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.
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.
A translation of the verses is given below: . The reference "Rsine-differences " is an allusion to Aryabhata's sine table.
Sine, Cosine and Tangent
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. Equivalent forms of Bhaskara I's formula have been given by almost all subsequent astronomers and mathematicians of India.
Lesson 1 - Basic Trig Identities Involving Sin, Cos, and Tan
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,. It shows that the maximum absolute error in using the formula is around 0. From a plot of the percentage value of the absolute error, it is clear that the maximum percentage error is less than 1.
The approximation formula thus gives sufficiently accurate values of sines for all practical purposes. However it was not sufficient for the more accurate computational requirements of astronomy.
Sin, Cos and Tan
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. Bhaskara I had not indicated any method by which he arrived at his formula. Historians have speculated on various possibilities. No definitive answers have as yet been obtained. 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.
Mathematicians have attempted to derive the rule using modern concepts and tools. Around half a dozen methods have been suggested, each based on a separate set of premises. Let the circumference of a circle be measured in degrees and let the radius R of the circle be also measured in degrees.
This gives. Letting x be the length of the arc AP , the length of the arc BP is - x. These arcs are much bigger than the respective chords. Hence one gets.
It is indeed not possible to obtain such constants. Assuming that x is in radians, one may seek an approximation to sin x in the following form:.
The general such parabola is. These give Bhaskara I's sine approximation formula. From Wikipedia, the free encyclopedia.
Archived from the original on 23 March Retrieved 22 April The mathematics of the heavens and the earth: the early history of trigonometry. Princeton University Press.
Gupta Indian Journal of History of Science. Archived from the original PDF on 16 March Retrieved 20 April A passage to infinity : Medieval Indian mathematics from Kerala and its impact.
Ken Ward's Mathematics Pages