# Transformation of Trigonometric Functions into Hyperbolic Functions Based on Cable Statics
## 1 Introduction  
[PERSON]'s theorem and trigonometric functions are widely used to solve right triangles. In contrast, the analysis of oblique triangles relies on alternative methodologies, such as the law of sines and the law of cosines. However, some approaches remain underutilized in education; for example, [PERSON]'s formulas define the relationships among a triangle's sides and angles [1]. Indeed, the tangent law can be used for the study of oblique triangles and is derived from [PERSON]'s formulas. In teaching the analysis of oblique triangles, sum and difference trigonometric identities and [PERSON]'s formulas are used [1,2]. However, the study of hyperbolic functions has been limited in the context of these triangles, with the focus primarily on their relationship with trigonometric functions. Trigonometric functions are fundamental for the calculation of hyperbolic functions due to their Eulerian properties, which are related through complex numbers. In fact, the identities that relate these functions are derived using [PERSON]'s number [3]. Other, less commonly explored studies have linked trigonometric functions with hyperbolic functions due to their geometric properties [4,5]. Thus, the Gudermann function is derived, which relates these trigonometric and hyperbolic functions without resorting to complex numbers. One application of the Gudermann function is the Mercator map projection,which involves mapping the Earth's surface by projecting the spherical surface of the Earth onto a cylindrical surface, preserving the true length of the parallels while distorting the equatorial lines in the projection [5; 6]. Traditionally, the solution of triangles is approached through trigonometry, overlooking the analysis of triangles with hyperbolic functions in educational contexts [1]. This manuscript is aimed at both mathematics education and research, as it integrates mathematical formulations derived from physical phenomena with practical applications in engineering. By linking traditional trigonometric concepts with innovative methods based on hyperbolic functions, it provides a powerful didactic tool and opens a new field of study that encourages future research, enriching both areas. The research proposes an original methodology that extracts a specific physical phenomenon, cable engineering, to establish a precise relationship between trigonometric and hyperbolic functions [7; 8; 9; 10]. This approach not only provides a robust mathematical framework for solving triangles in engineering problems but also facilitates the coherent integration of theory and practice. The results obtained have the potential to enrich mathematics education and drive research exploring the applications of hyperbolic functions derived from physical phenomena in undergraduate mathematics and engineering programs. The solution of triangles has traditionally been approached through trigonometry, a fundamental discipline in teaching mathematics at the high school and university levels. The handling of triangles relies on the use of trigonometric functions, equations, and identities that facilitate the resolution of angles and the application of classical theorems, such as [PERSON]'s theorem, which are foundational for calculus in engineering due to the geometric properties inherent in these figures [9]. However, in the context of engineering, particularly in the statics of cables, physical phenomena emerge where trigonometry is closely related to hyperbolic functions, enabling the approach of complex problems from an innovative perspective [7; 10]. This article shows the deduction of the hyperbolic transformation applied to right triangles. The theorem is an alternative method for solving engineering problems and can be easily used in industrial applications and in teaching calculations, for example, to determine distances, in problems of stability of rigid bodies, and in electricity [2; 7]. Other alternatives for the use of the proposed transformation are applied to [PERSON]'s formulas.