15.1 Definitions

There are definitely nice ways to think about these (including their relation to hyperbolic geometry, etc.).

The most efficient method is to cut to the chase and define the hyperbolic functions:

cosh⁡(x)\displaystyle\cosh(x) =ex+e−x2\displaystyle=\frac{e^{x}+e^{-x}}{2} (15.1)
=e2⁢x+12⁢ex⁢ multiplying by exex\displaystyle=\frac{e^{2x}+1}{2e^{x}}\text{ multiplying by $\frac{e^{x}}{e^{x}% }$} (15.2)
sinh⁡(x)\displaystyle\sinh(x) =ex−e−x2\displaystyle=\frac{e^{x}-e^{-x}}{2} (15.3)
=e2⁢x−12⁢ex⁢ multiplying by exex\displaystyle=\frac{e^{2x}-1}{2e^{x}}\text{ multiplying by $\frac{e^{x}}{e^{x}% }$} (15.4)

When plotted, they look like this:


−20-20−15-15−10-10−5-5551010151520201122⋅108\cdot 10^{8}xxcosh\coshcosh(x)
−20-20−15-15−10-10−5-555101015152020−2-222⋅108\cdot 10^{8}xxsinh\sinhsinh(x)
−20-20−15-15−10-10−5-555101015152020−1-1−0.5-0.50.50.511xxtanh\tanhtanh(x)