Hyperbolic functions are mathematical functions that are analogous to trigonometric functions but are based on hyperbolas instead of circles. The two primary hyperbolic functions are the hyperbolic sine, denoted as sinh, and the hyperbolic cosine, denoted as cosh. These functions are defined using exponential functions: sinh(x) = (e^x - e^(-x))/2 and cosh(x) = (e^x + e^(-x))/2, where e is the base of natural logarithms. Hyperbolic functions have various applications in mathematics, physics, and engineering, particularly in areas involving hyperbolic geometry and calculus. They also appear in the solutions to certain differential equations and in the study of special relativity. Additionally, the inverse hyperbolic functions, such as arsinh and arcosh, provide ways to solve equations involving hyperbolic functions.