Editing Trigonometric tables (section)

== 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 identity|trigonometric identities]] starting from a known value (such as sin(π/2)&nbsp;=&nbsp;1, cos(π/2)&nbsp;=&nbsp;0). This method was used by the ancient astronomer [[Ptolemy]], who derived them in the ''[[Almagest]]'', a treatise on [[History of astronomy|astronomy]]. In modern form, the identities he derived are stated as follows (with signs determined by the quadrant in which ''x'' lies):

:<math>\cos\left(\frac{x}{2}\right) = \pm \sqrt{\tfrac{1}{2}(1 + \cos x)}</math>

:<math>\sin\left(\frac{x}{2}\right) = \pm \sqrt{\tfrac{1}{2}(1 - \cos x)}</math>

:<math>\sin(x \pm y) = \sin(x) \cos(y) \pm \cos(x) \sin(y)\,</math>

:<math>\cos(x \pm y) = \cos(x) \cos(y) \mp \sin(x) \sin(y)\,</math>

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]].