Ancient accurate approximation for sine

This post started out as a Twitter thread. The text below is the same as that of the thread after correcting an error in the first part of the thread.

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The following approximation for sin(x) is remarkably accurate for 0 < x < π.

The approximation is so good that you can’t see the difference between the exact value and the approximation until you get outside the range of the approximation.

Here’s a plot of just the error.

This is a very old approximation, dating back to Aryabhata I, around 500 AD.

In modern terms, it is a rational approximation, quadratic in the numerator and denominator. It’s not quite optimal since the ripples in the error function are not of equal height, but the coefficients are nice numbers.