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 $\boxed{\cosh^{2}x-\sinh^{2}x=1.}$