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.

Enabling Khmer Unicode Keyboard on MacOS Sierra 10.12

You don't need to install anything on MacOS Sierra v10.12 or may be even on Mac OSX Mavericks v10.9. The Khmer Unicode keyboard layout and font (Kh Battambang) are built-in. It was created by Mr. Danh Hong (dnhhng@yahoo.com). To make it show up in the Menu bar, you have to open the Keyboard Preferences and add the keyboard named Khmer in the Input Sources as shown below.

Note that this keyboard layout does not work well with the Khmer fonts created by different developers. The issue can be missing or overlapping characters in Microsoft Offices. You have to use the fonts created by Mr. Danh Hong such as Kh Battambang (similar to Limon S1) and Kh Muol (similar to Limon R1).

Keyboard layout