Arccostanphi

Arccostanphi is a hypothetical mathematical function formed by composing the arccosine with the tangent of an angle phi, defined as arccostanphi(phi) = arccos(tan(phi)). It is not a standard or widely used notation in mainstream mathematics, but it serves as a convenient example for exploring the behavior of inverse-trigonometric compositions.

Domain and range. The function is real-valued when tan(phi) lies in the interval [-1, 1]. This occurs for phi in the union of intervals [kπ − π/4, kπ + π/4] for all integers k. At the endpoints, tan(phi) = ±1 and arccos(tan(phi)) equals 0 or π, respectively. Since tan(phi) has period π, arccostanphi also has period π. The range of arccostanphi on its domain is [0, π], because arccos maps [-1, 1] onto [0, π].

Properties. On each interval [kπ − π/4, kπ + π/4], tan(phi) increases monotonically from −1 to 1, and arccos is a decreasing function, so arccostanphi decreases from π to 0 across the interval. Consequently, arccostanphi is continuous on the closed interval and periodic with period π. The derivative exists on the interior of the domain where |tan(phi)| < 1 and is given by d/dphi arccostanphi(phi) = −sec^2(phi) / sqrt(1 − tan^2(phi)).

Special values and examples. At phi = 0, arccostanphi(0) = arccos(0) = π/2. At phi = π/4, arccostanphi(π/4) = arccos(1) = 0. At phi = −π/4, arccostanphi(−π/4) = arccos(−1) = π.

Relation to broader concepts. Arccostanphi illustrates how inverse functions can be composed with non-invertible or periodic functions, highlighting domain restrictions and piecewise behavior that arise in real-valued compositions. See also arccos, tan, and inverse-trigonometric functions.