cotangente

Cotangente (cotangent) is a trigonometric function defined for real numbers x by cot(x) = cos(x)/sin(x) = 1/tan(x). It is undefined where sin(x) = 0, i.e., at x = nπ. The function has period π and takes all real values (range: all real numbers). Its graph has vertical asymptotes at x = nπ and zeros at x = π/2 + nπ.

Identity relations: cot x = cos x / sin x; also cot x = 1/tan x for tan x ≠ 0. A Pythagorean identity states 1 + cot^2 x = csc^2 x. The derivative is d/dx[cot x] = -csc^2 x, and an antiderivative is ∫ cot x dx = ln|sin x| + C.

Geometric interpretation: in a right triangle, for a given acute angle θ, cot θ equals the ratio of the length of the adjacent side to the length of the opposite side (cot θ = adjacent/opposite).

Applications: cotangent appears in solving trigonometric equations and in calculus, where it arises in integrals and differential equations, as well as in physics and engineering contexts involving periodic phenomena and angular relationships.