Arctan

Arctan, denoted arctan x or tan^{-1} x, is the inverse function of the tangent when the latter is restricted to its principal branch. It maps real numbers to the interval (-pi/2, pi/2) and satisfies tan(arctan x) = x for all real x. The function is strictly increasing and smooth, with derivative d/dx arctan x = 1/(1 + x^2). In particular, arctan 0 = 0 and as x approaches ±∞, arctan x approaches ± pi/2.

Arctan can be expressed as an integral and by a power series. It satisfies arctan x = ∫_0^x dt/(1 + t^2). It also has the convergent series arctan x = x − x^3/3 + x^5/5 − x^7/7 + … for |x| ≤ 1, with arctan(1) = pi/4 and arctan(-1) = -pi/4 at the endpoints.

Addition formulas for arctan relate sums of arctan terms to a single arctan value, typically arctan x + arctan y = arctan((x + y)/(1 − xy)) with appropriate adjustments by ±π to keep the result in (-pi/2, pi/2). The function also has a basic relationship to the two-argument arctangent, atan2, which preserves the quadrant when computing the angle from a slope or a vector.

Applications of arctan include determining angles from slopes, evaluating certain integrals such as ∫ dx/(1 + x^2), and appearing in various formulas in physics and engineering. Notable values include arctan 1 = pi/4 and arctan ∞ = pi/2.