## Cosecant

(csc)

The **cosecant** of an angle is defined using the right-triangle formed by this angle: it is the ratio of the triangle's hypotenuse to its opposite side. Calcute implements it with the *csc* function.

csc(1)
1.18839510578

In the above example, the given angle is in radians.

The triangle involved in the definition of the **cosecant** function is commonly shown in the cartesian plane with its adjacent side on the X axis and its angle point located at the origin (0,0). The other end of the triangle's hypotenuse marks a point on the periphery of a circle centered at the origin; the circle's radius is the length of the hypotenuse. If the angle is increased from 0 to 2 pi radians (0 to 360 degrees) then the end point of the hypotenuse traces a full circle. As the angle of the hypotenuse changes, so does its **cosecant** value as the shape of the corresponding triangle changes to match.

The **cosecant** is undefined at multiples of 1/2 of a full circle (including zero). At these angles, the triangle geometrically vanishes into a simple line segment on the X axis. The value of the triangle's opposite side is zero and division by zero is undefined. For example, *csc* is undefined at pi radian, or equivalently at 180 degrees or at 200 gradiens.

The **cosecant** function is periodic. Except for those angles that represent multiples of 1/2 of a circle, it is defined for all real numbers. As angle values increase (or decrease) outside the range of [0, 2 pi] radians that represent one full rotation, the value of *csc* for these angles repeats itself periodically with each new rotation.