Bhaskara I's sine approximation formula
In mathematics, Bhaskara I's sine approximation formula is a rational expression in one variable for the computation of the approximate values of the trigonometric sines discovered by Bhaskara I (c. 600 – c. 680), a seventh-century Indian mathematician. This formula is given in his treatise titled Mahabhaskariya. It is not known how Bhaskara I arrived at his approximation formula. However, several historians of mathematics have put forward different hypotheses as to the method Bhaskara might have used to arrive at his formula. The formula is elegant, simple and enables one to compute reasonably accurate values of trigonometric sines without using any geometry whatsoever.
The formula is given in verses 17 – 19, Chapter VII, Mahabhaskariya of Bhaskara I. A translation of the verses is given below:
(The reference "Rsine-differences 225" is an allusion to Aryabhata's sine table.)
In modern mathematical notations, for an angle x in degrees, this formula gives
Bhaskara I's sine approximation formula can be expressed using the radian measure of angles as follows.
For a positive integer n this takes the following form:
The formula acquires an even simpler form when expressed in terms of the cosine rather than the sine. Using radian measure for angle, and putting , one gets
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