single argument and returns an angle in (-90°, 90°). It cannot distinguish between angles in different quadrants because it only receives the ratio y/x.
  • atan2(y, x) — The two-argument inverse tangent. Takes both y and x separately, preserving sign information. Returns an angle in (-180°, 180°], covering all four quadrants. This is generally preferred in programming and engineering.

For example, atan(-1) returns -45°, but atan2(-1, 1) = -45° (Quadrant IV) while atan2(1, -1) = 135° (Quadrant II). Both have a tangent ratio of -1, but atan2 correctly identifies the quadrant.

Key Properties

  • arctan(0) = 0°
  • arctan(1) = 45° = π/4
  • arctan(-1) = -45° = -π/4
  • arctan(√3) = 60° = π/3
  • arctan(1/√3) = 30° = π/6
  • As x → +∞, arctan(x) → 90°
  • As x → -∞, arctan(x) → -90°
  • arctan(x) is an odd function: arctan(-x) = -arctan(x)

Applications of Arctangent

  • Navigation and Surveying: Calculating bearing angles from coordinate differences. Given two points, atan2(delta_y, delta_x) gives the direction of travel.
  • Computer Graphics: Converting between Cartesian and polar coordinates, computing rotation angles, and determining the direction a sprite or camera should face.
  • Electrical Engineering: Calculating phase angles in AC circuits where impedance has both resistive and reactive components.
  • Physics: Finding launch angles in projectile motion, angles of incidence and refraction, and direction of resultant vectors.
  • Machine Learning: The arctan function and its scaled variants are used as activation functions in neural networks.

Derivative and Integral

d/dx [arctan(x)] = 1 / (1 + x²)

∫ arctan(x) dx = x · arctan(x) - ½ · ln(1 + x²) + C

Series Expansion

For |x| ≤ 1, the arctangent can be expressed as a power series (Gregory-Leibniz series):

arctan(x) = x - x³/3 + x⁵/5 - x⁷/7 + ...

Setting x = 1 gives the famous Leibniz formula for π/4: π/4 = 1 - 1/3 + 1/5 - 1/7 + ... This series converges slowly, but related identities (like Machin's formula) can compute π much more efficiently.