cosecante

The cosecant, abbreviated as csc, is a trigonometric function related to angles in a right-angled triangle. It is defined as the reciprocal of the sine function. Specifically, for an angle θ in a triangle, the cosecant is expressed as csc(θ) = 1 / sin(θ). Since sine represents the ratio of the length of the side opposite the angle to the hypotenuse, cosecant can be interpreted as the ratio of the hypotenuse to the opposite side.

Cosecant is undefined when sin(θ) equals zero, which occurs at angles where θ is an integer multiple of π (or 180 degrees), corresponding to points where the opposite side of the triangle collapses to zero length. Its domain therefore excludes these angles. The range of the cosecant function is all real numbers less than or equal to -1 or greater than or equal to 1.

In the unit circle, cosecant corresponds to the reciprocal of the y-coordinate of a point on the circle for a given angle θ. It has a period of 2π, meaning its graph repeats every full rotation. The cosecant function exhibits vertical asymptotes where sin(θ) is zero and has a characteristic wave-like shape in its graph, with branches approaching asymptotes near these points.

Cosecant finds application in various fields including engineering, physics, and mathematics for modeling periodic phenomena and solving triangle-related problems. It is often used alongside its reciprocal functions, sine, cosine, and tangent, to facilitate calculations involving angles and their relationships within a circle or triangle system.