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The Bhaskara-Aryabhata Approximation to the Sine Function

Shailesh A. Shirali
Mathematics Magazine
Vol. 84, No. 2 (April 2011), pp. 98-107
DOI: 10.4169/math.mag.84.2.098
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Page Count: 10
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The Bhaskara-Aryabhata Approximation to the Sine Function
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Summary In the seventh century ad the Indian mathematician Bh?skar? I gave a curious rational approximation to the sine function; he stated that if 0 ? x ? 180 then sin x deg is approximately equal to 4x(180 ? x)/(40500 ? x(180 ? x)). He stated this in verse form, in the style of the day, and attributed it to his illustrious predecessor ?ryabha?a (fifth century ad); however there is no trace of such a formula in ?ryabha?a’s known works. Considering the simplicity of the formula it turns out to be astonishingly accurate. Bh?skar? did not give any justification for the formula, nor did he qualify it in any way. In this paper we examine the formula from an empirical point of view, measuring its goodness of fit against various criteria. We find that the formula measures well, and indeed that these different criteria yield formulas that are very close to the one given by Bh?skar?.

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