käänteistangan

Käänteistangan is a Finnish term that translates to "inverse tangent" in English. It refers to the arctangent function, which is the inverse operation of the tangent trigonometric function. Mathematically, if y = tan(x), then x = arctan(y). The arctangent function takes a real number (the tangent value) and returns an angle in radians.

The principal value of the arctangent function, often denoted as arctan or tan⁻¹, is typically defined to return a value in the interval (-π/2, π/2) radians, which corresponds to (-90°, 90°) in degrees. This restriction ensures that the arctangent function is well-defined and produces a single output for each input.

In calculus, the derivative of arctan(x) is 1/(1+x²). This property makes it useful in integration, as the integral of 1/(1+x²) is arctan(x) + C, where C is the constant of integration. The arctangent function appears in various fields, including physics, engineering, and mathematics, particularly in problems involving angles and rotations. It is also a fundamental function in complex analysis and signal processing.