Transformation of Trigonometric Functions into Hyperbolic Functions Based on Cable Statics

Trigonometric functions are widely used to express relationships between the sides and angles of triangles, being fundamental in a wide variety of fields in science and engineering. Previous research indicates that the Gudermann function connects hyperbolic and trigonometric functions without requiring complex numbers, while the Mercator projection maps the Earth’s spherical surface onto a cylindrical plane. This article presents four key contributions derived from hyperbolic functions, with the main proof applying Newton’s first law in the static case of cables. First, a new method relates right triangle formulas to the sides of a right triangle, facilitating vector decomposition along the X and Y axes. Second, a right triangle function with a hyperbolic angle is proposed, relating the three sides of a right triangle and the hyperbolic angle, offering an alternative to the Pythagorean theorem. Third, the law of hyperbolic cosines and the law of hyperbolic tangents is applied to trigonometric problems. Fourth, the hyperbolic Mollweide’s formula is used to solve oblique triangles. These results demonstrate the potential of hyperbolic transformations in engineering and mathematical contexts, for both education and research. Future investigations should include experimental and analytical tests to further extend the applications to all branches based on mathematics.