Sinv

Sinv is a shorthand notation used in some mathematical and computing contexts to denote the inverse sine function, the inverse of the sine function on its principal branch. It is commonly understood as the function that returns an angle y for a given sine value x, with sin(y) = x and y constrained to the principal range.

Notation and equivalence: sinv(x) is typically written as arcsin(x) or asin(x) in many texts and programming libraries. For real values, x must lie in the interval [-1, 1], and sinv(x) yields a result y in [-π/2, π/2]. In many cases, sinv is used interchangeably with arcsin, though some software may prefer the explicit arcsin or asin naming.

Domain, range, and properties: The real inverse sine is defined only for x in [-1, 1]. It is an odd function, so sinv(-x) = -sinv(x). The derivative is sinv′(x) = 1/√(1 − x^2) for |x| < 1. In complex analysis, the inverse of sine is multi-valued, and sinv represents a chosen branch or principal value.

Notation caveats: The symbol sin^-1(x) is sometimes used to denote arcsin, but it can be misread as the reciprocal of sin(x) (cosecant). To avoid confusion, arcsin or asin is often preferred in formal writing.

Usage: In programming languages, the inverse sine is typically provided as asin or Math.asin, while some calculators or libraries may label it sinv. The concept remains the same: sinv(x) gives the angle whose sine equals x.