The infinite series approximations that have been used for many years to calculate the values of trigonometric functions have traditionally been attributed to Brook Taylor and Colin Maclaurin, European mathematicians of the early 18th century who were building on the work of Newton, Leibniz, James Gregory and Isaac Barrow among others.
However, I recently discovered that they were not the first to use these techniques. As author George Gheverghese Joseph points out in the previous link, the work of Newton and Leibniz was tremendous, however the Indian development of infinite series approximations for trigonometric functions was equally amazing and important. In addition, it came nearly 300 years before the European development of these techniques.
Madhava of Sangamagrama is generally recognized of the founder of the Kerala school of mathematics and astronomy in what is today the state of Kerala in southwest India. The work of the mathematicians of the Kerala school was based on a desire for accurate trigonometric values for use in navigation.
Madhava lived in the late 1300s and early 1400s and most of his original work has been lost. However, he is mentioned frequently in the surviving work of later mathematicians from the Kerala school. Madhava is credited with power series calculations for the sine, cosine, tangent and arctangent, and like Leibniz, he used the arctangent power series to approximate the value of to 13 decimal places.
Victor Katz’ A History of Mathematics (Brief Edition) has a wonderful and detailed derivation of the Kerala school trigonometric series, with diagrams showing how they used the relationships between the angles, radii, chords and arcs in a circle to arrive at these amazing calculations.
Katz also published this derivation in a paper for the MAA (Mathematics Magazine, vol. 68, n. 3, June 1995, pp. 163-174)
The derivation for the infinite series begins on page 169 (pg. 7 in the pdf).
I’ve just begun to unpack this derivation and will post a step-by-step explanation of Katz’ work in the “near” future.