What does cotangent mean?
Cotangent is one of the six fundamental trigonometric functions, usually written as cot. Defined as the reciprocal of the tangent, it gives, for an acute angle in a right-angled triangle, the ratio of the length of the adjacent side to that of the opposite side; more generally, in analytic terms, cot θ equals cos θ divided by sin θ. Like the tangent, it is periodic, repeating every 180 degrees, and its graph consists of repeating branches separated by vertical asymptotes where the function is undefined. The cotangent appears throughout geometry, calculus, wave analysis, and engineering, and it extends naturally to hyperbolic and complex settings. Though encountered less often than sine, cosine, and tangent, it is indispensable in certain identities and integrals, and understanding it completes the standard trigonometric toolkit.
nounIn mathematics, a trigonometric function equal to the ratio of the adjacent side to the opposite side in a right-angled triangle, or equivalently the reciprocal of the tangent. It is conventionally abbreviated as cot.
- A trigonometric function, abbreviated cot, defined as the reciprocal of the tangent: for an angle in a right-angled triangle, the ratio of the adjacent side to the opposite side.
"The cotangent of 45 degrees is exactly 1."
"To solve the triangle, she evaluated the cotangent of the unknown angle."
"The graph of the cotangent function repeats every 180 degrees, with vertical asymptotes at multiples of 180."
The plural refers either to multiple values of the function or to instances in different contexts.
"The table lists the cotangents of angles from 0 to 90 degrees."
Flip the tangent upside down and you get the cotangent — a function so tidy it equals 1 at exactly 45 degrees.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of cotangent
Cotangent derives from the Latin tangere, meaning 'to touch', via the related term tangent — literally 'touching', a fitting name since a tangent line touches a curve. The prefix co-, from Latin cum ('with' or 'together'), marks the cotangent as the companion function to the tangent: each cofunction equals the corresponding ordinary function applied to the complementary angle. The term entered English mathematical usage in the early modern period alongside the other trigonometric functions as their study was formalized.
How cotangent is actually used
A technical term used almost exclusively in mathematics education and higher mathematics, such as calculus, geometry, and complex analysis. In modern practice it appears less frequently than sine, cosine, and tangent, but it remains standard in textbooks and on scientific calculators.
Easily confused with cotangent
The tangent compares the opposite side to the adjacent side in a right triangle, whereas the cotangent is its reciprocal, comparing the adjacent side to the opposite side.