Defining the sine function
The sine function is one of the oldest mathematical functions. It was used in ancient Greece and India, and in 1140, R. de Chesters translated Abu Ja'far Muhammed ibn Musa al‐Khwarizme's works and used the word "sine" (in Latin, "sinus").
The classical definition of the sine function for real arguments is: "the sine of an angle in a right‐angle triangle is the ratio of the length of the opposite leg to the length of the hypotenuse." This description of is valid for when the triangle is nondegenerate. This approach to sine 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 sine function:
This series converges for all finite numbers .