arcseno

Arcseno is the inverse of the sine function, commonly denoted as arcsin or sin^-1 in mathematical notation. It yields the angle whose sine equals a given value, effectively inverting the sine function on a restricted domain.

In several Romance-language contexts, the function is referred to as arcseno, with arcsin as the standard mathematical symbol in English-speaking texts. The principal value convention is used: arcseno maps inputs in the interval [-1, 1] to angles in the interval [-π/2, π/2] (that is, [-90°, 90°]).

Domain, range and basic identities: The input x must satisfy -1 ≤ x ≤ 1. The output is an angle θ such that sin θ = x and θ ∈ [-π/2, π/2]. Consequently, sin(arcseno x) = x for all x in [-1, 1], and arcseno(sin θ) = θ whenever θ lies in the principal interval [-π/2, π/2]. Endpoints give arcseno(-1) = -π/2 and arcseno(1) = π/2.

Analytic properties: For -1 < x < 1, the derivative is d/dx arcseno x = 1 / sqrt(1 - x^2). The function has a power-series expansion around 0: arcseno x = x + (1/2) x^3/3 + (1·3)/(2·4) x^5/5 + ..., reflecting its role as the inverse of the sine. In many computational contexts, the function is implemented as asin or as sin^-1.

Applications: Arcseno is used to solve triangles, convert between sine values and angles, in geometry and trigonometry problems, and in physics, engineering, computer graphics, and signal processing where angle measures are required from sine data.