What Is the Range of Arctan?
The range of arctan—also written as atan or tan⁻¹—is the set of all possible output values for the inverse tangent function. Here's the thing — unlike the tangent function, which repeats its values infinitely due to its periodic nature, arctan is designed to return a single, consistent angle for every real input. In real terms, this range is crucial for understanding how arctan behaves, especially when solving equations, graphing functions, or working with calculus concepts like derivatives and integrals. This makes the range of arctan one of the first things students and professionals need to grasp when working with inverse trigonometric functions.
Introduction to Arctan and Its Domain
The arctangent function is the inverse of the tangent function. The tangent function, tan(x), maps angles to real numbers. To turn this into a proper function—one that assigns exactly one output to each input—we restrict the domain of tangent to a specific interval. On the flip side, tangent is periodic with a period of π radians (or 180°), meaning it repeats its values every π units. This periodicity means that for a given output y, there are infinitely many angles x that satisfy tan(x) = y. That's why for example, tan(π/4) = 1, so arctan(1) = π/4. That's why in simpler terms, if y = tan(x), then x = arctan(y). This restriction leads directly to the range of arctan.
The domain of arctan is all real numbers: (-∞, ∞). Even so, this is because tangent can take any real value, from negative infinity to positive infinity. As x approaches π/2 from the left, tan(x) grows without bound; as x approaches -π/2 from the right, tan(x) decreases without bound. Since every real number y corresponds to some angle x where tan(x) = y, arctan is defined for all real inputs Still holds up..
It sounds simple, but the gap is usually here.
The Range of Arctan: (-π/2, π/2)
The range of arctan is the open interval (-π/2, π/2). In degrees, this is (-90°, 90°). Simply put, for any real number y, the value arctan(y) will always lie strictly between
