## Friday, October 21, 2011

### Hyperbolic Tangent & Hyperbolic Cotangent

Hyperbolic tangent is defined as

$\fn_phv \tanh(x)=\frac{\sinh(x)}{\cosh(x)}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$,

while hyperbolic cotangent is the reciprocal of hyperbolic tangent, and it is defined as

$\fn_phv \coth(x)=\frac{\cosh(x)}{\sinh(x)}=\frac{e^x+e^{-x}}{e^x-e^{-x}}$.

If we want to sketch the graph of the earlier function, think of the y-intercept and limits below:

If we want to sketch the graph of the later function, think of the limits below:

The concept above applied for all values of a > 0. You may slide the value of 'a' below to look at the changes of $\fn_phv \tanh(ax)$ and $\fn_phv \coth(ax)$ for different values of 'a'.

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