arctangente

Arctangente, often written arctan in English, is the inverse function of the tangent when the latter is restricted to the interval (-π/2, π/2). For a real number x, arctan(x) is the unique angle y in (-π/2, π/2) such that tan(y) = x.

Its domain is all real numbers and its range is (-π/2, π/2). The function is strictly increasing and continuous, and it is odd: arctan(-x) = -arctan(x). As x approaches +∞, arctan(x) tends to π/2; as x approaches -∞, it tends to -π/2.

Key formulas include the derivative d/dx arctan(x) = 1/(1 + x^2), and the integral representation arctan(x) = ∫_0^x dt/(1 + t^2). A common series expansion for |x| ≤ 1 is arctan(x) = x − x^3/3 + x^5/5 − x^7/7 + ...

Typical identities involve arctan x + arctan y = arctan((x + y)/(1 − xy)) with appropriate adjustments by ±π when xy > 1 or when other quadrant considerations apply. Notable special values include arctan(0) = 0 and arctan(1) = π/4.

Applications and notation: arctangente is used throughout trigonometry, geometry, and calculus. In programming and calculators it is usually implemented as atan or arctan. It serves as the inverse of tan on the principal branch, providing a direct link between real numbers and angles measured in radians.