a bit of trigonometry
What is special about a regular pentagon?
Online you can find a wide range of websites on the golden section.
Some of them assign a special status to the pentagon because the golden section appears in it. Here's definitely some confusion between the number (or ) and the division of a segment according to the golden ratio. No one does assign a special status to the square because on exceptional metaphysical or esthetical grounds because the diagonal of a square with side 1 equals .
Likewise people don't ask question about an equality as sin 60°= sin = .
Whell, there's an analog equality for the sine of .
So no, there's nothing special on a pentagon. Starting from this equality it's even logic that in calculations in a pentagon the number (and also ) occurs.
You might as well skip the prove, just remembering the message that there's nothing special on a pentagon or a pentagram only because 'it reveals the golden section'.
on the next pages you can find some illustrations. If you are not a mathematician, you can leave the numbers and formulas for what they are. Just remember that it's logic itself that you can find and in geometrical applications with pentagons and decagons.
the relationship between phi and an angle
The equality is just a mathematical equality, just as , you learn in trigonometry.
This means that it's obvious that you'll meet where angles of (or mutiples) occcur. After all the length of the side of a regular polygon with n angles and radius r equals .
With the command SurdText you can easily check related equalities: