page_title: Bhaskara I's sine approximation formula
section_title: An elementary argument
hierarchical_section_title: Bhaskara I's sine approximation formula / Derivation of the formula / An elementary argument
caption_attribution_description: English: Graphs of the following functions: x*(180-x)/8100 x*(180-x)/9000 sin(x) (x in degrees)
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 part of the graph of sin(x) in the range from 0° to 180° "looks like" part of a parabola through the points (0, 0) and (180, 0). The general such parabola is
  The parabola that also passes through (90, 1) (which is the point corresponding to the value sin(90°) = 1) is
  The parabola which also passes through (30, 1/2) (which is the point corresponding to the value sin(30°) = 1/2) is
  These expressions suggest a varying denominator which takes the value 90 × 90 when x = 90 and the value 2 × 30 × 150 when x = 30. That this expression should also be symmetrical about the line ' x = 90' rules out the possibility of choosing a linear expression in x.  Computations involving x(180 − x) might immediately suggest that the expression could be of the form
  A little experimentation (or by setting up and solving two linear equations in a and b) will yield the values a = 5/4, b = −1/4. These give Bhaskara I's sine approximation formula.
caption: Comparison of graphs of the parabolas
   x(180 − x)/8100 and x(180 − x)/9000 
   with the graph of sin(x) (x in degrees).
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