arccosx

Arccosx, commonly written arccos(x) or acos(x), is the inverse cosine function. It returns the angle whose cosine equals the given value x. Because the cosine function is not one-to-one on its entire domain, arccos is defined as the inverse of cos restricted to the principal branch [0, π]. Consequently, arccos maps the real interval [-1, 1] to [0, π].

If y = arccos x, then cos y = x and y lies in the interval [0, π]. The function is continuous and strictly decreasing on [-1, 1]. Its derivative is -1 / sqrt(1 - x^2) for x in (-1, 1); it is undefined at x = ±1 due to vertical tangents.

Key values and identities help with computation. arccos(1) = 0, arccos(0) = π/2, and arccos(-1) = π. An important identity is arccos x = π/2 − arcsin x, valid for x in [−1, 1]. Another commonly used relation is cos(arccos x) = x, with the inverse equality arccos(cos y) = y only when y ∈ [0, π].

Applications of arccosx include solving triangles, physics problems involving angles, and computer graphics. In programming and scientific computing, acos is a standard function name, typically returning values in radians. In complex analysis, arccos can be extended to complex arguments, with the principal value often expressed as arccos z = π/2 − arcsin z = −i ln(z + sqrt(z^2 − 1)).