>Pratheesh sreenivasan at 08:53, 30 June 2021

2021-06-30T08:53:06Z

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{{short description|Trigonometric functions introduced by Indian mathematicians and astronomers}}
{{Use dmy dates|date=August 2019}}
{{Use Indian English|date=August 2019}}
'''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|access-date=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:Modern diagram for jya and kojya.svg|thumb|right|250px|Modern diagram for jyā and kojyā]]

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|right|250px|Literal meaning of jyā]]
[[File:Jya and kotijya.JPG|thumb|right|250px|Technical meaning of jyā and kojyā]]

An arc of a circle is like a bow and so is called a ''dhanu'' or ''chāpa'' 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", presumably translating [[Hipparchus]]'s {{lang|grc|[[chord (geometry)|χορδή]]}} with the same meaning{{citation needed|date=March 2017}}.
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'' could also mean the side a right triangle one of whose sides is the ''jyā''.{{clarify|date=October 2014}}<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.{{clarify|date=October 2014}} This is really{{clarify|date=October 2014}} 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==
{{main|History of trigonometry}}
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]]|access-date=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 |access-date=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=en }}</ref> When ''jyā'' became ''sinus'', by analogy ''kojyā'' became ''co-sinus''.

==See also==
*[[Versine]]

==References==

{{reflist}}

{{Indian mathematics}}
{{Trigonometric and hyperbolic functions}}

{{DEFAULTSORT:Jya, Koti-Jya And Utkrama-Jya}}
[[Category:Trigonometric functions]]
[[Category:Trigonometry]]
[[Category:Indian mathematics]]
[[Category:Sanskrit words and phrases]]
[[Category:Hindu astronomy]]
[[Category:Kerala school of astronomy and mathematics]]
[[Category:History of mathematics]]

Jyā, koti-jyā and utkrama-jyā - Revision history

>Pratheesh sreenivasan at 08:53, 30 June 2021