On elliptical trigonometry

Abstract

The ordinary trigonometric functions are defined by means of angles and lengths on the unit circle. There are several derivatives of classical trigonometric functions such as hyperbolic, polar, spherical, Fourier, inverse, log, complex, q-versions etc. In this paper, we add this list a new version of trigonometry which will be consistently called as elliptic trigonometry by working on an ellipse instead of a circle. Although the definitions of elliptic trigonometric functions remind us the classical trigonometry, there are several interesting differences between the two whence the relations and expansions are considered. Also as ellipses has applications in engineering, medicine, astronomy, architecture, etc., it is expected that this new notion will have new applications or improve the existing applications in these and other areas.