An elementary treatise on elliptic functions as trigonometry

Abstract

This article concerns the examination of trigonometric identities from an elliptic perspective. The treatment of elliptic functions presented herein adheres to a structure analogous to the traditional exposition of trigonometric functions, with the exception that an ellipse replaces the unit circle. The degree of similarity between the elliptic functions and their trigonometric counterparts is moderated by the periodicity of the so-called El- functions. These identities not only establish the values of the functions, but also establish a correlation between their ratios and the major and minor axes of the underlying ellipse. The resemblance between the functions is somewhat modified by the periodic nature of the El-identities, whereby each ratio is associated with the major and minor axis of the ellipse. This article adopts the notation (E) to denote the El- functions and distinguish them from the opposite circular functions.