 Ptolemy's theorem leads to the equivalent of the four sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine.Ptolemy further derived the equivalent of the half-angle formula

Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined.Neither the tables of Hipparchus nor those of Ptolemy have survived to the present day, although descriptions by other ancient authors leave little doubt that they once existed.