arccosine

Arccosine, written arccos(x) or cos^{-1}(x), is the inverse function of the cosine restricted to the interval [0, π]. It assigns to each x in [-1, 1] the unique angle θ in [0, π] such that cos θ = x. Consequently, arccos maps [-1, 1] onto [0, π].

Key identities: cos(arccos x) = x for x ∈ [-1, 1]. Conversely, arccos(cos θ) = θ for θ ∈ [0, π]. The function is decreasing on its domain, so larger x yields smaller angles. A common relation is arccos x + arcsin x = π/2 for x ∈ [-1, 1].

Derivative and range: for x ∈ (-1, 1), d/dx arccos x = -1 / √(1 - x^2). The derivative is undefined at x = ±1, reflecting vertical tangents at the endpoints. The range of arccos is [0, π].

Applications and context: arccosine is used to determine angles from cosine values in geometry and trigonometry, to solve equations involving inverse cosine, and in numerical methods that require inverse cosine computations. It is the standard means of extracting an angle in radians (or degrees, when converted) from a cosine value within the valid domain. Related functions include arccosine’s complementarity with arcsin, since arccos x + arcsin x = π/2.