Approximation of Sine formula by Bhaskara

Bhaskara (c. 600 — c. 680), one of the remarkable mathematician and an astronomer gave an unique rational approximation of the sine function in his commentary on Aryabhata’s work.

The formula is given in verses 17–19, Chapter VII, Mahabhaskariya

verses 17–19, Chapter VII, Mahabhaskariya

“(Now) I briefly state the rule (for finding the bhujaphala and the kotiphala, etc.) without making use of the Rsine-differences 225, etc. Subtract the degrees of a bhuja (or koti) from the degrees of a half circle (that is, 180 degrees). Then multiply the remainder by the degrees of the bhuja or koti and put down the result at two places. At one place subtract the result from 40500. 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). Thus is obtained the entire bahuphala (or, kotiphala) for the sun, moon or the star-planets. So also are obtained the direct and inverse Rsines.”

Bhaskara’s Formula for Sine in Modern Notation:

When you plot the function on graph, you will notice that the curve is completely indistinguishably with the curve of Sine function between(0–180°).

image taken from wikipedia

Source: Bhaskara I’ approximation to sine, R.C. Gupta, Indian Journal of History of Science

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