Introduction to Hyperbolic Functions (sinh, cosh, and tanh)

Terminology

A hyperbola (plural hyperbolas or hyperbolae) is an open curl lying on a plane. It has two pieces called branches or connected components. It's one of the three kinds of conic sections (such as parabola→1, the ellipse→2, and hyperbola→3), formed by the intersection of a plane and a double cone.

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called apex or vertex.

A plane is a flat two-dimensional survey that extends infinitely far.

A conic section is a curve obtained from the intersection of the surface of a cone with a plane.

Some equations produce hyperbolas on the coordinate system as shown in the image below.

Hyperbolic Functions

The ordinary Trigonometric functions (sin, cos, and tan) are constructed using a unit circle but the hyperbolic functions (sinh, cosh, and tanh) are constructed using unit hyperbola. The unit circle is a circle with a radius of one.

The unit hyperbolic is the set of points (x,y) in the Cartesian plane that satisfies the equation x²-y²=1. In other words,  x²-y² always equals 1 if the point (x,y) is on the curve.

A hyperbolic angle (the triangle contained the red area), that has an area a, has a hyperbolic sector with an area half the hyperbolic angle (the red area). The hyperbolic angle a is a real number that is the argument of the hyperbolic functions (sinh, cosh, and tanh). I don't know how to measure the area a, but the hyperbolic sector can be calculated (in radians) and then we just multiply it by 2 to get the area a. This video (I uploaded it to my server here) explains how to find the hyperbolic sector.


Below is the formula to calculate the hyperbolic functions. Unlike usual trigonometric functions, x is not an angle. It's the area.