Arctan Calculator
Enter any number and get its inverse tangent — the angle whose tangent equals that value — in degrees, radians, multiples of π and gradians.
arctan(1)
45°
- Degrees
- 45°
- Radians
- 0.785398 rad
- Radians as a multiple of π
- 0.25 × π
- Gradians
- 50 gon
arctan returns the angle strictly between −90° and 90° (−π/2 and π/2 radians) whose tangent equals your value. It is the inverse of tangent on that interval.
How to use this calculator
Type the tangent value you want to invert — any real number works, positive or negative, however large. Unlike arcsin and arccos there is no −1 to 1 restriction, because tangent itself can take any value. Pick whether the headline result shows degrees or radians; the breakdown always lists both, plus the radian value as a multiple of π and the angle in gradians. Negative inputs give negative angles, which is correct: arctan of a negative value lands between −90° and 0°. A common practical use is turning a slope into an angle — a ramp that rises 1 unit for every 4 along has gradient 0.25, and arctan(0.25) ≈ 14.04° is its angle of incline.
How the calculation works
Tangent takes an angle and returns the ratio of sine to cosine — equivalently, rise over run. Because tangent repeats every 180° and sweeps through every real value once per cycle, it has a true inverse when restricted to one interval: by convention the open interval −90° to 90° (−π/2 to π/2 radians). arctan(x) is therefore defined as the unique angle θ strictly between −π/2 and π/2 with tan θ = x — the principal value used by every scientific calculator and programming language. The interval is open because tangent is undefined at ±90° itself: as x grows without bound, arctan(x) approaches 90° ever more closely but never reaches it. Degrees convert from radians by multiplying by 180/π, and gradians (gons, where a right angle is 100) by 200/π. All other angles sharing the same tangent are θ plus whole multiples of 180°.
Worked example
Take x = 1. The angle strictly between −90° and 90° whose tangent is 1 is 45°, so arctan(1) = 45° — exactly π/4 radians (≈ 0.785398) or 50 gradians. Verify in reverse: tan(45°) = 1, because sine and cosine are equal there. For a slope example, a road that climbs 6 m over 100 m has gradient 0.06, and arctan(0.06) ≈ 3.43° — the sign-posted "6%" grade as an angle. And because tangent is odd, arctan(−1) = −45° = −π/4 radians.
Frequently asked questions
What is arctan?
Arctan (also written tan⁻¹ or atan) is the inverse of the tangent function. Where tangent takes an angle and gives you a ratio — rise over run — arctan takes the ratio and gives you back the angle. arctan(x) returns the unique angle strictly between −90° and 90° (−π/2 and π/2 radians) whose tangent equals x. For example, arctan(1) = 45° because tan(45°) = 1.
Why does arctan accept any number when arcsin and arccos stop at ±1?
Because the functions they invert have different ranges. Sine and cosine are ratios involving the hypotenuse, so they never leave [−1, 1]. Tangent is opposite over adjacent — the adjacent side can be arbitrarily small, so the ratio can be arbitrarily large in either direction. Since tangent takes every real value, arctan must accept every real value.
Why does arctan never return exactly 90°?
Because tan(90°) is undefined — at 90° the adjacent side of the triangle collapses to zero and the ratio blows up. As your input grows, arctan gets arbitrarily close to 90° (arctan(1,000,000) ≈ 89.99994°) but never reaches it. Mathematically, ±90° are horizontal asymptotes of the arctan curve, which is why its principal range is the open interval (−90°, 90°).
How do I use arctan to find the angle of a slope or gradient?
Divide the rise by the run, then take arctan of the result. A roof that rises 2 m over a 5 m span has gradient 2/5 = 0.4, and arctan(0.4) ≈ 21.8°. Road grades work the same way: a "10%" grade means 0.10 rise per unit run, which is arctan(0.1) ≈ 5.71° — note that a 100% grade is 45°, not vertical.
What are the exact arctan values worth remembering?
The unit-circle specials: arctan(−√3) = −60°, arctan(−1) = −45°, arctan(−1/√3) = −30°, arctan(0) = 0°, arctan(1/√3) = 30°, arctan(1) = 45°, and arctan(√3) = 60°. In radians those are −π/3, −π/4, −π/6, 0, π/6, π/4 and π/3. Handy identity: for positive x, arctan(x) + arctan(1/x) = 90°.
What is the difference between arctan and atan2?
arctan takes one number and can only tell you an angle in (−90°, 90°) — it cannot distinguish a point in the first quadrant from one in the third, because 3/4 and −3/−4 are the same ratio. atan2(y, x) takes the two coordinates separately, keeps their signs, and returns the full direction angle in (−180°, 180°]. Programming languages provide atan2 for exactly this reason; use it whenever you have both coordinates rather than a pre-divided slope.