What does arctangent mean?
Arctangent is the mathematical function that reverses the tangent function: given a real number, it returns the unique angle between −90° and 90° (−π/2 and π/2 radians) whose tangent equals that number. It is typically written arctan(x) or, on calculators, tan⁻¹(x), where the superscript denotes an inverse function rather than a reciprocal. Arctangent answers a practical question — if I know the ratio of opposite to adjacent sides of a right triangle, what is the angle? — which makes it indispensable in surveying, navigation, physics, and computer graphics. Programmers frequently use its two-argument cousin atan2, which determines the correct quadrant automatically. A technical term confined largely to mathematics and the sciences, arctangent carries no emotional connotation, but it embodies a foundational idea of trigonometry: that functions can be run backwards, recovering angles from ratios.
nounThe inverse of the tangent function: for a given real number x, arctangent is the angle whose tangent is x, expressed in radians or degrees. It is usually written as arctan(x) or tan⁻¹(x) and returns a value in the interval (−π/2, π/2).
- The angle θ in the range −90° to 90° (or −π/2 to π/2 radians) whose tangent equals a given number; the principal value of the inverse tangent function.
"To find the angle of elevation from the slope's gradient, take the arctangent of the rise over the run."
"The engineer computed the arctangent of 0.75 to determine that the ramp rose at an angle of about 36.9 degrees."
"In most programming languages, atan2(y, x) extends the ordinary arctangent by returning the correct quadrant for a point in the plane."
Rarely used; arctangent almost always appears in the singular when referring to the function itself.
"The textbook listed the derivatives of the arcsines and arctangents together."
Your calculator's 'tan⁻¹' button doesn't compute one divided by tangent — it computes the arctangent, undoing the tangent entirely.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of arctangent
The word arctangent combines the prefix arc-, borrowed from Latin arcus meaning 'bow' or 'arch' and long used in mathematics to denote an angular measure, with 'tangent', from Latin tangens, 'touching', so named because a tangent line touches a circle at a single point. The 'arc-' prefix was adopted across the inverse trigonometric functions — arcsine, arccosine, arctangent — to signal that the output is an arc, or angle, rather than a ratio. These terms were standardized as European analytic trigonometry matured in the seventeenth and eighteenth centuries, alongside the alternative notation of writing the function name with a superscript −1.
How arctangent is actually used
Arctangent is a technical term used almost exclusively in mathematics, engineering, physics, and computer science; it rarely appears in general conversation. The notation arctan is standard in mathematics, while tan⁻¹ is common on calculators — though the superscript −1 there denotes functional inverse, not reciprocal. In computing, the two-argument variant atan2 is often preferred because it resolves quadrant ambiguity.
Easily confused with arctangent
Tangent maps an angle to a ratio of side lengths, while arctangent does the reverse, mapping a ratio back to the angle.
Arcsine is the inverse of the sine function and returns the angle whose sine equals a given value, whereas arctangent is the inverse of tangent.