Arcsin

Arcsin, denoted arcsin or sin^{-1}, is the inverse function of the sine function when sine is restricted to the interval [-π/2, π/2]. It returns the unique angle y in that interval whose sine equals a given value x.

Domain and range: The domain of arcsin is x ∈ [-1, 1], and its range is y ∈ [-π/2, π/2]. By definition, y = arcsin(x) implies sin(y) = x with y in that principal value range.

Notational note: sin^{-1} x is commonly used, but may be confused with the reciprocal csc x in some contexts; arcsin is preferred for the inverse.

Properties: Arcsin is an odd function: arcsin(-x) = -arcsin(x). It composes with sine on its principal branch: sin(arcsin(x)) = x for x ∈ [-1, 1]. Conversely, arcsin(sin(y)) = y for y ∈ [-π/2, π/2].

Calculus: The derivative is d/dx arcsin(x) = 1/√(1 - x^2) for x ∈ (-1, 1). The integral ∫ arcsin(x) dx = x arcsin(x) + √(1 - x^2) + C.

Series: For |x| ≤ 1, arcsin(x) has the expansion arcsin(x) = ∑_{n=0}^∞ [(2n choose n) x^{2n+1}] / [4^n (2n+1)], with first terms x + x^3/6 + 3x^5/40 + …

Applications and notes: Arcsin is used to solve triangles, compute inverse trigonometric values, and in integration and probability. Its graph is an increasing, odd function mapping [-1,1] to [-π/2, π/2], with vertical tangents at the endpoints x = ±1.