Bhaskara I's sine approximation formula
Encyclopedia
In mathematics
, a certain rational expression in one variable
for the computation
of the approximate value
s of the trigonometric sine
s discovered by Bhaskara I
(c. 600 – c. 680), a seventh century India
n mathematician
, is known as Bhaskara I's sine approximation formula.
This formula
is given in his treatise titled Mahabhaskariya. It is not known how Bhaskara I arrived at his approximation formula. However, several historian
s of mathematics
have put forward different theories as to the method Bhaskara might have used to arrive at his formula. The formula is elegant and simple and enables one to compute reasonably accurate values of trigonometric sines without using any geometry whatsoever.
(The reference "Rsine-differences 225" is an allusion to Aryabhata's sine table
.)
In modern mathematical notations, for an angle x in degrees, this formula gives
measure of angle
s as follows.
For a positive integer n this takes the following form:
Equivalent forms of Bhaskara 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) gives the formula in the following form:
Also, Bhaskara II (1114 – 1185 CE) has given this formula in his Lilavati
(Kshetra-vyavahara, Soka No.48) in the following form:
in using the formula. It shows that the maximum absolute error in using the formula is around 0.0016. From a plot of the percentage value of the absolute error, it is clear that the maximum percentage error is less than 1.8. 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. 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. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics
.
of a circle
be measured in degree
s and let the radius
R of the circle
be also measured in degree
s. 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
.
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
.
One now seeks two constants α and β such that
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 Bhaskara I's sine approximation formula.
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). This procedure produces the formula expressed using radian
measure of angles.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a certain rational expression in one variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
for the computation
Computation
Computation is defined as any type of calculation. Also defined as use of computer technology in Information processing.Computation is a process following a well-defined model understood and expressed in an algorithm, protocol, network topology, etc...
of the approximate value
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...
s of the trigonometric sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
s discovered by Bhaskara I
Bhaskara I
Bhāskara was a 7th century Indian mathematician, who was apparently the first to write numbers in the Hindu-Arabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work...
(c. 600 – c. 680), a seventh century India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
n mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, is known as Bhaskara I's sine approximation formula.
This formula
Formula
In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....
is given in his treatise titled Mahabhaskariya. It is not known how Bhaskara I arrived at his approximation formula. However, several historian
Historian
A historian is a person who studies and writes about the past and is regarded as an authority on it. Historians are concerned with the continuous, methodical narrative and research of past events as relating to the human race; as well as the study of all history in time. If the individual is...
s of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
have put forward different theories as to the method Bhaskara might have used to arrive at his formula. The formula is elegant and simple and enables one to compute reasonably accurate values of trigonometric sines without using any geometry whatsoever.
The approximation formula
The formula is given in verses 17 – 19, Chapter VII, Mahabhaskariya of Bhaskara I. A translation of the verses is given below:- (Now) I briefly state the rule (for finding the bhujaphala and the kotiphala, etc.) without making use of the Rsine-differences 225, etc. Subtract the degrees of a bhuja (or koti) from the degrees of a half circle (that is, 180 degrees). Then multiply the remainder by the degrees of the bhuja or koti and put down the result at two places. At one place subtract the result from 40500. By one-fourth of the remainder (thus obtained), divide the result at the other place as multiplied by the anthyaphala (that is, the epicyclic radius). Thus is obtained the entire bahuphala (or, kotiphala) for the sun, moon or the star-planets. So also are obtained the direct and inverse Rsines.
(The reference "Rsine-differences 225" is an allusion to Aryabhata's sine table
Āryabhaṭa's sine table
Āryabhaṭa's sine table is a set of twenty-four of numbers given in the astronomical treatise Āryabhaṭiya composed by the fifth century Indian mathematician and astronomer Āryabhaṭa , for the computation of the half-chords of certain set of arcs of a circle...
.)
In modern mathematical notations, for an angle x in degrees, this formula gives
Equivalent forms of the formula
Bhaskara I's sine approximation formula can be expressed using the radianRadian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
measure of angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
s as follows.
For a positive integer n this takes the following form:
Equivalent forms of Bhaskara I's formula have been given by almost all subsequent astronomers and mathematicians of India. For example, Brahmagupta
Brahmagupta
Brahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...
's (598 – 668 CE)
Brhma-Sphuta-Siddhanta (verses 23 – 24, Chapter XIV) gives the formula in the following form:
Also, Bhaskara II (1114 – 1185 CE) has given this formula in his Lilavati
Lilavati
Lilavati was Indian mathematician Bhāskara II's treatise on mathematics. It is the first volume of his main work Siddhānta Shiromani, Sanskrit for "Crown of treatises," alongside Bijaganita, Grahaganita and Golādhyāya.- Name :The name comes from his daughter Līlāvatī...
(Kshetra-vyavahara, Soka No.48) in the following form:
Accuracy of the formula
The formula is applicable for values of x° in the range from 0 to 180. 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,in using the formula. It shows that the maximum absolute error in using the formula is around 0.0016. From a plot of the percentage value of the absolute error, it is clear that the maximum percentage error is less than 1.8. 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. The search for more accurate formulas by Indian astronomers eventually led to the discovery the power series
Madhava series
In mathematics, a Madhava series is any one of the series in a collection of infinite series expressions all of which are believed to have been discovered by Sangamagrama Madhava the founder of the Kerala school of astronomy and mathematics...
expansions of sin x and cos x by Madhava of Sangamagrama
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...
(c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics
Kerala school
Kerala school may refer to*Kerala school of astronomy and mathematics, a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which flourished between the 14th and 16th centuries CE....
.
Derivation of the formula
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. Most of these derivations use only elementary concepts.Derivation based on elementary geometry
Let the circumferenceCircumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....
of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
be measured in degree
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
s and let the radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
R of the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
be also measured in degree
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
s. 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
.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
.One now seeks two constants α and β such that
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 Bhaskara I's sine approximation formula.
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;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). This procedure produces the formula expressed using radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
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 isThe 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.
Further references
- R.C..Gupta, On derivation of Bhaskara I's formula for the sine, Ganita Bharati 8 (1-4) (1986), 39-41.
- T. Hayashi, A note on Bhaskara I's rational approximation to sine, Historia Sci. No. 42 (1991), 45-48.