Defining the secant function
The word "secant" was introduced by Alhabaš Alhãsib around 800. Later on Th. Fincke (1583) used the word "secans" in Latin for notation of the corresponding function. The secant function also appeared in the works of A. Magini (1592) and B. Cavalieri (1643). J. Kresa (1720) used the symbol "sec" that was later widely used by L. Euler (1748).
The classical definition of the secant function for real arguments is: "the secant of an angle in a right‐angle triangle is the ratio of the length of the hypotenuse to the adjacent leg." This description of is valid for when this triangle is nondegenerate. This approach to the secant 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 .
Comparing the definition with the definition of the cosine function shows that the following formula can also be used as a definition of the secant function: