Trigonometric Tables - Half-angle and Angle-addition Formulas

Half-angle and Angle-addition Formulas

Historically, the earliest method by which trigonometric tables were computed, and probably the most common until the advent of computers, was to repeatedly apply the half-angle and angle-addition trigonometric identities starting from a known value (such as sin(π/2) = 1, cos(π/2) = 0). This method was used by the ancient astronomer Ptolemy, who derived them in the Almagest, a treatise on astronomy. In modern form, the identities he derived are stated as follows (with signs determined by the quadrant in which x lies):

These were used to construct Ptolemy's table of chords, which was applied to astronomical problems.

Various other permutations on these identities are possible: for example, some early trigonometric tables used not sine and cosine, but sine and versine).