Coordinates of the vertices of a five-pointed star

For the tips you want to have 5 points evenly spread around a cycle with radius $r$, so take $$\{(r\cos(2\pi k/5+\pi/2),r\sin(2\pi k/5+\pi/2)) \mid k=0,...,4\}$$Do the same for the inner five points, but use a smaller radius and an additional $+2\pi/10$ within $\cos$ and $\sin$ to make them half way between the tips. (I have added $\pi/2$ to make the first tip point up, instead of right...)

Prett much the same approach provided already, but I've included a diagram and gave the information in degrees rather than radians.

As you can see in the image, the coordinates are generated on an inner and outer circle. The radius of the smaller circle in the image is 2 units and the radius of the larger outer circle is 6 units. Each coordinate on the outer circle is $72^\circ$ around the circle from the previous coordinate. I started at $18^\circ$ so that a star point would be at $18^\circ+72^\circ=90^\circ$. The inner circle coordinates started at $54^\circ$ because it is the median angle of $18^\circ$ and $90^\circ$.

Every coordinate is written $(r~cos(\Theta),~r~sin(\Theta))$