Short and Sweet Calculus

3.11 Hyperbolic Functions

The hyperbolic sine, the hyperbolic cosine, and the hyperbolic tangent are related to the exponential functions through the following formulas:

\[\boxed{\sinh x=\frac{e^{x}-e^{-x}}{2},\quad\cosh x=\frac{e^{x}+e^{-x}}{2},\quad\tanh x=\frac{\sinh x}{\cosh x}.}\]

  • An easy way to remember which one includes “\(e^{x}\) plus \(e^{-x}\)” and each one includes “\(e^{x}\) minus \(e^{-x}\)” is to notice that similar to \(\sin x\), \(\sinh x\) is an odd function so it must include “\(e^{x}\) minus \(e^{-x}\)” and similar to \(\cos x\), \(\cosh x\) is an even function so it must include “\(e^{x}\) plus \(e^{-x}\).”

The graphs of the hyperbolic functions are illustrated in Figure [fig:Ch5-Hyperbolic-Fig1].

In Figure [fig:Ch5-Hyperbolic-Fig1], notice the following features:

  • \(\sinh0=0\).

  • \(\sinh x\to+\infty\) as \(x\to+\infty\) and \(\sinh x\to-\infty\) as \(x\to-\infty\).

  • \(\cosh x\geq1\)

  • \(\cosh0=1\) is the minimum value of the hyperbolic cosine function.

  • \(\cosh x\to+\infty\) as \(x\to\pm\infty\).

  • \(\tanh0=0\).

  • \(\tanh x\to1\) as \(x\to+\infty\) and \(\tanh x\to-1\) as \(x\to-\infty\).

  • The hyperbolic function is an odd function.

An important identity is \[\boxed{\cosh^{2}x-\sinh^{2}x=1.}\]


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