### Introduction

The hyperbolic functions $sinhx,\phantom{\rule{1em}{0ex}}coshx$ , $tanhx$ 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 $sinx,\phantom{\rule{1em}{0ex}}cosx$ , $tanx$ etc. The close relationship is algebraic rather than geometrical. For example, the functions $coshx$ and $sinhx$ satisfy the relation

$\phantom{\rule{2em}{0ex}}{cosh}^{2}x-{sinh}^{2}x\equiv 1$

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 $tan,\phantom{\rule{1em}{0ex}}sec,\phantom{\rule{1em}{0ex}}cosec,cot$ 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