Bhaskara's approximation for the sine

(ProQuest: ... denotes formulae omitted.)

The 7th century Indian mathematician Bhaskara (c.600 - c.680) obtained a remarkable approximation for the sine function. Many subsequent ancient authors have given versions of this rule, but none provided a proof or described how the result was obtained. Grover [1] provides a possible explanation, but I think the rule can be explained more clearly. Rather than give the rule first, we will derive it, and then discuss its accuracy, and explore some alternative approximations. Our derivation is simply an exercise in modeling.

Approximating the Sine: A Possible Derivation of Bhaskara's Approximation. Our goal is to approximate the sine function on the interval [0°,180°] (I have seen the rule formulated for radians, in conjunction with certain approximations of 7r; we will see that none of this is necessary when we simply work with degrees. I will be careful to use degrees in the notation, so that we can translate some of the formulas for use with radians without ambiguity). Graph of the function y = sin(0°) over the interval [0,180]:

We will use that the above graph is symmetric with respect to 6 = 90. It is not hard to find a polynomial with the same symmetry which takes values 0 for 6 = 0 and 6 = 180: p{6) = 0(180 - 0). The value of p at 0 = 90 is p(90) = 90(180 - 90) = 8100, so the polynomial p/8100 has the same symmetry about 0 = 90, and the same values as sin(0°) at the points 0 = 0, 90,180. This is all as in Grover [1], who states that Bhaskara referred to the quantity 0(180 - 0) as 'prathama' (Grover does not provide the meaning of this word; it is Sanskrit for "first"), from which we can infer that this is also in Bhaskara's work. The quadratic polynomial p(0)/81OO = 0(180 - 0)/81OO can be viewed as a crude (first) approximation of sin(0°) on the interval [0,180]. The following graph shows how the two functions compare:

Note that ... To get a better approximation, Bhaskara must have interpolated also the value sin(30°) = 1/2. To do this, consider the following function

...

Clearly f has the same symmetry as p about θ =...