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 1

which is very similar to the trigonometric identity cos 2 x + sin 2 x 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 .


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