History of infinite series representations of $\sin(x)$ and $\cos(x)$

Question

When did the famous infinite series representations for $\sin(x)$ and $\cos(x)$ came about?

To be specific when did people realise that the ratio of the two sides of a right triangle with one angle being the size of $x$ radians can be expressed as a sum of an infinite number of quantities that depend on $x$.

What was the explanation/justification for them? Is there any nice geometric way of seeing that those identities are in fact true?

Answer 1

It is not possible to be certain, but close relatives of the series for sine and cosine were probably first obtained by Madhava (1340-1425) of the Kerala school. This work, and related work by other mathematicians of the Kerala school, predated European work on series by more than two centuries.

For a condensed technical discussion, please see the first chapter of Ranjan Roy's wonderful Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-first Century.

Answer 2

This answer of mine describes a geometric interpretation of the series for sine and cosine, discovered by math teacher Y. S. Chaikovsky at least as early as 1935. The result was presented in this American Mathematical Monthly article in 1996 by Chaikovsky's student, Leo Gurin.

Answer 3

The power series for sine and cosine are certainly already in Euler if not earlier. He obtained them from the power series for the exponential function. His approach used the binomial formula for an infinite exponent. Euler used both infinite numbers and infinitesimals to obtain correct series developments.