Functionsarcsin

The term functionsarcsin refers to the inverse sine function, commonly written as arcsin(x) or sin^{-1}(x). It returns an angle y in radians such that sin(y) = x, with the angle chosen in the principal value range.

Domain and range: The input x must lie in [-1, 1]. The output y lies in [-π/2, π/2]. This restriction defines the principal branch of the inverse sine, ensuring uniqueness.

Properties: Functionsarcsin is an increasing function on its domain, and it is odd, meaning arcsin(-x) = -arcsin(x). Its derivative exists for |x| < 1 and is given by d/dx arcsin(x) = 1/√(1 - x^2); at the endpoints x = ±1 the derivative diverges. The identity sin(arcsin(x)) = x holds for all x in [-1, 1].

Relations and identities: For any real θ, arcsin(sin(θ)) equals θ only when θ lies in the principal range [-π/2, π/2]. In general, arcsin(sin(θ)) yields the corresponding principal value. The function is the inverse of the sine function restricted to [-π/2, π/2].

Series and computation: arcsin(x) has a power series about x = 0: arcsin(x) = ∑_{n=0}^∞ [(2n)! / (4^n (n!)^2 (2n+1))] x^{2n+1}, valid for |x| ≤ 1. The series begins arcsin(x) = x + (1/6)x^3 + (3/40)x^5 + (5/112)x^7 + ...

Applications: Functionsarcsin is used in solving triangles, integrating certain expressions, and in fields requiring inverse trigonometric evaluation or principal-value determination.