Bhāskara I's sine approximation formula

In mathematics, Bhāskara 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 Bhāskara I (c. 600 – c. 680), a seventh-century Indian mathematician.

However, several historians of mathematics have put forward different hypotheses as to the method Bhāskara might have used to arrive at his formula.

The formula is elegant and simple, and it enables the computation of reasonably accurate values of trigonometric sines without the use of geometry.

A translation of the verses is given below:[3] (Now) I briefly state the rule (for finding the bhujaphala and the kotiphala, etc.)

Then multiply the remainder by the degrees of the bhuja or koti and put down the result at two places.

By one-fourth of the remainder (thus obtained), divide the result at the other place as multiplied by the anthyaphala (that is, the epicyclic radius).

one can use Equivalent forms of Bhāskara I's formula have been given by almost all subsequent astronomers and mathematicians of India.

The approximation formula thus gives sufficiently accurate values of sines for most practical purposes.

The search for more accurate formulas by Indian astronomers eventually led to the discovery of the power series expansions of sin x and cos x by Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics.

Beyond its historical importance of being a prime example of the mathematical achievements of ancient Indian astronomers, the formula is of significance from a modern perspective also.

Around half a dozen methods have been suggested, each based on a separate set of premises.

Choosing 30° and 90° as these values and solving the resulting equations, one immediately gets Bhāskara I's sine approximation formula.

[4] Karel Stroethoff (2014) offers a similar, but simpler argument for Bhāskara I's choice.