If you look at a regular hexagon inscribed in a circle (i.e. whose vertices are determined by 6 rotations of 60 degrees whose end point returns to the starting position), the length of one edge is equal to the radius (this could be demonstrated based on symmetry, because in the triangles formed by 1 edge and 2 radii all angles are equal, therefore also all edges).

The cosine of 60 degrees is one half of the edge divided by the radius, i.e. divided by the edge, therefore it is 1/2.