cscx

Csc x, or the cosecant of x, is a trigonometric function defined as the reciprocal of the sine: csc x = 1/sin x, for angles x where sin x ≠ 0. It is commonly written as csc x or, less commonly, cscx.

The domain of csc x is all real numbers x except multiples of π, where sin x = 0. The range consists of all real numbers with absolute value at least 1, i.e., (-∞, -1] ∪ [1, ∞). The function has period 2π and is an odd function, satisfying csc(-x) = -csc x. It has vertical asymptotes at x = kπ and attains ±1 at x = π/2 + kπ.

Csc x relates to other trigonometric functions through identities such as csc x = 1/sin x and csc^2 x = 1 + cot^2 x. These connections make it useful in various trigonometric and calculus contexts, especially when expressions involve reciprocals of sine.

In calculus, the derivative of the cosecant is d/dx [csc x] = -csc x cot x. An antiderivative is ∫ csc x dx = -ln|csc x + cot x| + C, with an alternative form often given as ∫ csc x dx = ln|tan(x/2)| + C.

Applications of csc x appear in triangle problems, trigonometric equations, and problems involving integrals and Fourier analysis, wherever reciprocal sine expressions arise.