The Hyperbolic 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

${cosh}^{2} x - {sinh}^{2} x \equiv 1$

which is very similar to the trigonometric identity ${cos}^{2} x + {sin}^{2} x \equiv 1$ . (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 .

Prerequisites

have a good knowledge of the exponential function
have knowledge of odd and even functions
have familiarity with the definitions of $tan, sec, c o s e c, cot$ and of trigonometric identities

Learning Outcomes

explain how hyperbolic functions are defined in terms of exponential functions
obtain and use hyperbolic function identities
manipulate expressions involving hyperbolic functions

1 Even and odd functions

1.1 Constructing even and odd functions

1.2 The odd and even parts of the exponential function

2 Hyperbolic identities

3 Related hyperbolic functions