# Transformation of Trigonometric Functions into Hyperbolic Functions Based on Cable Statics
## 5 Discussion of Results  
In this section, the results obtained from hyperbolic angle functions are discussed, along with key observations regarding the study and its correlation with functions reported in the literature. This article deems it necessary to clarify that hyperbolic angles are being used in the context of hyperbolic functions, as trigonometric functions are being developed in parallel, in order to avoid confusion between the two types of angles. However, for simpler use, in other research, the term \"angle\" may be employed when working with hyperbolic functions. The trigonometric functions are related by Gudermann to the hyperbolic functions. The angle associated with the trigonometric function is referred to as the Gudermann angle, denoted as gd \(x\), in honor of [PERSON]. This angle represents the same value as the trigonometric functions, but in this article, the term is simplified to \"hyperbolic angle\" in order to facilitate understanding. An innovative approach to the demonstration of the Gudermann function is presented in this work, based on engineering concepts. The statics of cables, used in transmission lines, is employed as the foundation for the solution to the Gudermann function, which enables a better understanding of mathematics through the analogy with physical phenomena [5]. Previous research has shown that in hyperbolic triangles, the sum of their interior angles differs from \(\pi\), and the relationship between these angles is addressed through Euclidean trigonometry. In other words, hyperbolic trigonometry is solved using Euclidean geometry. Unlike these studies, the focus of this work is on the resolution of Euclidean triangles, where the sum of the vertex angles is equal to \(\pi\). The proposed solution employs non-Euclidean functions, specifically hyperbolic functions, which constitutes an innovative finding and establishes a bridge between the resolution of triangles in Euclidean trigonometry, where Euclidean triangles are solved using hyperbolic functions [28]. The application of the Mercator projection involves representing the spherical surface of the Earth on a plane, which produces a distortion of the areas as they move away from the equator. This research presents an innovative approach to deriving the Mercator projection from cable statics, emphasizing the advantage that trigonometric angles are converted into hyperbolic angles without the need for hyperbolic functions. Another benefit of this variant of the Mercator projection is that hyperbolic angles are transformed into trigonometric angles without the need for trigonometric functions. Indeed, for both cases, Euler's number is replaced by the natural logarithm introduced in this study. In this research, several applications are carried out in which numerical results are obtained by comparing the use of hyperbolic functions with trigonometric functions, storing all decimal values of the angles. Exact values for the sides are obtained. That is, the only numerical errors that could arise when using hyperbolic functions are attributable to rounding errors in the computing systems used. A significant contribution of this research is the use of hyperbolic functions to solve trigonometric problems related to triangles, with applications that can be extended to teaching in undergraduate and graduate programs. In general terms, it is found that both methods are valid, and it cannot be stated that one method is superior to the other.