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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>

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>

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.

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) ==

== 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]].<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" />

===Definition===

===Definition===

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:

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

* ''jyā'' of arc AB  = MB

* ''koti-jyā'' of arc AB  = CM = R cos θ

* ''utkrama-jyā'' of arc AB = MA

* ''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:

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'' )

* ''jyā'' ( arc AB ) = ''R'' sin θ

* ''koti-jyā'' ( arc AB ) = ''R'' cos ( ''s'' / ''R'' )

* ''koti-jyā'' ( arc AB ) = ''R'' cos θ

* ''utkrama-jyā'' ( arc AB ) = ''R'' ( 1 - cos ( ''s'' / ''R'' ) ) = ''R'' [[Versine|versin]] ( ''s'' / ''R'' )

* ''utkrama-jyā'' ( arc AB ) = R - R cos θ

 

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".  

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";  

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>

''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&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".

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]]".

In trigonometry, it came to denote "the complement of an arc to 90°". Thus  

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"/>

''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".  

''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" />

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"/>

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"/>

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===

===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>   

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''.

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:  

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]]

* [[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]]

* [[Gerard of Cremona]]'s c. 1150 translation of the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwārizmī]]

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.<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."

* "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.

This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function.

[[Category:Trigonometry]]

[[Category:Trigonometry]]

[[Category:Indian mathematics]]

[[Category:Indian mathematics]]