arkustangentti

Arkustangentti is the inverse function of the tangent function. Also known as arctangent, it is commonly denoted as arctan(x), atan(x), or tan⁻¹(x). The arkustangent function returns the angle whose tangent is a given number. For any real number x, the arkustangent of x, arctan(x), is the angle θ in the interval (-π/2, π/2) such that tan(θ) = x.

The domain of the arkustangent function is all real numbers, meaning it can accept any real number as input. The range of the principal value of the arkustangent function is the open interval from -π/2 radians to π/2 radians, or equivalently, from -90 degrees to 90 degrees. This restriction to a specific interval is necessary because the tangent function is periodic, and its inverse must be single-valued.

Arkustangent has applications in various fields, including trigonometry, calculus, and physics. In calculus, the derivative of arctan(x) is 1/(1+x²), which is a fundamental result used in integration. It also appears in formulas for calculating areas and in solving differential equations. Geometrically, it can be used to find the angle of elevation or depression.