Equivalent forms of the formula

Bhāskara I's sine approximation formula can be expressed using the radian measure of angles as follows:^[1]

${\displaystyle \sin x\approx {\frac {16x(\pi -x)}{5\pi ^{2}-4x(\pi -x)}}.}$

For a positive integer n this takes the following form:^[4]

${\displaystyle \sin {\frac {\pi }{n}}\approx {\frac {16(n-1)}{5n^{2}-4n+4}}.}$

The formula acquires an even simpler form when expressed in terms of the cosine rather than the sine. Using radian measure for angles from ${\displaystyle -{\frac {\pi }{2}}}$ to ${\displaystyle {\frac {\pi }{2}}}$ and putting ${\displaystyle x={\tfrac {1}{2}}\pi +y}$, one gets

${\displaystyle \cos y\approx {\frac {\pi ^{2}-4y^{2}}{\pi ^{2}+y^{2}}}.}$

To express the previous formula with the constant ${\displaystyle \tau =2\pi ,}$ one can use

${\displaystyle \cos y\approx 1-{\frac {20y^{2}}{4y^{2}+\tau ^{2}}}.}$

Equivalent forms of Bhāskara I's formula have been given by almost all subsequent astronomers and mathematicians of India. For example, Brahmagupta's (598–668 CE) Brhma-Sphuta-Siddhanta (verses 23–24, chapter XIV)^[3] gives the formula in the following form:

${\displaystyle R\sin x^{\circ }\approx {\frac {Rx(180-x)}{10125-{\frac {1}{4}}x(180-x)}}.}$

Also, Bhāskara II (1114–1185 CE) has given this formula in his Lilavati (Kshetra-vyavahara, Soka No. 48) in the following form:

${\displaystyle 2R\sin x^{\circ }\approx {\frac {4\times 2R\times 2Rx\times (360R-2Rx)}{{\frac {1}{4}}\times 5\times (360R)^{2}-2Rx\times (360R-2Rx)}}={\frac {5760Rx-32Rx^{2}}{162000-720x+4x^{2}}}}$

Derivation based on elementary geometry

Let the circumference of a circle be measured in degrees and let the radius R of the circle be also measured in degrees. Choosing a fixed diameter AB and an arbitrary point P on the circle and dropping the perpendicular PM to AB, we can compute the area of the triangle APB in two ways. Equating the two expressions for the area one gets (1/2) AB × PM = (1/2) AP × BP. This gives

${\displaystyle {\frac {1}{PM}}={\frac {AB}{AP\times BP}}.}$

Letting x be the length of the arc AP, the length of the arc BP is 180 − x. These arcs are much bigger than the respective chords. Hence one gets

${\displaystyle {\frac {1}{PM}}>{\frac {2R}{x(180-x)}}.}$

One now seeks two constants α and β such that

${\displaystyle {\frac {1}{PM}}=\alpha {\frac {2R}{x(180-x)}}+\beta .}$

It is indeed not possible to obtain such constants. However, one may choose values for α and β so that the above expression is valid for two chosen values of the arc length x. Choosing 30° and 90° as these values and solving the resulting equations, one immediately gets Bhāskara I's sine approximation formula.^[2][3]

Derivation starting with a general rational expression

Assuming that x is in radians, one may seek an approximation to sin x in the following form:

${\displaystyle \sin x\approx {\frac {a+bx+cx^{2}}{p+qx+rx^{2}}}.}$

The constants a, b, c, p, q and r (only five of them are independent) can be determined by assuming that the formula must be exactly valid when x = 0, π/6, π/2, π, and further assuming that it has to satisfy the property that sin(x) = sin(π − x).^[2][3] This procedure produces the formula expressed using radian measure of angles.

An elementary argument

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

${\displaystyle kx(180-x).}$

The parabola that also passes through (90, 1) (which is the point corresponding to the value sin(90°) = 1) is

${\displaystyle {\frac {x(180-x)}{90\times 90}}={\frac {x(180-x)}{8100}}.}$

The parabola which also passes through (30, 1/2) (which is the point corresponding to the value sin(30°) = 1/2) is

${\displaystyle {\frac {x(180-x)}{2\times 30\times 150}}={\frac {x(180-x)}{9000}}.}$

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

${\displaystyle 8100a+bx(180-x).}$

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 Bhāskara I's sine approximation formula.^[4]