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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?

Timotej
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  • The first question (history of the infinite series) is very different from the last (geometric visualization). They should be two different questions. (And the second one has almost certainly been asked a few hundred times on this site before.) –  Apr 22 '14 at 03:43
  • The the first written record of the series representations of trigonometric functions comes from the Indian mathematician Madhava in the 1300's. – David H Apr 22 '14 at 03:59
  • James Gregory (1638-1675) is generally credited with discovering the infinite series representations for some of the trig functions (by extension of Archimedes' "method of exhaustion", no less). Brook Taylor (1685-1731) would not be far behind in advancing this approach. – colormegone Apr 22 '14 at 03:59
  • OK, I see my sources are again too Eurocentric; I should have figured that if infinities were involved, someone in India had already been thinking about this. Thanks for the historical pointer! – colormegone Apr 22 '14 at 04:01

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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.

André Nicolas
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  • Nice one. I have a question for you that is not related to this. If two bernoulli trials go through to infinity and that you want the probability that one of them has more successes than the other at all time, can you use ratio of expected number of successes of one in the numerator and sum of expected number of successes of both of them in the denominator to arrive at the probability that one will stay ahead in the number of successes than the other. – Satish Ramanathan Apr 22 '14 at 06:20
  • I think so, my guess is that it can be done by a variant of the gambler's ruin analysis. Have not done the calculation, however. – André Nicolas Apr 22 '14 at 07:07
  • Thanks. If you happen to solve this for anyone, let me know. – Satish Ramanathan Apr 22 '14 at 07:28
  • You should ask it as a question, preferably at a time when the site has a fair bit of activity. Someone will answer. – André Nicolas Apr 22 '14 at 07:30
  • No problem, I will do so. I solved a similar problem this way and was kind of quickly verify if it is OK. – Satish Ramanathan Apr 22 '14 at 07:33
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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.

José Carlos Santos
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Mikhail Katz
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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.

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