Consider a polygon with E A and C as three successive vertices.
Angle EAC is a vertex angle. If the polygon is centred on B then angle BAC (a) is half of the vertex angle.
If D is the midpoint of line AC then line DB forms the perpendicular from B onto AC.
Angle ABC is the central angle of the polygon and so angle ABD (b) is half of the central angle.
As angle ABD is a right angle then a + b = pi/2, sin(a) = cos(b) and cos(a) = sin(b).
The central angle of an n-gon is given by 2*pi/n. The half central angle is thus given by pi/n.
Where the polygon is an n/d-gon, the half central
angle is given by pi*d/n