arccoscos

Arccoscos is a nonstandard notation sometimes used to denote the composition f(x) = arccos(cos x). In this form, arccos is the inverse cosine restricted to its principal value, so its domain is [-1, 1] and its range is [0, π]. Consequently, f maps every real number to the interval [0, π].

Unlike x, which can take any real value, f(x) equals x only when x lies in the interval [0, π]. More generally, arccos(cos x) returns the principal angle whose cosine equals cos x, collapsing all real inputs onto [0, π].

Periodicity and symmetry: The function is even, f(-x) = f(x), since cos is even. It is periodic with period 2π, and on each interval [2kπ, 2kπ+2π] it is linear with slopes ±1: for x in [2kπ, 2kπ+π], f(x) = x - 2kπ; for x in [2kπ+π, 2kπ+2π], f(x) = 2π - (x - 2kπ) = 2(k+1)π - x.

Derivative: The derivative is 1 on the first subinterval and -1 on the second, with nondifferentiability at x = kπ for all integers k.

Examples: arccos(cos 1) = 1, since 1 ∈ [0, π]. For x = 4, arccos(cos 4) ≈ 2.283... which equals 2π - 4.

See also: arccos, cos, inverse function, triangular wave.