arccos3q2p

arccos3q2p is not a standard named function, but in many contexts it is shorthand for the inverse cosine applied to the expression 3 q^2 p, where p and q are real numbers. This follows the usual arccos function, which is the inverse of cosine restricted to the interval from 0 to pi.

The key mathematical property is the domain of arccos: its input must lie in [-1, 1]. For real p and q, the argument is a = 3 q^2 p. Since q^2 is nonnegative, the sign of a matches the sign of p (or is zero). If q ≠ 0, the domain requirement is -1 ≤ 3 q^2 p ≤ 1, equivalently -1/(3 q^2) ≤ p ≤ 1/(3 q^2). If q = 0, then a = 0 for all p, and arccos(0) = π/2, so the expression is defined for all p in that case. The range of arccos is [0, π], and arccos is a decreasing function of its input.

Special values help illuminate the behavior: if p = 0, a = 0 and arccos(0) = π/2. If a = 1, arccos(1) = 0; if a = -1, arccos(-1) = π. When p > 0 (with q ≠ 0), the value lies in [0, π/2); when p < 0, it lies in (π/2, π].

Notes on notation and usage: because arccos3q2p is not a standard function name, interpret it as arccos(3 q^2 p). In computations, the result is normally in radians unless degrees are specified; always verify the input lies within [-1, 1] to avoid domain errors.