arccossin

Arccossin refers to the composition arccos(sin x), i.e., the inverse cosine applied to the sine of x. It is a real-valued function defined for all real x, with output in the interval [0, π].

Properties:

- Domain: all real numbers; Range: [0, π].

- Periodicity: the function is 2π-periodic, since sin x and arccos have compatible periodic behavior.

- Behaviour within one period: on the central interval from -π/2 to 3π/2, arccos(sin x) can be expressed piecewise as

- arccos(sin x) = π/2 − x for x ∈ [−π/2, π/2],

- arccos(sin x) = x − π/2 for x ∈ [π/2, 3π/2].

These pieces join smoothly at x = π/2.

Equivalent compact forms:

- On the central interval [-π/2, 3π/2], arccos(sin x) = |π/2 − x|.

- More generally, the function can be described periodically by extending the central piece with period 2π, yielding the same range [0, π].

Relation to other functions:

- Since sin x = cos(π/2 − x), arccos(sin x) = arccos(cos(π/2 − x)). The standard expression for arccos(cos y) yields a sawtooth-like or V-shaped curve, reflected and shifted according to y = π/2 − x.

Examples:

- x = 0: sin 0 = 0, arccos(0) = π/2.

- x = π/2: sin π/2 = 1, arccos(1) = 0.

- x = π: sin π = 0, arccos(0) = π/2.

- x = 3π/2: sin 3π/2 = −1, arccos(−1) = π.

Notes:

Arccossin is not a standard elementary function but a common composition studied for its simple linear behavior within each period and its periodic, piecewise definition.