About

The inverse tangent function arctan(x) returns the angle whose tangent equals x. Its output is bounded to the open interval (−π2, π2), or equivalently (−90°, 90°). Misidentifying the quadrant is the single most common error in applied trigonometry. Navigation systems, robotics kinematics, and signal-processing phase detectors all depend on the two-argument variant atan2(y, x) precisely because the single-argument form discards quadrant information. A sign error here can rotate a robotic arm 180° in the wrong direction or invert a radar bearing.

This calculator computes both arctan(x) and atan2(y, x), reports results in radians, degrees, and gradians, and displays the instantaneous derivative 11 + x². The interactive graph plots y = arctan(x) and marks your input point. Note: the tool uses IEEE 754 double-precision arithmetic, so results near the asymptotes (±π2) carry approximately 15 significant digits of precision.

Formulas

The principal inverse tangent for a single argument:

θ = arctan(x), θ ∈ (−π2, π2)

The two-argument form that preserves quadrant information:

θ = atan2(y, x), θ ∈ (−π, π]

Unit conversions from radians:

θ_deg = θ_rad × 180π

θ_grad = θ_rad × 200π

The derivative of the inverse tangent:

ddx arctan(x) = 11 + x²

Where θ = resulting angle, x = tangent value (input), y = vertical component, x = horizontal component (for atan2), π ≈ 3.14159265358979.

Reference Data

x	arctan(x) rad	arctan(x) °	arctan(x) grad	Exact Form
−∞	−π2	−90°	−100grad	−π/2
−√3	−1.04720	−60°	−66.667grad	−π/3
−1	−0.78540	−45°	−50grad	−π/4
−√33	−0.52360	−30°	−33.333grad	−π/6
0	0	0°	0grad	0
√33	0.52360	30°	33.333grad	π/6
1	0.78540	45°	50grad	π/4
√3	1.04720	60°	66.667grad	π/3
2	1.10715	63.435°	70.483grad	arctan(2)
3	1.24905	71.565°	79.517grad	arctan(3)
5	1.37340	78.690°	87.433grad	arctan(5)
10	1.47113	84.289°	93.654grad	arctan(10)
100	1.56080	89.427°	99.363grad	arctan(100)
+∞	π2	90°	100grad	π/2

Frequently Asked Questions

The single-argument arctan(x) returns an angle in the range (−90°, 90°), covering only quadrants I and IV. It cannot distinguish between the point (1, 1) and (−1, −1) because both yield a ratio of 1. The two-argument atan2(y, x) examines the signs of both components independently, returning angles across the full (−180°, 180°] range across all four quadrants.

When both y = 0 and x = 0, no unique angle exists because the point lies at the origin with no directional information. IEEE 754 defines atan2(0, 0) as 0 in most implementations, but mathematically the result is undefined. This calculator returns 0 with a warning note.

As x approaches ±∞, arctan(x) approaches ±π/2 but never reaches it. For inputs like x = 10¹⁵, the result is accurate to about 15 significant digits due to IEEE 754 double-precision limits. The last few decimal places may differ from symbolic computation systems.

Gradians (also called gons) divide a right angle into exactly 100 parts, making them standard in European surveying, geodesy, and land measurement. A full circle equals 400 grad. The conversion is θ_grad = θ_rad × 200/π. Use gradians when working with European topographic maps or civil engineering specifications that reference the metric angular system.

The derivative ddxarctan(x) = 11 + x² produces the Cauchy (Lorentzian) distribution shape, fundamental in spectroscopy, resonance analysis, and probability theory. This calculator displays the derivative value at your input point, useful for sensitivity analysis in control systems.

The Taylor expansion arctan(x) = x − x³3 + x⁵5 − … converges for |x| ≤ 1. Setting x = 1 yields the Leibniz formula π/4 = 1 − 1/3 + 1/5 − …, one of the classical methods for approximating π. Machin-like formulas use arctan identities for faster convergence.