page_title: Bhaskara I's sine approximation formula
section_title: Accuracy of the formula
hierarchical_section_title: Bhaskara I's sine approximation formula / Accuracy of the formula
caption_attribution_description: English: Graphs of the following functions: sin(x)+0.2 sin(x) 4*x*(180-x)/(40500-x*(180-x))
context_page_description: 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, 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.
context_section_description: The formula is applicable for values of x° in the range from 0 to 180. The formula is remarkably accurate in this range. The graphs of sin ( x ) and the approximation formula are indistinguishable and are nearly identical. One of the accompanying figures gives the graph of the error function, namely the function,
  in using the formula. It shows that the maximum absolute error in using the formula is around 0.0016. From a plot of the percentage value of the absolute error, it is clear  that the maximum percentage error is less than 1.8. The approximation formula thus gives sufficiently accurate values of sines for most practical purposes. However it was not sufficient for the more accurate computational requirements of astronomy. The search for more accurate formulas by Indian astronomers eventually led to the discovery 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.
caption: Figure illustrates the level of accuracy of the Bhaskara I's sine approximation formula. The shifted curves 4 x ( 180 - x ) / ( 40500 - x ( 180 - x ) ) - 0.2 and sin ( x ) + 0.2 look like exact copies of the curve sin ( x ).
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