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Arctan(x), written tan⁻¹(x) or atan(x), is the inverse of the tangent function: it answers "which angle has this tangent value?" Unlike arcsin and arccos, which only accept inputs between -1 and 1, tangent can produce any real number — so arctan accepts any real input at all, from very large negative numbers to very large positive ones — but it always returns an angle strictly between -90° and 90° (its "principal value" range). Arctan shows up constantly wherever a ratio of two lengths needs to become an angle: finding the angle of a slope or ramp from its rise-over-run, computing the angle of elevation to an object from its height and distance, converting between rectangular and polar coordinates in engineering and physics, and working out phase angles in AC circuit analysis from a reactance-to-resistance ratio. Programming languages implement it as `atan()`, and most also provide a two-argument `atan2(y, x)` that fixes the quadrant ambiguity a plain arctan can't resolve on its own.
How it works
Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle (equivalently, sin(θ)/cos(θ)). Arctan reverses that: given a ratio x, it finds the angle θ such that tan(θ) = x. Because tangent repeats every 180°, there are infinitely many angles with any given tangent value — arctan always reports the one in the principal range of -90° to 90°, the branch closest to zero. To recover an angle in a different quadrant (for example, when you know both the horizontal and vertical components of a vector, not just their ratio), you'd add or subtract 180° as appropriate, or use the two-argument atan2 function instead.
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Examples
arctan(1)
arctan(1) = 45°, because tan(45°) = sin(45°)/cos(45°) = 1 (the two legs of a 45-45-90 triangle are equal).
arctan(√3) ≈ arctan(1.732)
arctan(1.732) = 60°, since tan(60°) = √3 ≈ 1.732 — one of the standard reference angles.
arctan(-1)
arctan(-1) = -45°, the negative counterpart of the first example, since tangent is an odd function: arctan(-x) = -arctan(x).
Common mistakes to avoid
- Assuming arctan can return exactly 90° or -90° — it only ever approaches those values, never reaches them.
- Using arctan(y/x) when the correct tool is atan2(y, x), and getting an angle in the wrong quadrant as a result.
- Confusing arctan(x) with the reciprocal function 1/tan(x) (which is cotangent) — arctan is an inverse function that outputs an angle, cotangent is a ratio that outputs a number.
- Forgetting that tangent (and therefore arctan) is undefined for the ratio "infinity", i.e. when the adjacent side is zero — that limiting case corresponds to a 90° angle and needs to be handled separately.