Title: 6_2 The Hyperbolic Functions
Course: HELM Workbooks 2023-2024 > Section: HELM 6: Exponential and Logarithmic Functions

Introduction

The hyperbolic functions \(\sinh x, \cosh x\), \(\tanh x\) etc are certain combinations of the exponential functions \(e^x\) and \(e^{-x}\). The notation implies a close relationship between these functions and the trigonometric functions \(\sin x, \cos x\), \(\tan x\) etc. The close relationship is algebraic rather than geometrical. For example, the functions \(\cosh x\) and \(\sinh x\) satisfy the relation \(\cosh^2x-\sinh^2x\equiv 1\); which is very similar to the trigonometric identity \( \cos^2x+\sin^2x \equiv 1\). (In fact every trigonometric identity has an equivalent hyperbolic function identity.) The hyperbolic functions are not introduced because they are a mathematical nicety. They arise naturally and sufficiently often to warrant sustained study. For example, the shape of a chain hanging under gravity is well described by \( \cosh \) and the deformation of uniform beams can be expressed in terms of \( \tanh \).