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Indian sine tables were constructed and improved upon by several ancient Indian mathematicians including the authors of [[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 [[Jya|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 [[Finite difference|first differences]] of the values of [[Trigonometric functions|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>
[[File:Jya and ardhajya.jpg|thumb]]
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 [[Madhava series|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 [[Madhava's sine table|sine table by Madhava]] with values accurate to seven or eight decimal places.
== Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==
'''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three [[trigonometric functions]] introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is [[Surya Siddhanta]].<ref name="Datta">{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39–108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}</ref> These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern [[trigonometric functions]] of [[sine]] and [[cosine]]. In fact, the origins of the modern terms of "sine" and "cosine" have been traced back to the [[Sanskrit]] words jyā and kotijyā.<ref name="Datta"/>
===Definition===
[[File:1920px-Modern diagram for jya and kojya.svg.png|thumb]]
Let 'arc AB' denote an [[Arc (geometry)|arc]] whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:
* ''jyā'' of arc AB = BM
* ''koti-jyā'' of arc AB = OM
* ''utkrama-jyā'' of arc AB = MA
If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:
* ''jyā'' ( arc AB ) = ''R'' sin ( ''s'' / ''R'' )
* ''koti-jyā'' ( arc AB ) = ''R'' cos ( ''s'' / ''R'' )
* ''utkrama-jyā'' ( arc AB ) = ''R'' ( 1 - cos ( ''s'' / ''R'' ) ) = ''R'' [[Versine|versin]] ( ''s'' / ''R'' )
==={{anchor|Rsin|Rcos}}Terminology===
[[File:Jya and ardhajya.jpg|thumb]]
[[File:Jya and kotijya.jpg|thumb]]
An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means "a bow".
The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means "a bow-string".
The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.<ref>According to lexicographers, it is a synonym also meaning "bow-string", but only its
geometrical meaning is attested in literature. Monier-Williams, ''A Sanskrit Dictionary'' (1899): "'' jīvá'' n. (in geom. = ''jyā'') the chord of an arc; the sine of an arc ''Suryasiddhanta'' 2.57";
''jīvá'' as a generic adjective has the meaning of "living, alive" ([[:wikt:Appendix:Proto-Indo-European/gʷih₃wós|cognate]] with English ''[[:wikt:quick|quick]]'')</ref>
At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.<ref name="Datta"/><ref name="Glen">{{cite book|last=Glen Van Brummelen|title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=95–97|isbn=978-0-691-12973-0}}</ref> The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant "half of".
The Sanskrit word ''koṭi'' has the meaning of "point, cusp", and specifically "the [[Recurve bow|curved end of a bow]]".
In trigonometry, it came to denote "the complement of an arc to 90°". Thus
''koṭi-jyā'' is "the ''jyā'' of the complementary arc". In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes "the side of a right angled triangle". Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .<ref name="Datta"/>
''Utkrama'' means "inverted", thus ''utkrama-jyā'' means "inverted chord".
The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning "arrow".<ref name="Datta"/>
An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of "the ''jyā'' of three signs". The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning "semi-diameter".<ref name="Datta"/>
According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by "Rsin" and "Rcos" treated as single words.<ref name="Datta"/> Others denote ''jyā'' and ''koti-jyā'' respectively by "Sin" and "Cos" (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).<ref name="Glen"/>
===From jyā to sine===
The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',<ref>{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}</ref><ref>{{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=2 March 2010}}</ref>
or more specifically to its synonym ''jīva''.
This term was [[Indian influence on Islamic science|adopted in medieval Islamic mathematics]], transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means "bosom". The text's 12th-century [[Medieval Latin|Latin]] translator used the Latin equivalent for "bosom", ''[[wikt:sinus|sinus]]''.<ref> Various sources credit the first use of ''sinus'' to either:
* [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]
* [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]
* [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī
See Merlet, [https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1 ''A Note on the History of the Trigonometric Functions''] in Ceccarelli (ed.), ''International Symposium on History of Machines and Mechanisms'', Springer, 2004<br>See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.<br>See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=Pearson |page=210 (sidebar) |isbn= 978-0321387004 |language=English }}</ref> When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.
==The Indian sine tables==
=== Surya Siddhanta sine table ===
The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as
{{Quote
|text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''<ref>Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.</ref>
}}
The verse 17-22 translates as
{{Quote
|text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.
Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''<ref>Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.</ref>
}}
The verse 23-27 translates as
{{Quote
|text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''<ref>Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.</ref>
}}
===Āryabhaṭa's sine table===
The stanza in Āryabhaṭiya describing the sine table is reproduced below:
मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||
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"/>
* "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 [[Degree (angle)|degrees]], [[arcminutes|minutes]], [[arcseconds|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.
The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the [[Aryabhata numeration|numerical scheme]] explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in [[ISO 15919]] transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''[[jya]]'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.
===Comparing different sine tables===
[[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.
{| class="wikitable" style="margin: 1em auto 1em auto;"
!Sl. No
!Angle<br />(in [[Degree (angle)|degrees]],<br />[[arcminutes]])
!Āryabhaṭa's <br />value of Jyā (R.sine)
![[Surya Siddhanta]] value of Jyā (R.sine)
!Surya Siddhanta versed sines
!Modern value <br />of ''[[Jyā]]'' R.sine
!Madhava's sine values
!Madhava's derived sine values
!Modern sine values
|-
| 1
|{{center|03° 45′}}
|{{center|225′}}
|{{center|225′}}
|{{center|7'}}
|{{center|224.8560}}
|<center>224ʹ 50ʹʹ 22ʹʹʹ</center>
|0.06540314
|0.06540313
|-
| 2
|{{center|07° 30′}}
|{{center|449′}}
|{{center|449′}}
|{{center|29'}}
|{{center|448.7490}}
|<center>448ʹ 42ʹʹ 58ʹʹʹ</center>
|0.13052623
|0.13052619
|-
| 3
|{{center|11° 15′}}
|{{center|671′}}
|{{center|671′}}
|{{center|66'}}
|{{center|670.7205}}
|<center>670ʹ 40ʹʹ 16ʹʹʹ</center>
|0.19509032
|0.19509032
|-
| 4
|{{center|15° 00′}}
|{{center|890′}}
|{{center|890′}}
|{{center|117′}}
|{{center|889.8199}}
|<center>889ʹ 45ʹʹ 15ʹʹʹ</center>
|0.25881900
|0.25881905
|-
| 5
|{{center|18° 45′}}
|{{center|1105′}}
|{{center|1105′}}
|{{center|182′}}
|{{center|1105.1089}}
|<center>1105ʹ 01ʹʹ 39ʹʹʹ</center>
|0.32143947
|0.32143947
|-
| 6
|{{center|22° 30′}}
|{{center|1315′}}
|{{center|1315′}}
|{{center|261′}}
|{{center|1315.6656}}
|<center>1315ʹ 34ʹʹ 07ʹʹʹ</center>
|0.38268340
|0.38268343
|-
| 7
|{{center|26° 15′}}
|{{center|1520′}}
|{{center|1520′}}
|{{center|354′}}
|{{center|1520.5885}}
|<center>1520ʹ 28ʹʹ 35ʹʹʹ</center>
|0.44228865
|0.44228869
|-
| 8
|{{center|30° 00′}}
|{{center|1719′}}
|{{center|1719′}}
|{{center|460′}}
|{{center|1719.0000}}
|<center>1718ʹ 52ʹʹ 24ʹʹʹ</center>
|0.49999998
|0.50000000
|-
| 9
|{{center|33° 45′}}
|{{center|1910′}}
|{{center|1910′}}
|{{center|579′}}
|{{center|1910.0505}}
|<center>1718ʹ 52ʹʹ 24ʹʹʹ</center>
|0.55557022
|0.55557023
|-
| 10
|{{center|37° 30′}}
|{{center|2093′}}
|{{center|2093′}}
|{{center|710′}}
|{{center|2092.9218}}
|<center>2092ʹ 46ʹʹ 03ʹʹʹ</center>
|0.60876139
|0.60876143
|-
| 11
|{{center|41° 15′}}
|{{center|2267′}}
|{{center|2267′}}
|{{center|853′}}
|{{center|2266.8309}}
|<center>2266ʹ 39ʹʹ 50ʹʹʹ</center>
|0.65934580
|0.65934582
|-
| 12
|{{center|45° 00′}}
|{{center|2431′}}
|{{center|2431′}}
|{{center|1007′}}
|{{center|2431.0331}}
|<center>2430ʹ 51ʹʹ 15ʹʹʹ</center>
|0.70710681
|0.70710678
|-
| 13
|{{center|48° 45′}}
|{{center|2585′}}
|{{center|2585′}}
|{{center|1171′}}
|{{center|2584.8253}}
|<center>2584ʹ 38ʹʹ 06ʹʹʹ</center>
|0.75183985
|0.75183981
|-
| 14
|{{center|52° 30′}}
|{{center|2728′}}
|{{center|2728′}}
|{{center|1345′}}
|{{center|2727.5488}}
|<center>2727ʹ 20ʹʹ 52ʹʹʹ</center>
|0.79335331
|0.79335334
|-
| 15
|{{center|56° 15′}}
|{{center|2859′}}
|{{center|2859′}}
|{{center|1528′}}
|{{center|2858.5925}}
|<center>2858ʹ 22ʹʹ 55ʹʹʹ</center>
|0.83146960
|0.83146961
|-
| 16
|{{center|60° 00′}}
|{{center|2978′}}
|{{center|2978′}}
|{{center|1719′}}
|{{center|2977.3953}}
|<center>2977ʹ 10ʹʹ 34ʹʹʹ</center>
|0.86602543
|0.86602540
|-
| 17
|{{center|63° 45′}}
|{{center|3084′}}
|{{center|3084′}}
|{{center|1918′}}
|{{center|3083.4485}}
|<center>3083ʹ 13ʹʹ 17ʹʹʹ</center>
|0.89687275
|0.89687274
|-
| 18
|{{center|67° 30′}}
|{{center|3177′}}
|{{center|3177′}}
|{{center|2123′}}
|{{center|3176.2978}}
|<center>3176ʹ 03ʹʹ 50ʹʹʹ</center>
|0.92387954
|0.92387953
|-
| 19
|{{center|71° 15′}}
|{{center|3256′}}
|{{center|3256′}}
|{{center|2333′}}
|{{center|3255.5458}}
|<center>3255ʹ 18ʹʹ 22ʹʹʹ</center>
|0.94693016
|0.94693013
|-
| 20
|{{center|75° 00′}}
|{{center|3321′}}
|{{center|3321′}}
|{{center|2548′}}
|{{center|3320.8530}}
|<center>3320ʹ 36ʹʹ 30ʹʹʹ</center>
|0.96592581
|0.96592583
|-
| 21
|{{center|78° 45′}}
|{{center|3372′}}
|{{center|3372′}}
|{{center|2767′}}
|{{center|3371.9398}}
|<center>3371ʹ 41ʹʹ 29ʹʹʹ</center>
|0.98078527
|0.98078528
|-
| 22
|{{center|82° 30′}}
|{{center|3409′}}
|{{center|3409′}}
|{{center|2989′}}
|{{center|3408.5874}}
|<center>3408ʹ 20ʹʹ 11ʹʹʹ</center>
|0.99144487
|0.99144486
|-
| 23
|{{center|86° 15′}}
|{{center|3431′}}
|{{center|3431′}}
|{{center|3213′}}
|{{center|3430.6390}}
|<center>3430ʹ 23ʹʹ 11ʹʹʹ</center>
|0.99785895
|0.99785892
|-
| 24
|{{center|90° 00′}}
|{{center|3438′}}
|{{center|3438′}}
|{{center|3438′}}
|{{center|3438.0000}}
|<center>3437ʹ 44ʹʹ 48ʹʹʹ</center>
|0.99999997
|1.00000000
|-
|}
==See also==
* [[Madhava's sine table]]
* [[Bhaskara I's sine approximation formula]]
==References==
{{reflist}}
{{DEFAULTSORT:Aryabhata's Sine Table}}
[[Category:Trigonometry]]
[[Category:Indian mathematics]]
[[File:Jya and ardhajya.jpg|thumb]]
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 [[Madhava series|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 [[Madhava's sine table|sine table by Madhava]] with values accurate to seven or eight decimal places.
== Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==
'''Jyā''', '''koti-jyā''' and '''utkrama-jyā''' are three [[trigonometric functions]] introduced by [[Indian mathematics|Indian mathematician]]s and astronomers. The earliest known Indian treatise containing references to these functions is [[Surya Siddhanta]].<ref name="Datta">{{cite journal|last=B.B. Datta and A.N. Singh|date=1983|title=Hindu Trigonometry|journal=Indian Journal of History of Science|volume=18|issue=1|pages=39–108|url=http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol18_1_5_BDatta.pdf|accessdate=1 March 2010}}</ref> These are functions of arcs of circles and not functions of angles. Jyā and kotijyā are closely related to the modern [[trigonometric functions]] of [[sine]] and [[cosine]]. In fact, the origins of the modern terms of "sine" and "cosine" have been traced back to the [[Sanskrit]] words jyā and kotijyā.<ref name="Datta"/>
===Definition===
[[File:1920px-Modern diagram for jya and kojya.svg.png|thumb]]
Let 'arc AB' denote an [[Arc (geometry)|arc]] whose two extremities are A and B of a circle with center O. If a perpendicular BM be dropped from B to OA, then:
* ''jyā'' of arc AB = BM
* ''koti-jyā'' of arc AB = OM
* ''utkrama-jyā'' of arc AB = MA
If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to modern trigonometric functions as follows:
* ''jyā'' ( arc AB ) = ''R'' sin ( ''s'' / ''R'' )
* ''koti-jyā'' ( arc AB ) = ''R'' cos ( ''s'' / ''R'' )
* ''utkrama-jyā'' ( arc AB ) = ''R'' ( 1 - cos ( ''s'' / ''R'' ) ) = ''R'' [[Versine|versin]] ( ''s'' / ''R'' )
==={{anchor|Rsin|Rcos}}Terminology===
[[File:Jya and ardhajya.jpg|thumb]]
[[File:Jya and kotijya.jpg|thumb]]
An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means "a bow".
The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a ''jyā'' which in [[Sanskrit]] means "a bow-string".
The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.<ref>According to lexicographers, it is a synonym also meaning "bow-string", but only its
geometrical meaning is attested in literature. Monier-Williams, ''A Sanskrit Dictionary'' (1899): "'' jīvá'' n. (in geom. = ''jyā'') the chord of an arc; the sine of an arc ''Suryasiddhanta'' 2.57";
''jīvá'' as a generic adjective has the meaning of "living, alive" ([[:wikt:Appendix:Proto-Indo-European/gʷih₃wós|cognate]] with English ''[[:wikt:quick|quick]]'')</ref>
At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs.<ref name="Datta"/><ref name="Glen">{{cite book|last=Glen Van Brummelen|title=The mathematics of the heavens and the earth : the early history of trigonometry|publisher=[[Princeton University Press]]|date=2009|pages=95–97|isbn=978-0-691-12973-0}}</ref> The half-chords were called ''ardha-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant "half of".
The Sanskrit word ''koṭi'' has the meaning of "point, cusp", and specifically "the [[Recurve bow|curved end of a bow]]".
In trigonometry, it came to denote "the complement of an arc to 90°". Thus
''koṭi-jyā'' is "the ''jyā'' of the complementary arc". In Indian treatises, especially in commentaries, ''koṭi-jyā'' is often abbreviated as ''kojyā''. The term ''koṭi'' also denotes "the side of a right angled triangle". Thus ''koṭi-jyā'' is the base/Run of a right triangle with ''jyā'' being the perpendicular/rise .<ref name="Datta"/>
''Utkrama'' means "inverted", thus ''utkrama-jyā'' means "inverted chord".
The tabular values of ''utkrama-jyā'' are derived from the tabular values of ''jyā'' by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called ''bāṇa'', ''iṣu'' or ''śara'' all meaning "arrow".<ref name="Datta"/>
An arc of a circle which subtends an angle of 90° at the center is called a ''vritta-pāda'' (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a ''vritta-pāda''. The ''jyā'' of a ''vritta-pāda'' is the radius of the circle. The Indian astronomers coined the term ''tri-jyā'' to denote the radius of the base circle, the term ''tri-jyā'' being indicative of "the ''jyā'' of three signs". The radius is also called ''vyāsārdha'', ''viṣkambhārdha'', ''vistarārdha'', etc., all meaning "semi-diameter".<ref name="Datta"/>
According to one convention, the functions ''jyā'' and ''koti-jyā'' are respectively denoted by "Rsin" and "Rcos" treated as single words.<ref name="Datta"/> Others denote ''jyā'' and ''koti-jyā'' respectively by "Sin" and "Cos" (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions).<ref name="Glen"/>
===From jyā to sine===
The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',<ref>{{cite web|url=http://mathforum.org/library/drmath/view/54053.html|title=How the Trig Functions Got their Names|work=Ask Dr. Math|publisher=[[Drexel University]]|accessdate=2 March 2010}}</ref><ref>{{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=2 March 2010}}</ref>
or more specifically to its synonym ''jīva''.
This term was [[Indian influence on Islamic science|adopted in medieval Islamic mathematics]], transliterated in Arabic as ''jība'' ([[:wikt:جيب|جيب]]). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homographic ''jayb'', which means "bosom". The text's 12th-century [[Medieval Latin|Latin]] translator used the Latin equivalent for "bosom", ''[[wikt:sinus|sinus]]''.<ref> Various sources credit the first use of ''sinus'' to either:
* [[Plato Tiburtinus]]'s 1116 translation of the ''Astronomy'' of [[Al-Battani]]
* [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]
* [[Robert of Chester]]'s 1145 translation of the tables of al-Khwārizmī
See Merlet, [https://link.springer.com/chapter/10.1007/1-4020-2204-2_16#page-1 ''A Note on the History of the Trigonometric Functions''] in Ceccarelli (ed.), ''International Symposium on History of Machines and Mechanisms'', Springer, 2004<br>See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.<br>See {{cite book |last=Katx |first=Victor |date=July 2008 |title=A history of mathematics |edition=3rd |location=Boston |publisher=Pearson |page=210 (sidebar) |isbn= 978-0321387004 |language=English }}</ref> When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.
==The Indian sine tables==
=== Surya Siddhanta sine table ===
The ''Surya Siddhanta'' provides methods to calculate the sine value in chapter 2. It uses an Indian standard circle of radius 3438 units. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and a length of 225 minutes. The verse 15-16 translates as
{{Quote
|text = ''The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 units)''<ref>Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.</ref>
}}
The verse 17-22 translates as
{{Quote
|text = ''The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.
Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ''<ref>Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.</ref>
}}
The verse 23-27 translates as
{{Quote
|text = ''The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.''<ref>Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.</ref>
}}
===Āryabhaṭa's sine table===
The stanza in Āryabhaṭiya describing the sine table is reproduced below:
मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||
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"/>
* "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 [[Degree (angle)|degrees]], [[arcminutes|minutes]], [[arcseconds|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.
The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the [[Aryabhata numeration|numerical scheme]] explained in [[Āryabhaṭīya]], and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in [[Devanagari]] script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in [[ISO 15919]] transliteration. The next column contains these numbers in the [[Hindu-Arabic numerals]]. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of ''[[jya]]'') can be obtained by summing up the differences up to that difference. Thus the value of ''jya'' corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of ''jya''s are given in the last column of the table.
===Comparing different sine tables===
[[Surya Siddhanta]] is the earliest known text to contain the sine table. Aryabhata's sine table calculates the same sine values as calculated in the ''Surya Siddhanta''. It is evident that several Indian mathematicians improved upon the sine tables. Aryabhata calculates the sine values which are same as the ones in ''Surya Siddhanta'' but Madhava has not rounded off the Jyā values and have given them further in minutes, seconds and thirds. Madhava's Jyā values derived into modern sine values provide astonishing accuracy upto 6 decimals places when compared with modern sine values.
{| class="wikitable" style="margin: 1em auto 1em auto;"
!Sl. No
!Angle<br />(in [[Degree (angle)|degrees]],<br />[[arcminutes]])
!Āryabhaṭa's <br />value of Jyā (R.sine)
![[Surya Siddhanta]] value of Jyā (R.sine)
!Surya Siddhanta versed sines
!Modern value <br />of ''[[Jyā]]'' R.sine
!Madhava's sine values
!Madhava's derived sine values
!Modern sine values
|-
| 1
|{{center|03° 45′}}
|{{center|225′}}
|{{center|225′}}
|{{center|7'}}
|{{center|224.8560}}
|<center>224ʹ 50ʹʹ 22ʹʹʹ</center>
|0.06540314
|0.06540313
|-
| 2
|{{center|07° 30′}}
|{{center|449′}}
|{{center|449′}}
|{{center|29'}}
|{{center|448.7490}}
|<center>448ʹ 42ʹʹ 58ʹʹʹ</center>
|0.13052623
|0.13052619
|-
| 3
|{{center|11° 15′}}
|{{center|671′}}
|{{center|671′}}
|{{center|66'}}
|{{center|670.7205}}
|<center>670ʹ 40ʹʹ 16ʹʹʹ</center>
|0.19509032
|0.19509032
|-
| 4
|{{center|15° 00′}}
|{{center|890′}}
|{{center|890′}}
|{{center|117′}}
|{{center|889.8199}}
|<center>889ʹ 45ʹʹ 15ʹʹʹ</center>
|0.25881900
|0.25881905
|-
| 5
|{{center|18° 45′}}
|{{center|1105′}}
|{{center|1105′}}
|{{center|182′}}
|{{center|1105.1089}}
|<center>1105ʹ 01ʹʹ 39ʹʹʹ</center>
|0.32143947
|0.32143947
|-
| 6
|{{center|22° 30′}}
|{{center|1315′}}
|{{center|1315′}}
|{{center|261′}}
|{{center|1315.6656}}
|<center>1315ʹ 34ʹʹ 07ʹʹʹ</center>
|0.38268340
|0.38268343
|-
| 7
|{{center|26° 15′}}
|{{center|1520′}}
|{{center|1520′}}
|{{center|354′}}
|{{center|1520.5885}}
|<center>1520ʹ 28ʹʹ 35ʹʹʹ</center>
|0.44228865
|0.44228869
|-
| 8
|{{center|30° 00′}}
|{{center|1719′}}
|{{center|1719′}}
|{{center|460′}}
|{{center|1719.0000}}
|<center>1718ʹ 52ʹʹ 24ʹʹʹ</center>
|0.49999998
|0.50000000
|-
| 9
|{{center|33° 45′}}
|{{center|1910′}}
|{{center|1910′}}
|{{center|579′}}
|{{center|1910.0505}}
|<center>1718ʹ 52ʹʹ 24ʹʹʹ</center>
|0.55557022
|0.55557023
|-
| 10
|{{center|37° 30′}}
|{{center|2093′}}
|{{center|2093′}}
|{{center|710′}}
|{{center|2092.9218}}
|<center>2092ʹ 46ʹʹ 03ʹʹʹ</center>
|0.60876139
|0.60876143
|-
| 11
|{{center|41° 15′}}
|{{center|2267′}}
|{{center|2267′}}
|{{center|853′}}
|{{center|2266.8309}}
|<center>2266ʹ 39ʹʹ 50ʹʹʹ</center>
|0.65934580
|0.65934582
|-
| 12
|{{center|45° 00′}}
|{{center|2431′}}
|{{center|2431′}}
|{{center|1007′}}
|{{center|2431.0331}}
|<center>2430ʹ 51ʹʹ 15ʹʹʹ</center>
|0.70710681
|0.70710678
|-
| 13
|{{center|48° 45′}}
|{{center|2585′}}
|{{center|2585′}}
|{{center|1171′}}
|{{center|2584.8253}}
|<center>2584ʹ 38ʹʹ 06ʹʹʹ</center>
|0.75183985
|0.75183981
|-
| 14
|{{center|52° 30′}}
|{{center|2728′}}
|{{center|2728′}}
|{{center|1345′}}
|{{center|2727.5488}}
|<center>2727ʹ 20ʹʹ 52ʹʹʹ</center>
|0.79335331
|0.79335334
|-
| 15
|{{center|56° 15′}}
|{{center|2859′}}
|{{center|2859′}}
|{{center|1528′}}
|{{center|2858.5925}}
|<center>2858ʹ 22ʹʹ 55ʹʹʹ</center>
|0.83146960
|0.83146961
|-
| 16
|{{center|60° 00′}}
|{{center|2978′}}
|{{center|2978′}}
|{{center|1719′}}
|{{center|2977.3953}}
|<center>2977ʹ 10ʹʹ 34ʹʹʹ</center>
|0.86602543
|0.86602540
|-
| 17
|{{center|63° 45′}}
|{{center|3084′}}
|{{center|3084′}}
|{{center|1918′}}
|{{center|3083.4485}}
|<center>3083ʹ 13ʹʹ 17ʹʹʹ</center>
|0.89687275
|0.89687274
|-
| 18
|{{center|67° 30′}}
|{{center|3177′}}
|{{center|3177′}}
|{{center|2123′}}
|{{center|3176.2978}}
|<center>3176ʹ 03ʹʹ 50ʹʹʹ</center>
|0.92387954
|0.92387953
|-
| 19
|{{center|71° 15′}}
|{{center|3256′}}
|{{center|3256′}}
|{{center|2333′}}
|{{center|3255.5458}}
|<center>3255ʹ 18ʹʹ 22ʹʹʹ</center>
|0.94693016
|0.94693013
|-
| 20
|{{center|75° 00′}}
|{{center|3321′}}
|{{center|3321′}}
|{{center|2548′}}
|{{center|3320.8530}}
|<center>3320ʹ 36ʹʹ 30ʹʹʹ</center>
|0.96592581
|0.96592583
|-
| 21
|{{center|78° 45′}}
|{{center|3372′}}
|{{center|3372′}}
|{{center|2767′}}
|{{center|3371.9398}}
|<center>3371ʹ 41ʹʹ 29ʹʹʹ</center>
|0.98078527
|0.98078528
|-
| 22
|{{center|82° 30′}}
|{{center|3409′}}
|{{center|3409′}}
|{{center|2989′}}
|{{center|3408.5874}}
|<center>3408ʹ 20ʹʹ 11ʹʹʹ</center>
|0.99144487
|0.99144486
|-
| 23
|{{center|86° 15′}}
|{{center|3431′}}
|{{center|3431′}}
|{{center|3213′}}
|{{center|3430.6390}}
|<center>3430ʹ 23ʹʹ 11ʹʹʹ</center>
|0.99785895
|0.99785892
|-
| 24
|{{center|90° 00′}}
|{{center|3438′}}
|{{center|3438′}}
|{{center|3438′}}
|{{center|3438.0000}}
|<center>3437ʹ 44ʹʹ 48ʹʹʹ</center>
|0.99999997
|1.00000000
|-
|}
==See also==
* [[Madhava's sine table]]
* [[Bhaskara I's sine approximation formula]]
==References==
{{reflist}}
{{DEFAULTSORT:Aryabhata's Sine Table}}
[[Category:Trigonometry]]
[[Category:Indian mathematics]]