Mathematica 13.1:
CirclePoints[{0, 0}, {1, 0}, 9]
returns the coordinates of 9 points equally spaced around a unit circle,
a distance of 2 Pi/9 from each other along the circumference.
{ {1, 0}, {Cos[(2 Pi/9], Sin[(2 Pi/9]}, {Sin[Pi/18], Cos[Pi/18]}, .......
The given coordinates of the 3rd point, {Sin[Pi/18], Cos[Pi/18]} , are correct as {Sin[Pi/18], Cos[Pi/18]} is equal to {Cos[4 Pi/9], Sin[ 4 Pi/9]}, but why does Mathematica return the values in the form it does? What expansion is being used so that a multiple of 2 Pi/9 is not used? Why is Sin[Pi/18] chosen over Cos[2 Pi/9]?
