Although there were many who devoted their lives to pioneering trigonometry, there were two men in particular that touched on the Sum and Difference and Multiple Angle formulae in their work: Ptolemy and Abu'l-Wafa.

Ptolemy was an influential Greek astronomer and geographer during his time, and propounded the geocentric theory in a form that prevailed for 1400 years. Of all of Ptolemy's major works, the most important is the Almagest, a treatise in thirteen books. The Almagest is the earliest of Ptolemy's works and gives in great detail his mathematical theory of the motions of the sun, moon, and planets. It was not superseded until a century after Copernicus presented his heliocentric theory in the De revolutionibus of 1543.

In the Almagest, Ptolemy first justifies his description of universe based on the geocentric universe described by Aristotle. Ptolemy used geometric models to predict the positions of the sun, moon, and planets using combinations of circular motions known as "epicycles." Having presented this model, Ptolemy then goes on to describe the mathematics which he needs in the rest of the work. In particular, he introduces trigonometric methods based on the chord function Crd (which is related to the sine function by sin a = Crd(2a)/120). Ptolemy used the Crd funtion to approximate pi as 3 17/120 or 3.14166, and, using sqrt(3) = chord 60°, sqrt(3)= 1.73205. Ptolemy also used formulas for the Crd function that are equivalent to the modern Sum-and-Difference formulas for sine and cosine, and, in the category of Multiple-Angle formulas, a formula for Crd(a/2) that resembles the Half Angle formula for sine. Ptolemy used these formulas to create a table for the Crd function and incremented every half-degree.

Abu'l-Wafa, an arabic scientist during the Buyid Islamic dynasty of 945 to 1055, wrote the Kitab al-Kamil, which was a simplified version of Ptolemy's Almagest. In it, Abu'l-Wafa uses the Multiple Angle formula for sine: sin2x = 2 sin x cos x, which he very easily could have derived from Ptolemy's sin(x + y) = sin x cos y + cos x sin y, by providing that x=y. Abu'l-Wafa also discovered a way to calculate and build sine and cosine tables that were accurate to eight decimal places, whereas Ptolemy's tables were accurate to only three.

Researched at the MacTutor History of Mathematics Archive.