3 Related hyperbolic functions

Given the trigonometric functions $cosx,sinx$ related functions can be defined; $tanx,secx,cosecx$ through the relations:

$tanx\equiv \frac{sinx}{cosx}secx\equiv \frac{1}{cosx}cosecx\equiv \frac{1}{sinx}cotx\equiv \frac{cosx}{sinx}$

In an analogous way, given $coshx$ and $sinhx$ we can introduce hyperbolic functions $tanhx,sechx$ , $cosechx$ and $cothx$ . These functions are defined in the following Key Point:

Task!

Show that $1-{tanh}^{2}x\equiv {\mathrm{\text{sech}}}^{2}x$

Use the identity ${cosh}^{2}x-{sinh}^{2}x\equiv 1$ :

Answer

Exercises

1. Express
   1. $2sinhx+3coshx$ in terms of ${\text{e}}^x$ and ${\text{e}}^{-x}$ .
   2. $2sinh4x-7cosh4x$ in terms of ${\text{e}}^{4x}$ and ${\text{e}}^{-4x}$ .
2.  Express
   1. $2{\text{e}}^x-{\text{e}}^{-x}$ in terms of $sinhx$ and $coshx$ .
   2. $\frac{7{\text{e}}^x}{\left({\text{e}}^x-{\text{e}}^{-x}\right)}$ in terms of $sinhx$ and $coshx$ , and then in terms of $cothx$ .
   3. $4{\text{e}}^{-3x}-3{\text{e}}^{3x}$ in terms of $sinh3x$ and $cosh3x$ .

      3. Using only the $cosh$ and $sinh$ keys on your calculator (or ${\text{e}}^x$ key) find the values of

3. $tanh0.35$ ,
4. $cosech2$ ,
5. $\mathrm{\text{sech}}0.6$ .

Answer