functionsarcsine

The arcsine function, denoted arcsin or sin^{-1}, is the inverse of the sine function when the latter is restricted to the interval [-π/2, π/2]. It maps the domain [-1, 1] to the range [-π/2, π/2]. Formally, y = arcsin(x) if and only if sin(y) = x and y lies in [-π/2, π/2].

The function is continuous and strictly increasing on its domain. It is differentiable for all x in (-1, 1) with derivative dy/dx = 1 / sqrt(1 - x^2). The derivative becomes unbounded as x approaches ±1, where arcsin(±1) = ±π/2. The value at the endpoints are arcsin(1) = π/2 and arcsin(-1) = -π/2.

Key identities describe its relationships with other trigonometric functions. For x in [-1, 1], sin(arcsin(x)) = x, and for y in [-π/2, π/2], arcsin(sin(y)) = y. A common complementary relation is arcsin(x) + arccos(x) = π/2, valid for x in [-1, 1].

In computation and applications, arcsin is used to solve equations involving sine inverses, in geometry, and in various fields of science. In programming languages, the function is typically named asin or sin^{-1} and is defined only for inputs within [-1, 1], producing a principal value in [-π/2, π/2]. The function has a convergent power series about 0, providing a local analytic expansion for small |x|.