Cotangent

Cotangent, denoted cot x, is the trigonometric function defined as the ratio of the cosine to the sine: cot x = cos x / sin x. Equivalently, cot x = 1 / tan x. In a right triangle, for a non-right angle θ, cot θ equals the length of the adjacent side divided by the opposite side. The function is undefined wherever sin x = 0, i.e., at x = kπ (k ∈ Z).

Cotangent has period π. Its graph on each interval (kπ, (k+1)π) is strictly decreasing and has vertical asymptotes at x = kπ. As x approaches kπ from the right, cot x tends to +∞; from the left, it tends to −∞. The range on each interval is all real numbers.

Key identities include cot^2 x + 1 = csc^2 x and cot x = cos x / sin x. It is also related to tangent by cot x = tan(π/2 − x). Calculus provides cot’s derivative and antiderivative: d/dx cot x = −csc^2 x, and ∫ cot x dx = ln|sin x| + C.

Applications of cotangent arise in trigonometric equations, right-triangle problems, and certain integrals. It frequently appears in conjunction with cosecant and tangent functions in identities and transformations.