Arccos

Arccos, denoted arccos x or cos^{-1} x, is the inverse function of the cosine function restricted to the interval [0, π]. It maps real numbers x in [-1, 1] to values in [0, π]. By definition, arccos x is the unique angle θ in [0, π] such that cos θ = x. Consequently, cos(arccos x) = x for x in [-1, 1], and arccos(cos θ) = θ for θ in [0, π].

Endpoints and basic behavior: arccos(1) = 0 and arccos(-1) = π. The function is continuous and strictly decreasing on [-1, 1]. Its derivative is arccos'(x) = -1 / sqrt(1 - x^2) for x in (-1, 1), with the derivative undefined at x = ±1 due to vertical tangents.

Relationships and notation: For x in [-1, 1], arcsin x + arccos x = π/2. The arccos notation cos^{-1} x is commonly used, but may be confused with 1/cos x; the arccos function is the inverse of cos on the principal domain [0, π], not a reciprocal.

Extensions and applications: Arccos is widely used in trigonometry, geometry, and solving equations where a cosine value must be translated back to an angle. It is a standard function in mathematical software and programming libraries, enabling the recovery of angle measures from cosine values. In complex analysis, arccos can be extended with a more general branch structure, but the real-valued arccos is defined on [-1, 1] with range [0, π].