## 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 \[\boxed{\cosh^{2}x-\sinh^{2}x=1.}\]

The derivatives of the hyperbolic functions are tabulated in Section [sec:Ch5-How-to-Differentiate].

[up][previous][table of contents][next]