Defining the cosecant function

The cosecant function is an old mathematical function. It was mentioned in the works of G. J. von Lauchen Rheticus (1596) and E. Gunter (around 1620). It was widely used by L. Euler (1748) and T. Olivier, Wait, and Jones (1881).

The classical definition of the cosecant function for real arguments is: "the cosecant of an angle in a right‐angle triangle is the ratio of the length of the hypotenuse to the length of the opposite leg." This description of is valid for when this triangle is nondegenerate. This approach to the cosecant can be expanded to arbitrary real values of if the arbitrary point in the ,‐Cartesian plane is considered and is defined as the ratio assuming that α is the value of the angle between the positive direction of the ‐axis and the direction from the origin to the point .

Comparing the classical definition with the definition of the sine function shows that the following formula can also be used as a definition of the cosecant function: