Author(s)

HaoLin Fu

Affiliation(s)

School of Mathematics and Statistics, Hubei University of Education, Wuhan 430205, Hubei, China.

Corresponding Author

HaoLin Fu

ABSTRACT

This paper investigates some fundamental properties of hyperbolic trigonometric functions on the hyperbolic number plane. Hyperbolic numbers form a commutative ring with zero divisors, generated by two real numbers via the Cartesian basis. Within this algebraic framework, this paper systematically establish the system of hyperbolic trigonometric functions. By fully leveraging the decomposition properties of hyperbolic numbers and their associated trigonometric functions, this work effectively overcomes the mathematical difficulties arising from the presence of zero divisors in the hyperbolic number ring. On this basis, this paper for the first time in the context of a hyperbolic number ring containing zero divisors, systematically derived and rigorously proved the addition theorems for the hyperbolic sine, cosine, tangent, and cotangent functions. On this basis, we have successfully derived and rigorously proved the addition theorems for the hyperbolic sine, cosine, tangent, and cotangent functions, establishing a complete system of angle addition formulas and laying a solid foundation for the theory of hyperbolic functions. The addition theorems established in this research will provide an important theoretical foundation for the further development of hyperbolic analysis in function theory, while simultaneously injecting new research momentum into the further study of the properties and expansions of hyperbolic trigonometric series. These theoretical achievements are expected to play significant roles in the study of hyperbolic differential equations, geometric analysis, and related physical problems.

KEYWORDS

Hyperbolic numbers; Hyperbolic trigonometric functions; Identity transformation; Addition theorems

CITE THIS PAPER

HaoLin Fu. Some properties of hyperbolic trigonometric functions. World Journal of Engineering Research. 2025, 3(6): 42-49. DOI: https://doi.org/10.61784/wjer3070.

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