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 = 90, and f(0) = f(180) = 0. To get value 1 at θ = 90 it is required that = 1, that is, a = 1 + b, so...To get value 1/2 at 0 = 30 it is required that...which is easily solved to yield b = -\. It follows that...This gives Bhaskara's approximation formula for the sine function.Bhaskara's Approximation Formula: ...Alternative Derivation. There is an even simpler argument to obtain Bhaskara's formula. I have no doubt now that Bhaskara must have reasoned as follows.The polynomial p{0) = 0(180 - 0)/81OO is symmetric with respect to 90, and agrees with the values of the sine for 0, 90, and 180 degrees, but not for 30 degrees (and 150 degrees). In fact we have ... Now, it is easy to construct a polynomial q symmetric about 0 = 90 which takes value ... at 30 (the reason for this value will become clear momentarily) and value 1 at 90: the polynomial...will do the job. Since p(90) = 1 and c(90) = 1, we have ... Note that...Also = 0. So the following function is symmetric about 90 and has the same values as the sine at 0, 30, and 90 degrees (as well as 150 and 180 degrees by symmetry):...which is Bhaskara's rational function! The simplicity of this argument makes it likely that the formula was derived along these lines.Accuracy of Bhaskara's Approximation. To see how good of an approximation this is we plot the above rational function in the same window as the sine function:Since the two graphs are not distinguishable, we plot the difference on a larger scaJe:The error is bounded by 0.00165 over the entire interval (in the above graph the minimum of about -0.001631765 occurs near 0 = 11.543848 and 0 = 168.4561524, and the maximum of about 0.0013436967 occurs near 0 = 51.34589377 and 0 = 128.6541062) so this is a very good approximation. …