Generalized Trigonometric Functions

Abstract

A new trigonometric functions called generalized trigonometric functions are defined by system of first order nonlinear ordinary differential equations with initial conditions. This system is related to the Hamilton system. In this thesis, we identify new trigonometric functions using the equation │x│m + │y│m = 1, for m > 0. The main purpose of this thesis is generalizing the rules of the usual trigonometric functions to the new functions. We prove some properties and identities of these functions and graph these functions when 0<m<1, m = 1, and m > 1. When m=2, we notice that new functions coincide with the usual trigonometric functions. Moreover, we give relation formulas of these functions which depend on the usual trigonometric functions. We find the first derivative of these functions and prove that for m = 1, the generalized trigonometric functions y = sin1 θ and y = cos1 θ have no derivatives at θ = n; n ∈ Z. We find generalizing form of the first derivative of these functions when m is odd and when m is even. We notice that the derivatives are different. Throughout this study, we understand more about the theoretical framework of trigonometric functions.