Arctangent Calculator

What is Arctan?

As the inverse of the tangent, arctan (arc tangent) takes a value x and returns the angle θ that satisfies tan(θ) = x. The result always stays within (−90°, 90°).

Some reference values: arctan(0) = 0°, arctan(1) = 45°, arctan(√3) = 60°. Any real number works as input, since the domain runs from −∞ to +∞.

As x grows toward +∞, arctan(x) approaches 90°; and as x heads to −∞, arctan(x) tends to −90°.

The arctangent function

The arctangent undoes the tangent: atan(x) = θ such that tan(θ) = x. You can hand it any real number ℝ, and the answer always lands inside the open interval (−π/2, π/2), so the graph flattens toward horizontal asymptotes at ±π/2 without ever touching them. Take atan(1) = π/4 = 45°. The derivative is (atan x)' = 1/(1 + x²), and the Maclaurin series reads atan(x) = x − x³/3 + x⁵/5 − ... (Gregory–Leibniz). It converges slowly, yet plugging in x = 1 gives the famous Leibniz series π/4 = 1 − 1/3 + 1/5 − .... When you code, reach for atan2(y, x), the two-argument cousin that picks the right quadrant and returns an angle in (−π, π]. You will want it for polar coordinates and plotting.

Applications: computer vision, navigation and signals

Arctangent, and especially atan2, shows up all over the place. In computer vision it gives the angle between a camera and a point. In navigation it turns a heading vector into a compass bearing. Signal processing uses it to read the phase of a complex number, and terrain analysis leans on it to turn a gradient into an inclination angle.

FAQ

Why does the range never reach ±π/2? At those angles the tangent itself blows up to ±∞. The inverse keeps creeping toward them as asymptotes but can never land on them.

What is the difference between atan and atan2? Once you divide, atan(y/x) has thrown away whether x and y were positive or negative, so it can only answer in (−π/2, π/2). atan2(y, x) keeps both signs around to nail the right quadrant, returning a value in (−π, π].

How does the Leibniz series for π work? Integrate the geometric series 1/(1+t²) = 1 − t² + t⁴ − ... from 0 to 1 and you land on atan(1) = π/4. It does converge, just painfully slowly, which makes it a nice curiosity rather than a practical way to compute π.

Is the arctangent an odd function? It is: atan(−x) = −atan(x), a direct consequence of the tangent being odd too.