Hyperbolic Functions

Hyperbolic functions are functions that are built up by exponential functions, but have similar properties with trigonometry function, yet is based on hyperbola equation. Sounds complicated? Let's take a look at two basic hyperbolic functions:

	----------(1)
	----------(2)

These are the definition for these function, and we can see clearly that these are built up by exponential functions.

Next, let's square both equations above and find the difference as below:

From the outcome, we can assign x = cosh t and y = sinh t and we get the hyperbola equation x² - y² = 1.

As such, let's assign x = a cosh t and y = a sinh t as shown in the graph below. By default, a = 1. Try to move point A and look at the calculation in red at the bottom of the graph. The calculation is and you will find that the answer is always |a|. Try to slide the value of a at the slider and observe the calculation again when you move Point A. You will find that the point  always satisfies the hyperbola equation x² + y² = a².



As how the trigonometry function is defined, the hyperbolic function tanh(t) is also defined as
.
Similar with trigonometry functions, there are six hyperbolic functions:

1. sinh(t) - hyperbolic sine function
2. cosh(t) - hyperbolic cosine function
3. tanh(t) - hyperbolic tangent function
4. csch(t) - hyperbolic cosecant function
5. sech(t) - hyperbolic secant function
6. coth(t) - hyperbolic cotangent function

Betty, Created with GeoGebra