asec

Asec, short for arcsecant, is the inverse function of the secant function. It takes a real number x with |x| ≥ 1 and returns an angle y in the interval [0, π], with y ≠ π/2, such that sec(y) = x. Equivalently, asec(x) = arccos(1/x) for |x| ≥ 1. The principal value convention is chosen so that the inverse is well defined.

Domain and range: The domain of asec is all real numbers x with |x| ≥ 1. The range is 0 ≤ y ≤ π, y ≠ π/2. This makes asec the inverse of the secant function restricted to [0, π], y ≠ π/2.

Relationships and properties: asec(x) can be expressed as arccos(1/x), linking it to the arccosine function. The function is even in the sense that asec(−x) = π − asec(x) for |x| ≥ 1, reflecting the symmetry of the cosine on [0, π]. The derivative is asec′(x) = 1/(|x| sqrt(x^2 − 1)) for |x| > 1, and it is undefined at x = ±1. An antiderivative is ∫ asec(x) dx = x·asec(x) − sqrt(x^2 − 1) + C, valid for |x| > 1.

Examples: asec(2) = π/3, since sec(π/3) = 2. asec(-2) = 2π/3, since sec(2π/3) = -2.

See also: arcsec, inverse trigonometric functions, trigonometric identities.

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