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        "intro": "<h3>Introduction<\/h3><p>The hyperbolic functions <span class=\"math inline\">\\(\\sinh x, \\cosh x\\)<\/span>, <span class=\"math inline\">\\(\\tanh x\\)<\/span> etc are certain combinations of the exponential functions <span class=\"math inline\">\\(e^x\\)<\/span> and <span class=\"math inline\">\\(e^{-x}\\)<\/span>. The notation implies a close relationship between these functions and the trigonometric functions <span class=\"math inline\">\\(\\sin x, \\cos x\\)<\/span>, <span class=\"math inline\">\\(\\tan x\\)<\/span> etc. The close relationship is algebraic rather than geometrical. For example, the functions <span class=\"math inline\">\\(\\cosh x\\)<\/span> and <span class=\"math inline\">\\(\\sinh x\\)<\/span> satisfy the relation <span class=\"math inline\">\\(\\cosh^2x-\\sinh^2x\\equiv 1\\)<\/span>; which is very similar to the trigonometric identity <span class=\"math inline\">\\( \\cos^2x+\\sin^2x \\equiv 1\\)<\/span>. (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 <span class=\"math inline\">\\( \\cosh \\)<\/span> and the deformation of uniform beams can be expressed in terms of <span class=\"math inline\">\\( \\tanh \\)<\/span>.<\/p>",
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