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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''.

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. 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.

== Ancient Indian concepts of Jya (chord) and Chaap or Dhanu (arc) ==

== 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&ndash;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" />

'''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. 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" />

[[File:Jya Figure.jpg|thumb|419x419px]]

[[File:Jya Figure.jpg|thumb|419x419px]]

An arc of a circle is like a bow and so is called a ''dhanu'' or ''cāpa'' which in [[Sanskrit]] means "a bow".  

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 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

The word ''jīvá'' is also used as a synonym for ''jyā'' in geometrical literature.

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" /> 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".

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&ndash;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]]".

The Sanskrit word ''koṭi'' has the meaning of "point, cusp", and specifically "the [[Recurve bow|curved end of a bow]]".

===From jyā to sine===

===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 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  Latin translator used the Latin equivalent for "bosom", ''[[wikt:sinus|sinus]]''.<ref>Various sources credit the first use of ''sinus'' to either:

The origins of the modern term sine have been traced to the Sanskrit word ''jyā'',  or more specifically to its synonym  ''jīva''. This term was 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  Latin translator used the Latin equivalent for "bosom", ''[[wikt:sinus|sinus]]''.  When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.

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==

==The Indian sine tables==

==References==

==References==

[[Category:Indian Astronomy]]

[[Category:Indian Astronomy]]

[[Category:Indian Mathematics]]

[[Category:Indian Mathematics]]

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