Cosecant

The cosecant, abbreviated csc, is a trigonometric function defined as the reciprocal of sine: csc x = 1/sin x, for angles x where sin x ≠ 0. In a right triangle, csc θ equals the hypotenuse divided by the opposite side, the reciprocal of sine.

Domain and range: csc x is defined for all real x except multiples of π (x ≠ nπ). Its range is (-∞, -1] ∪ [1, ∞).

Period and symmetry: The cosecant function has period 2π and is an odd function, meaning csc(-x) = -csc(x).

Relations to other functions: csc x is the reciprocal of sine, so sin x = 1/csc x. It can also be expressed as csc x = sec(π/2 − x), since sin x = cos(π/2 − x).

Calculus: The derivative of csc x is -csc x cot x. An antiderivative is ∫ csc x dx = -ln|csc x + cot x| + C.

Special values and graph: At x = π/2 + kπ, sin x = ±1, so csc x = ±1. The graph has vertical asymptotes at x = kπ and oscillates between the two branches with the same 2π period as sine.

Applications: Cosecant appears in trigonometric identities, triangle calculations, and various integrals in calculus, often alongside its reciprocal counterparts.