Arctand

Arctand is a hypothetical mathematical operator that serves as a generalized inverse of the tangent function, with explicit control over the branch of the inverse. It is defined to produce an angle x such that tan x = y, with x chosen from a specified interval according to an integer parameter.

Formally, arctand(y; k) denotes the value x for which tan x = y and x lies in the interval (−π/2 + kπ, π/2 + kπ), where k is any integer. The parameter k selects the branch of the inverse tangent, reflecting the periodicity of the tangent function. When k = 0, arctand coincides with the principal value of the arctan function, yielding x ∈ (−π/2, π/2).

Notationally, arctand is sometimes written as arctand(y, k) or arctand(y; k). In practice, the standard arctan function corresponds to arctand(y; 0). For other branches, arctand returns angle representatives that differ by integer multiples of π.

Relationship to other functions: arctand is functionally equivalent to arctan when the branch is fixed to the principal interval. It provides a framework for solving trigonometric equations where the solution must be stated with explicit branch information. In computational contexts, similar goals are achieved with functions like atan2, which determine a quadrant-aware angle from coordinates; arctand can be viewed as a one-argument analogue with explicit branch selection.

See also: arctan, inverse trigonometric functions, arctan with branches, atan2.