Arccosines

Arccosines refer to the inverse cosine function, denoted arccos or cos^-1. It is the inverse of the cosine function restricted to the interval [0, pi], where cosine is strictly decreasing from 1 to -1, ensuring a one-to-one correspondence and a well-defined inverse.

Domain and range: For real inputs, arccos x is defined for x in [-1, 1] and yields values in [0, pi]. Outside this domain, arccos may be defined only as a complex-valued function in many contexts; in real-valued mathematics, it is typically left undefined for x outside [-1, 1].

Definition and principal value: If y = arccos x, then cos y = x and y ∈ [0, pi]. This y is the principal value of the inverse cosine.

Key properties: arccos(-x) = pi - arccos x. Also arccos x + arcsin x = pi/2 for x in [-1, 1]. Another common form is arccos x = 2 arctan( sqrt((1 - x)/(1 + x)) ), valid for x in (-1, 1].

Derivative and behavior: d/dx arccos x = -1/√(1 - x^2) for x ∈ (-1, 1). The derivative is unbounded as x approaches ±1. The function is continuous and strictly decreasing on [-1, 1].

Special values: arccos(1) = 0, arccos(0) = pi/2, arccos(-1) = pi.

Series and relations: arccos x = pi/2 − arcsin x, and arcsin x has the usual power series around 0: x + x^3/6 + 3x^5/40 + …. Arccosines play a central role in trigonometry, geometry, and numerical methods.