OICC Press Communications in Nonlinear Analysis 2371-7920 12 1 2025 11 09 Convex Trigonometric Functions and Their Applications EN Morteza Bayat Journal Article 2025 11 09 In this paper, we first introduce a new kind of trigonometric functions called convex trigonometric functions. These functions are natural generalization of the ordinary trigonometric and hyperbolic functions. Then investigate their properties and their connections with the trigonometric Jacobian elliptic functions. Also we find the period of circular trigonometric function and power series of circular and hyperbolic trigonometric functions, using Maple package. Finally, we present some applications of these functions for obtaining the closed from of some special integrals and finding the solution of the differential equation of simple harmonic pendulum.