There are many interesting ways of representing . Here is one of the more approachable and fun representations.
We will prove the above formula using little more than High School mathematical concepts and techniques.
Some of the notation may look a bit strange and the level of conceptual difficulty is quite high. The main concepts that we will be using here are:
- Trigonometric Identities
- Recursion Formulas
- Summation/Product terminology
The last one will look very strange.
We will be using to show an infinite product of terms.
We will start with a fundamental concept about the function .
It can be proved that if is very small, then .
Another way of writing this is to say that
We can also write this as
Next up come the three main Trigonometric Identities that we will be using.
These are the two double angle formulas and the fundamental link between and .
With a bit of rewriting and redefining we get the following:
We are now ready to begin:
Now, with a bit of abstraction, we get the following:
Now we consider the Product terms.
First off, note that
Note that as above,
Thus we get a recursive formula for all of the terms.
This recursive formula can be written as
Now, we can rewrite our Formula as
, , .
As we let get bigger and bigger we see that the gets smaller and smaller and if we let we get:
, , .
, , as when is small.
Or, finally and succinctly,
Or in “headache” form