Hyperbolic functions are a family of functions defined using exponential functions. The hyperbolic sine, sinh of x, equals e to the x minus e to the negative x, all divided by 2. The hyperbolic cosine, cosh of x, equals e to the x plus e to the negative x, divided by 2. These functions combine the exponential curves in different ways to create new mathematical relationships.
The complete family of hyperbolic functions includes six functions. Hyperbolic tangent, tanh, equals sinh over cosh. Hyperbolic cotangent, coth, equals cosh over sinh. Hyperbolic secant, sech, equals one over cosh. And hyperbolic cosecant, csch, equals one over sinh. These functions have important asymptotic behaviors and domain restrictions, with tanh approaching plus or minus one, and the reciprocal functions having vertical asymptotes where their denominators equal zero.
Hyperbolic functions have many important identities. The fundamental identity states that cosh squared x minus sinh squared x equals one. This is analogous to the Pythagorean identity for circular functions, but with a minus sign. There are also addition formulas for sinh and cosh, and even-odd properties. The fundamental identity can be visualized using the unit hyperbola x squared minus y squared equals one, where points on the hyperbola correspond to cosh t and sinh t values.
Hyperbolic trigonometric functions are mathematical functions that are analogs of the ordinary trigonometric functions. While circular trigonometric functions are based on the unit circle with equation x squared plus y squared equals 1, hyperbolic functions are based on the unit hyperbola with equation x squared minus y squared equals 1. These functions appear frequently in calculus, physics, and engineering applications.
The three basic hyperbolic functions are sinh, cosh, and tanh. Sinh x is defined as e to the x minus e to the negative x, all divided by 2. Cosh x is defined as e to the x plus e to the negative x, all divided by 2. Tanh x is the ratio of sinh x to cosh x. These definitions using exponential functions make hyperbolic functions particularly useful in solving differential equations and modeling exponential growth and decay.
Hyperbolic functions satisfy many important identities, similar to trigonometric functions. The fundamental hyperbolic identity is cosh squared x minus sinh squared x equals 1, which is analogous to the Pythagorean identity for circular functions. There are also addition formulas and double angle formulas. These identities are essential for solving hyperbolic equations and simplifying complex expressions involving hyperbolic functions.
The derivatives of hyperbolic functions have elegant patterns. The derivative of sinh x is cosh x, and the derivative of cosh x is sinh x. This creates a cyclic relationship similar to trigonometric functions. The derivative of tanh x is sech squared x, and the derivative of sech x is negative sech x times tanh x. These derivative relationships make hyperbolic functions particularly useful in calculus and differential equations.
Hyperbolic functions have numerous practical applications across science and engineering. The most famous is the catenary curve, which describes the shape of a hanging chain or cable under its own weight. This curve follows the equation y equals a times cosh of x over a. Hyperbolic functions also appear in special relativity, where they describe Lorentz transformations. They are essential in solving many differential equations, particularly those involving exponential growth and decay, heat conduction, and wave propagation. In engineering and architecture, catenary curves are used to design stable arches and suspension bridges.
Inverse hyperbolic functions can be expressed in terms of natural logarithms. The inverse sinh of x equals the natural log of x plus the square root of x squared plus 1. The inverse cosh of x equals the natural log of x plus the square root of x squared minus 1, but this is only defined for x greater than or equal to 1. The inverse tanh of x equals one half times the natural log of 1 plus x over 1 minus x, defined only for x between negative 1 and 1. These inverse functions are reflections of the original functions across the line y equals x.