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

Conic Sections

Conic section is the intersection between a cone and a plane. To make the explanation simple, I use an upright cone for the following examples. There are basically three types of conic sections:

1. Ellipse
2. Parabola
3. Hyperbola

Let's take a look at how the conic sections are formed:

1. Ellipse
   An ellipse is formed when a slanted plane ax + by + cz = d intersects with the upright cone. Take note the angle of the plane from the z-axis is bigger than the angle of the cone from the z-axis. Otherwise, a hyperbola is form.
   When a, b = 0, the plane becomes a horizontal plane, cz = d. This makes the curve of intersection changes from the ellipse to a circle.
2. Parabola
   A parabola is formed when a plane that is "parallel" to the cone intersects with the cone as shown in the first graph below.
3. Hyperbola
   When a plane ax + by + cz = d (slanted/vertical) that forms an angle from the z-axis is smaller than the angle of the cone from the z-axis (as mentioned in ellipse just now), a hyperbola is formed.
   

Betty, Created with GeoGebra