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Jyā, koti-jyā and utkrama-jyā

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Jyā,koṭi-jyāandutkrama-jyāare threetrigonometric functionsintroduced byIndian mathematiciansand astronomers. The earliest known Indian treatise containing references to these functions isSurya Siddhanta.[1]These are functions of arcs of circles and not functions of angles. Jyā and koti-jyā are closely related to the moderntrigonometric functionsofsineandcosine.In fact, the origins of the modern terms of "sine" and "cosine" have been traced back to theSanskritwords jyā and koti-jyā.[1]

Definition

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Modern diagram for jyā and kojyā

Let 'arc AB' denote anarcwhose two extremities are A and B of a circle with center O. If a perpendicular BM is dropped from B to OA, then:

  • jyāof arc AB = BM
  • koti-jyāof arc AB = OM
  • utkrama-jyāof arc AB = MA

If the radius of the circle isRand the length of arc AB iss,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 ) =Rsin (s/R)
  • koti-jyā( arc AB ) =Rcos (s/R)
  • utkrama-jyā( arc AB ) =R( 1 - cos (s/R) ) =Rversin(s/R)

Terminology

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Literal meaning of jyā
Technical meaning of jyā and kojyā

An arc of a circle is like a bow and so is called adhanuorchāpawhich inSanskritmeans "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 ajyāwhich inSanskritmeans "a bow-string", presumably translatingHipparchus'sχορδήwith the same meaning[citation needed]. The wordjīváis also used as a synonym forjyāin geometrical literature.[2] 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.[1][3]The half-chords were calledardha-jyās orjyā-ardhas. These terms were again shortened tojyāby omitting the qualifierardhawhich meant "half of".

The Sanskrit wordkoṭihas the meaning of "point, cusp", and specifically "thecurved end of a bow". In trigonometry, it came to denote" the complement of an arc to 90° ". Thus koṭi-jyāis "thejyāof the complementary arc ". In Indian treatises, especially in commentaries,koṭi-jyāis often abbreviated askojyā.The termkoṭialso denotes "the side of a right angled triangle". Thuskoṭi-jyācould also mean the othercathetusof a right triangle, the first cathetus being thejyā.[clarification needed][1]

Utkramameans "inverted", thusutkrama-jyāmeans "inverted chord". The tabular values ofutkrama-jyāare derived from the tabular values ofjyāby subtracting the elements from the radius in the reversed order.[clarification needed]This is really[clarification needed]the arrow between the bow and the bow-string and hence it has also been calledbāṇa,iṣuorśaraall meaning "arrow".[1]

An arc of a circle which subtends an angle of 90° at the center is called avritta-pāda(a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines avritta-pāda.Thejyāof avritta-pādais the radius of the circle. The Indian astronomers coined the termtri-jyāto denote the radius of the base circle, the termtri-jyābeing indicative of "thejyāof three signs ". The radius is also calledvyāsārdha,viṣkambhārdha,vistarārdha,etc., all meaning "semi-diameter".[1]

According to one convention, the functionsjyāandkoti-jyāare respectively denoted by "Rsin" and "Rcos" treated as single words.[1]Others denotejyāandkoti-jyārespectively 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).[3]

From jyā to sine

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Main article:Sine and cosine § Etymology

The origins of the modern term sine have been traced to the Sanskrit wordjyā,[4][5] or more specifically to its synonymjīvá. This term wasadopted in medieval Islamic mathematics,transliterated in Arabic asjība(جيب). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted withyāʾ– this was interpreted as thehomographjaib,jayb(جيب), which means "bosom". The text's 12th-centuryLatintranslator used the Latin equivalent for "bosom",sinus.[6]Whenjyābecamesinus,it has been suggested that by analogykojyābecameco-sinus.However, in early medieval texts, the cosine is called thecomplementi sinus"sine of the complement", suggesting the similarity tokojyāis coincidental.[7]

See also

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References

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  1. ^abcdefgB.B. Datta and A.N. Singh (1983)."Hindu Trigonometry"(PDF).Indian Journal of History of Science.18(1): 39–108.Retrieved2 June2022.
  2. ^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): "jīván. (in geom. =jyā) the chord of an arc; the sine of an arcSuryasiddhanta2.57 "; jīváas a generic adjective has the meaning of "living, alive" (cognatewith Englishquick)
  3. ^abGlen Van Brummelen (2009).The mathematics of the heavens and the earth: the early history of trigonometry.Princeton University Press.pp. 95–97.ISBN978-0-691-12973-0.
  4. ^"How the Trig Functions Got their Names".Ask Dr. Math.Drexel University.Retrieved2 March2010.
  5. ^J J O'Connor and E F Robertson (June 1996)."The trigonometric functions".Retrieved2 March2010.
  6. ^Various sources credit the first use ofsinusto either: See Merlet,A Note on the History of the Trigonometric Functionsin Ceccarelli (ed.),International Symposium on History of Machines and Mechanisms,Springer, 2004
    See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.
    SeeKatx, Victor (July 2008).A history of mathematics(3rd ed.). Boston: Pearson. p. 210 (sidebar).ISBN978-0321387004.
  7. ^"cosine".
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