Basic hyperbolic functions: geometric and analytical construction of their definitions from the fundamental unit hyperbola

Abstract

This article focuses on the study of the principal or basic hyperbolic trigonometric functions, aiming to provide a rigorous understanding that integrates geometric and analytical approaches. The fundamental objective is to develop a formal construction of these functions based on the unit hyperbola, establishing precise definitions and demonstrating their assignment laws in terms of the natural exponential function. To this end, a geometric interpretation is initially introduced through the construction of segments called hyperbolic lines, which arise from the direct relationship with the unit hyperbola and are named hyperbolic sine, hyperbolic cosine, and hyperbolic tangent. Subsequently, a structured mathematical study is carried out, integrating tools such as parametrizations, integration methods, and analytical reasoning, demonstrating that these hyperbolic lines represent well-defined functions. This process not only consolidates the link between geometry and analysis but also provides a coherent formalization that strengthens the theoretical rigor of the treatment of hyperbolic functions. This work offers a significant contribution from both a mathematical and pedagogical perspective, proposing an integrative and progressive approach that benefits students, teachers, and researchers interested in mathematical analysis. In conclusion, although the principal hyperbolic functions are usually presented in the literature in a predominantly analytical manner, expressed in terms of the exponential function, they have a fundamental geometric origin. It also opens new possibilities for their didactic application and for the development of future research in the field of mathematical analysis.