asin

asin is the inverse sine function, denoting the principal value of arcsin. For real inputs x in [-1, 1], asin(x) yields the unique y in [-π/2, π/2] such that sin(y) = x. Thus the domain of the real function is [-1, 1], and its range is [-π/2, π/2].

Properties include that asin is an odd function and strictly increasing on its domain. Its derivative is 1/√(1 - x^2) for |x| < 1, and the slope grows without bound as x approaches ±1. The identity asin(x) + acos(x) = π/2 holds, where acos is the arccosine function.

In programming and calculators, asin(x) is typically the function that returns the principal value in radians. If |x| > 1, most implementations produce NaN or raise a domain error. For complex inputs, asin can be extended to complex values, with principal value definitions such as asin(z) = -i ln(i z + √(1 - z^2)).

Applications include solving triangles, integral calculations, and trigonometric substitutions in calculus. A common series expansion for |x| < 1 is asin(x) = x + x^3/6 + 3x^5/40 + 5x^7/112 + ..., useful for small-x approximations.