Arctan Calculator

Arctan reverses the tangent function. Feed it a ratio and you get back the angle that produces that ratio. The notation shifts depending on the source: arctan(x), tan⁻¹(x), and atan(x) all describe the same operation. This calculator works in either direction and supports degrees, radians, gradians, and turns.

What arctan actually does

If $\tan(\theta) = z \implies \arctan(z) = \theta$. The complication is that tangent repeats every 180°, so any ratio matches infinitely many angles. tan(45°) and tan(225°) both equal 1, but arctan(1) returns only 45°. The function commits to its principal range, between -90° and 90°, which is what lets it qualify as a proper function in the first place.

Arctan is the friendlier cousin of arcsin and arccos. Those two reject anything outside [-1, 1] because sine and cosine never leave that band. Tangent runs from negative infinity to positive infinity, so arctan happily accepts any real number you hand it.

Using the calculator

Type a number into the x field to get the matching angle. Type an angle into the output field to recover the ratio. Switch units using the selector beside the angle field.

Where you'll see arctan

Anywhere you have a ratio of two perpendicular quantities and want the angle between them, arctan is the tool. Roof pitch is rise over run, and arctan turns that ratio into the slope angle a builder can measure with a level. In a vector problem, if you know the x and y components, arctan gives you the direction. Electrical engineers use it to find the phase angle in an AC circuit from the ratio of reactance to resistance.

The version you'll meet most often in code is atan2(y, x). It takes the two components separately and returns angles across the full [-π, π] range. Regular arctan cannot distinguish (1, -1) from (-1, 1) because the ratio is identical in both cases, even though those points sit 180° apart. atan2 looks at the sign of each component and resolves which quadrant the angle belongs in.

Quick reference

A few numbers worth keeping in your head. arctan(0) is 0°. arctan(1) is 45°. arctan(√3) is 60°. As the input grows toward infinity the output approaches 90° but never reaches it, which produces the horizontal asymptote you see when you plot the function. Negative inputs produce negative angles, and the curve is symmetric about the origin.

For calculus and physics, work in radians. For carpentry, surveying, or anything you'd discuss with a non-mathematician, stick with degrees.

Frequently asked questions

What is the domain of arctan?

Every real number. Arcsin and arccos only accept inputs between -1 and 1, but arctan handles whatever you throw at it, from a tiny fraction up to a number with more digits than your screen can fit.

How does atan2 differ from regular arctan?

atan2 takes the y and x components as separate arguments and uses the sign of each to figure out the correct quadrant. Plain arctan only sees the ratio, so it cannot tell a positive-positive pair from a negative-negative pair. If you're converting Cartesian coordinates to polar, you almost always want atan2.

Why is the output limited to -90° to 90°?

Because tangent repeats every 180°, the inverse cannot be single-valued unless we pick one branch. Mathematicians settled on the branch around zero by convention. It keeps the function continuous and makes the derivative clean:

Can the calculator accept very large numbers?

Yes. Arctan is defined for every real number, and the result simply gets closer to ±90° as the input grows. A value like 1,000,000 returns roughly 89.99994°.