### Hyperbolic Functions

Functions that are defined in
terms of the exponential function or its derivatives are called hyperbolic
functions.

The exponential function and its derivatives take these forms.

**E.g**

**sinh x = (e**^{x} - e^{-x}) / 2

**cosh x = (e**^{x} + e^{-x}) / 2

**tanh x = sinh x / cosh x = (e**^{x} - e^{-x}) / (e^{x} + e^{-x})

In order to obtain the values of hyperbolic functions, the following table can be used.

Let's see the graphical representation of the
three hyperbolic functions.

Now, the basic properties of
these functions can easily be derived:

**E.g.1**

**cosh-x = cosh x**

cosh x = (e^{x} + e^{-x}) / 2

cosh -x = (e^{(-x)} + e^{-(-x)}) / 2 = (e^{x} + e^{-x}) / 2 = cosh x

**E.g.2**

**sinh -x = -sinh x**

sinh x = (e^{x} - e^{-x}) / 2

sinh -x = (e^{(-x)} - e^{-(-x)}) / 2 = (e^{-x} - e^{x}) / 2 = - sinh x

**E.g.3**

**tanh -x = -tanh x**

tanh x = (e^{x} - e^{-x}) / (e^{x} + e^{-x})

tanh -x = (e^{-x} - e^{-(-x)}) / (e^{-x} + e^{-(-x)}) = (e^{-x} - e^{x}) / (e^{-x} + e^{x}) = -tanh x

The functions can be extended to bring about the following identities.

- cosh(a + b) = cosh(a)cosh(b) +
sinh(a)sinh(b)
- sinh(a + b) = sinh(a)cosh(b) +
cosh(a)sinh(b)
- cosh(a - b) = cosh(a)cosh(b) -
sinh(a)sinh(b)
- sinh(a - b) = sinh(a)cosh(b) -
cosh(a)sinh(b)
- tanh (a+b) = tanh a + tanh b /(1 + tanh a.tanh b)

Please work out the following questions to complement what you have just learnt.

- Show that tanh x is an odd function.
- Show that cosh
^{2}x - sinh^{2}x = 1
- Differentiate y = sinh x and y = cosh x
- Prove that cosh x is even and sinh x is odd.
- Find the point on y = cosh x curve, where the gradient is one.

### Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level.
They are organized according to these specific levels.

The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.

This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

**Email:**
### Stand Out - from the crowd

~~"There's no such thing as a free lunch."~~

The best things in **nature** are free with no strings attached - fresh air, breathtakingly warm sunshine, scene of meadow on the horizon...

Vivax Solutions, while mimicking nature, offers a huge set of tutorials along with interactive tools for free.

Please use them and excel in the sphere of science education.

Everything is free; not even **registration** is required.