How this instrument works
Arctan answers a specific question: given a ratio x, what angle has that ratio as its tangent? Picture a right triangle where x is opposite divided by adjacent, or a line on a graph where x is the slope — arctan(x) hands back the angle that produced it. Because tangent's range covers every real number (a nearly vertical line has an enormous slope, a nearly horizontal one a tiny one), arctan's domain is unrestricted too: unlike arcsin and arccos, which choke on anything outside [−1, 1], arctan happily takes x = 1, x = 1000, or x = −0.0003.
The output is restricted, though, and for a clean reason. Tangent repeats every 180° and shoots off to infinity as the angle approaches 90° from below, then reappears from negative infinity just past 90°. A function needs exactly one output per input, so arctan is defined to report only the branch closest to zero — the open interval strictly between −90° and 90°. That interval is open on purpose: tangent never actually reaches infinity at a finite angle, so arctan never actually reaches 90°, no matter how large x grows.
A consequence worth sitting with: as x runs off toward positive or negative infinity, arctan(x) creeps toward 90° or −90° but the curve flattens rather than arriving — a horizontal asymptote, not a wall. Contrast that with arcsin and arccos, whose graphs simply stop at x = ±1 because sine and cosine can't produce anything past that. Arctan's boundary is a limit you approach forever; theirs is an edge you fall off.
- Type your value into the x field — any real number: positive, negative, zero, or well outside the −1 to 1 range that trips up arcsin.
- Pick the unit for the answer from the dial next to the result: degrees, radians, or turns.
- Read the arctan(x) field for the principal angle — the one value between −90° and 90° whose tangent equals x.
- Switch the unit dial at any time; x stays put and the angle is recomputed in the newly chosen unit instantly.
Worked example — the ratio 1 and the 45° it hides
A carpenter checks a brace where the rise and run are equal, so x = 1. Entering 1 into the x field with the dial on degrees returns arctan(1) = 45° exactly, because tan(45°) = 1 exactly: an isosceles right triangle where the two legs match, so opposite over adjacent is 1 and the angle splitting a square along its diagonal is 45°.
Flip the dial to radians and the same computation reads 0.78539816, matching arctan(1) = 0.7853981633974483 radians to full precision — that number is π ⁄ 4. It is not a coincidence worth shrugging off: arctan(1) = π ⁄ 4 is the seed of the Gregory–Leibniz series, π = 4(1 − 1⁄3 + 1⁄5 − 1⁄7 + …), a 17th-century result that turns this single arctangent value into an infinite recipe for π, however slowly it converges.
Questions
What is the difference between arctan and atan2?
Arctan(x) takes one number and only ever returns an angle between −90° and 90°, so it cannot tell x = 1 apart from a point in the opposite direction that has the same ratio — arctan(1) is 45°, whether the underlying point is (1, 1) in the first quadrant or (−1, −1) in the third, where the true direction is 225°. Atan2 takes the rise and run separately and uses their individual signs to return the full angle, all 360° of it.
Why is arctan(1) exactly 45 degrees?
Because tan(45°) = 1 exactly: a right triangle with equal legs has opposite over adjacent equal to 1, and 45° is precisely half of the 90° corner split by that triangle's diagonal. Arctan simply reverses the lookup, so feeding in the ratio 1 hands back the 45° that produced it, with no rounding involved anywhere in the identity.
How is arctan different from arcsin and arccos?
Arcsin and arccos only accept inputs between −1 and 1, since sine and cosine never leave that range; feed either one a 2 and it has no answer. Arctan takes any real number at all, because a tangent ratio (opposite over adjacent) can be arbitrarily large or small. Their outputs differ too: arcsin and arccos span a closed 180°, while arctan's output stays strictly inside an open 180° span, never touching either end.
What mistake do people make with arctan and negative angles?
Assuming the single output is the only angle with that tangent. Tangent repeats every 180°, so if arctan(x) returns θ, then θ + 180° and θ − 180° share the exact same tangent and are equally valid depending on which quadrant the original situation actually sits in. Arctan reports the principal value only; recovering the true direction needs the signs of the original rise and run, which is exactly what atan2 supplies.
Why does arctan's output never reach 90 degrees?
Because tangent itself never reaches a finite value at 90° — it grows without bound as the angle approaches that mark from below, and jumps to unbounded negative values just past it. There is no finite ratio that arctan could return for 90° itself, so the function approaches that boundary as x grows huge but never arrives, however large a number you enter.