Another quite slick construction here.
Expand a line by the ratio 1:1.618 (using the golden rectangle from a square method) and then extend arcs as shown in the first image. You can then produce a golden triangle by joining the corners – the magenta arc makes sure two sides are of equal length, and the blue arc makes sure it’s a 72˚ angle – meaning the third edge will be to the others as 1 is to 1.618.
Find the mid-points of each edge to the triangle. By joining these points we have the all-important triangle with which to create a pentagon/pentagram.
As we know the sides are going to be of equal length we can just use the compass, like in the fourth image, to find the two remaining points to the desired shape.
I added a last image to show the relation of one side of the pentagon to the longer side of the triangle – if the triangle edge is 1 unit, a side of the pentagon is 0.5/1.618.