Ming Antu's infinite series expansion of trigonometric functions

Fig. 1: The Ming Antu Model
Fig. 3: Ming Antu independently discovered Catalan numbers.

Ming Antu's infinite series expansion of trigonometric functions. Ming Antu, a court mathematician of the Qing dynasty did extensive work on the infinite series expansion of trigonometric functions in his masterpiece Geyuan Milü Jiefa (Quick Method of Dissecting the Circle and Determination of The Precise Ratio of the Circle). Ming Antu built geometrical models based on a major arc of a circle and the nth dissection of the major arc. In Fig 1, AE is the major chord of arc ABCDE, and AB, BC, CD, DE are its nth equal segments. If chord AE = y, chord AB = BC = CD = DE = x, the task was to find chord y as the infinite series expansion of chord x. He studied the cases of n = 2, 3, 4, 5, 10, 100, 1000 and 10000 in great detail in volumes 3 and 4 of Geyuan Milü Jiefa.