Arccosqw

Arccosqw is the notation for the inverse cosine function applied to the argument qw. In mathematics, arccos denotes the inverse of the cosine function on a chosen domain. For a real input x, arccos x is the unique angle θ in the interval [0, π] such that cos θ = x. When the argument is written as qw, the function is read as arccos(qw) and can be applied to real or complex inputs, with appropriate interpretation of the domain.

For real inputs, arccosqw is defined for qw in the closed interval [-1, 1], yielding values in radians within [0, π]. The function is decreasing on (-1, 1) and satisfies key identities such as cos(arccos qw) = qw and arccos(cos θ) = θ for θ in [0, π]. Special values include arccos(1) = 0, arccos(-1) = π, and arccos(0) = π/2.

In the broader complex setting, arccos z is extended by principal value formulas, for example arccos z = π/2 − arcsin z and arccos z = −i log(z + sqrt(z^2 − 1)). These definitions involve choice of branch cuts for the square root and the logarithm, leading to potential multi-valuedness that is resolved by selecting a principal branch.

Derivative and behavior: for real x in (-1, 1), the derivative is d/dx arccos x = −1/√(1 − x^2). The function is analytic on the complex plane with cut lines extending from −∞ to −1 and from 1 to ∞.

Applications of arccosqw include solving trigonometric equations, computing angles from cosine values, and various fields in physics and computer science where inverse cosine calculations are required. See also arccos, cosine, and inverse trigonometric functions.