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Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the  [[Indian mathematician]] and astronomer  [[Aryabhata|Āryabhaṭa]] (476–550 CE),  for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.<ref>{{cite journal|doi=10.1006/hmat.1997.2160|last=Takao Hayashi|first1=T|date=November 1997|title=Āryabhaṭa's rule and table for sine-differences|journal=Historia Mathematica |volume=24|issue=4|pages=396–406 }}</ref><ref>{{cite journal|doi=10.1007/BF00329978|last=B. L. van der Waerden|date=March 1988|first1=B. L.|title=Reconstruction of a Greek table of chords|journal=Archive for History of Exact Sciences|volume=38|issue=1|pages=23–38|title-link=table of chords}}</ref>

Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[Surya Siddhanta (सूर्य सिद्धांता)|Surya Siddhanta]] and [[Āryabhaṭa]]. Earliest sine table is found in Surya Siddhanta and another text is the astronomical treatise Āryabhaṭīya which was composed during the fifth century by the  [[Indian mathematician]] and astronomer  [[Aryabhata|Āryabhaṭa]] (476–550 CE),  for the computation of the half-chords of certain set of arcs of a circle. The table found in Surya Siddhanta is a table (in modern terms) of values of R.sinθ where R is the Indian standard radius of 3438 minutes. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as ''Āryabhaṭa's table of sine-differences''.

Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.<ref name="mcs">{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}</ref> Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.

Some believe that the Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.<ref name="mcs">{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |accessdate=4 March 2010}}</ref> Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.

The second section of Āryabhaṭiya titled Ganitapādd

The second section of Āryabhaṭiya titled Ganitapādd

a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.<ref name="Katz" />

a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.

* "When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord."

* "When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord."

This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table.  The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes  = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle.  

This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table.  The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes  = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value [[Pi|π]] = 3.1416 known to [[Aryabhata]] one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle.  

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