Arctangent Calculator

The arctangent (arctan, atan, tan⁻¹) is the inverse of the tangent function. Since tan is periodic, the inverse is made single-valued by restricting the output to the principal branch: (−π/2, π/2) rad, equivalently (−90°, 90°). For any real x, arctan(x) returns the unique angle θ in that open interval such that tan(θ) = x. The endpoints ±90° are approached asymptotically but never reached, because tan(±90°) is undefined.

The one-argument arctan only tells you an angle modulo 180°, so it cannot distinguish the quadrant. atan2(y, x) takes both coordinates and returns an angle in (−π, π] that identifies the full quadrant of the point (x, y). For example, arctan(1/1) and arctan(−1/−1) both give 45°, but atan2(1, 1) = 45° while atan2(−1, −1) = −135°. Use atan2 whenever you convert Cartesian to polar coordinates.

arctan(x) returns the angle whose tangent equals x. If tan(θ) = x, then arctan(x) = θ (within the principal range −90° to 90°). For example, arctan(1) = 45° because tan(45°) = 1. The function is also written atan(x) or tan⁻¹(x) on calculators. It is defined for every real number, including very large or very negative values.

tan maps an angle to a ratio (opposite ÷ adjacent in a right triangle). arctan goes the other way: given the ratio, recover the angle. tan takes an angle as input and can output any real number (it has vertical asymptotes at ±90°, ±270°, etc.). arctan takes any real number as input and always outputs an angle strictly between −90° and 90°. They are inverse on that principal branch: arctan(tan(θ)) = θ only when −90° < θ < 90°.

Because tan is periodic with period 180°, infinitely many angles share the same tangent (e.g., tan(45°) = tan(225°) = 1). To make arctan a function (one output per input), mathematicians restrict the output to a single monotonic branch of tan: the one on (−90°, 90°). That branch covers every real ratio exactly once. If you need all angles with a given tangent, add 180° × k for any integer k to the principal value.

In a right triangle with legs a (opposite) and b (adjacent), the acute angle at the base equals arctan(a / b). This is the core tool for finding angles when you know two sides. Example: a roof with a 3 m rise over a 5 m run has a pitch angle of arctan(3 / 5) ≈ 30.96°. In surveying, the elevation angle to a tower of height h at horizontal distance d is arctan(h / d). In robotics and computer graphics, arctan (via atan2) converts (x, y) positions into heading angles.

As x grows without bound, arctan(x) approaches 90° (π/2 rad) but never reaches it. For example, arctan(10) ≈ 84.2894°, arctan(100) ≈ 89.4271°, arctan(1000) ≈ 89.9427°. Symmetrically, arctan(−x) = −arctan(x), so arctan(−1000) ≈ −89.9427°. The limits lim x→+∞ arctan(x) = π/2 and lim x→−∞ arctan(x) = −π/2 make arctan a bounded function — useful for squashing unbounded inputs into a finite range in signal processing and neural networks.

The calculator uses JavaScript's built-in Math.atan, which implements IEEE 754 double-precision floating point (about 15–17 significant decimal digits). Results are displayed to 8 decimal places. For engineering or educational purposes this is more than sufficient. For symbolic results — for example recognizing arctan(1) = π/4 exactly — the calculator matches common fractions of π (π/6, π/4, π/3, etc.) within a small tolerance and labels them.