Defining the cosine function
The cosine function is one of the oldest mathematical functions. It was first used in ancient Egypt in the book of Ahmes (c. 2000 B.C.). Much later F. Viète (1590) evaluated some values of , E. Gunter (1636) introduced the notation "Cosi" and the word "cosinus" (replacing "complementi sinus"), and I. Newton (1658, 1665) found the series expansion for .
The classical definition of the cosine function for real arguments is: "the cosine of an angle in a right‐angle triangle is the ratio of the length of the adjacent leg to the length of the hypotenuse." This description of is valid for when the triangle is nondegenerate. This approach to the cosine can be expanded to arbitrary real values of if consideration is given to the arbitrary point in the ,‐Cartesian plane 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 .
The following formula can also be used as a definition of the cosine function:
This series converges for all finite numbers .