Why is Ptolemy's chord table more precise than Hipparchus'? The 7 1/2° steps suggest that he arrived at the angles by successive division 60° by 2 (7 1/2° = 15°/2). Subdivisions of 72° do not exist so he appears not to have know the chord of 72° which is the side of the pentagon.
As mentioned before, one can use the golden ratio and a method for the chord of a half angle to find the side of a pentagon. In the figure above c_1 is the chord of the angle and c_2 that of half the angle. The sides, x and y, are the required sides of the right triangles. Here the circle is assumed to have a unit radius.
One can then use the Pythagorean Theorem to find a formula for the chord of the half angle in terms of the chord of the original angle. This along with the golden ratio will allow us to find an expression for the side of a pentagon.