Fun with Math!: hyperbolic

Hyperbolic Tangent & Hyperbolic Cotangent

Hyperbolic tangent is defined as
$\tanh(x)=\frac{\sinh(x)}{\cosh(x)}=\frac{e^x - e^{-x}}{e^x +e^{-x}}$ ,
while hyperbolic cotangent is the reciprocal of hyperbolic tangent, and it is defined as
$\coth(x)=\frac{\cosh(x)}{\sinh(x)}=\frac{e^x + e^{-x}}{e^x - e^{-x}}$ .
If we want to sketch the graph of the earlier function, think of the y-intercept and limits below:

As $x \rightarrow \infty$ , $\tanh(x)=\frac{\sinh(x)}{\cosh(x)}=\frac{e^x - e^{-x}}{e^x + e^{-x}}\rightarrow 1^{-}$ ;
when x = 0, $\tanh(0)=\frac{\sinh(0)}{\cosh(0))}=\frac{e^0 - e^{0}}{e^0 + e^{0}}=\frac{0}{2}=0$ ;
as $x \rightarrow -\infty$ , $\tanh(x)=\frac{\sinh(x)}{\cosh(x)}=\frac{e^x - e^{-x}}{e^x + e^{-x}}\rightarrow 1^{+}$ .

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

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

Betty, Created with GeoGebra

Hyperbolic Cosine & Hyperbolic Secant

The hyperbolic cosine function is defined as
$\cosh(x)=\frac{e^x + e^{-x}}{2}$
or generally,
$\cosh(ax)=\frac{e^{ax} + e^{-ax}}{2}$ .

Look at the graphs below, the blue solid curve is the curve of $\cosh(ax)=\frac{e^{ax} + e^{-ax}}{2}$ , whereas the dotted curves are the graphs of its terms. If you can't remember the graph of $\cosh(ax)=\frac{e^{ax} + e^{-ax}}{2}$ , you can try to add up the y-values of the two terms (dotted curves) and get the y-value of $\cosh(ax)=\frac{e^{ax} + e^{-ax}}{2}$ .

Try to slide the value of a to different values to view the changes of the graphs.

Now, take a look at the graph of the reciprocal of hyperbolic cosine function, i.e. hyperbolic secant function, which is defined as
$\mathrm{sech}(ax)=\frac{1}{\cosh(ax)}=\frac{2}{e^{ax}+e^{-ax}}$
Observe that when a = 1,

as $x\rightarrow \infty$ , $\cosh(x)\rightarrow\infty$ , thus $\mathrm{sech}(x)\rightarrow 0^{+}$ ;
as $x\rightarrow -\infty$ , $\cosh(x)\rightarrow\infty$ , thus $\mathrm{sech}(x)\rightarrow 0^{+}$ ;
as $x\rightarrow 0$ , $\cosh(x)\rightarrow 1$ , thus $\mathrm{sech}(x)\rightarrow 1$ .

Again, slide the value of a to view the changes of the graphs.

Betty, Created with GeoGebra

Hyperbolic Sine & Hyperbolic Cosecant

The hyperbolic sine function is defined as
$\sinh(x)=\frac{e^{x}-e^{-x}}{2}$

or generally,
$\sinh(ax)=\frac{e^{ax}-e^{-ax}}{2}$

Look at the graphs below, the blue solid curve is the curve of $\sinh(ax)=\frac{e^{ax}-e^{-ax}}{2}$ , whereas the dotted curves are the graphs of its terms. If you can't remember the graph of $\sinh(ax)=\frac{e^{ax}-e^{-ax}}{2}$ , you can try to add up the y-values of the two terms (dotted curves) and get the y-value of $\sinh(ax)=\frac{e^{ax}-e^{-ax}}{2}$ .

Look at the graph below, try to slide the value of a to different values to view the changes on graph.

Now, take a look at the graph of the reciprocal of hyperbolic sine function, i.e. hyperbolic cosecant function, which is defined as
$\mathrm{csch}(ax)=\frac{1}{\sinh(ax)}=\frac{2}{e^{ax}-e^{-ax}}$ .
Observe that when a =1,