Cosarccos

Cosarccos is the inverse of the cosine function when the latter is restricted to a domain on which it is one-to-one. It is commonly identified with the arccosine function, often written arccos x, and in some contexts as cosarccos(x). The function returns an angle whose cosine is a given value.

Definition and domain. For x in the interval [-1, 1], cosarccos(x) yields the unique y in the interval [0, π] such that cos y = x. Thus, cosarccos(x) is defined only for -1 ≤ x ≤ 1, and its range is 0 ≤ cosarccos(x) ≤ π.

Key identities. The composition cosarccos(x) and x satisfies cos(cos arccos x) = x, and arccos(cos θ) = θ whenever θ lies in [0, π]. The notation cosarccos is sometimes used interchangeably with arccos, though some authors reserve arccos for the standard inverse cosine function.

Derivative and behavior. The derivative of cosarccos x with respect to x is -1 / sqrt(1 - x^2) for x in (-1, 1). At the endpoints x = ±1, the derivative is undefined, reflecting the vertical tangent behavior of the inverse near the domain boundaries. Graphically, cosarccos is a monotonically decreasing function from π at x = -1 to 0 at x = 1.

Applications. Cosarccos is used to determine angles from cosine values, solve trigonometric equations, and in integration and analysis where inverse trigonometric functions appear. It is a fundamental tool in geometry, physics, engineering, and computer science.

Notation notes. Depending on the context, cosarccos, arccos, and acos may be used to denote the same principal inverse cosine function, with arccos and acos more common in mathematical and programming contexts, and cosarccos appearing in some texts and software documentation.