Jyā,koṭi-jyā andutkrama-jyā are threetrigonometric functions introduced byIndian mathematicians and 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 functions ofsine andcosine. In fact, the origins of the modern terms of "sine" and "cosine" have been traced back to theSanskrit words jyā and koti-jyā.[1]
Let 'arc AB' denote anarc whose two extremities are A and B of a circle with center 'O'. If a perpendicular BM is dropped from B to OA, then:
If the radius of the circle isR and 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:
An arc of a circle is like a bow and so is called adhanu orchāpa which inSanskrit 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 ajyā which inSanskrit means "a bow-string", presumably translatingHipparchus'sχορδή with the same meaning[citation needed]. The wordjīvá is also used as a synonym forjyā in 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 tojyā by omitting the qualifierardha which meant "half of".
The Sanskrit wordkoṭi has 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°". Thuskoṭi-jyā is "thejyā of the complementary arc". In Indian treatises, especially in commentaries,koṭi-jyā is often abbreviated askojyā. The termkoṭi also denotes "the side of a right angled triangle". Thuskoṭi-jyā could also mean the othercathetus of a right triangle, the first cathetus being thejyā.[clarification needed][1]
Utkrama means "inverted", thusutkrama-jyā means "inverted chord". The tabular values ofutkrama-jyā are derived from the tabular values ofjyā by 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,iṣu orśara all 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. Thejyā of avritta-pāda is the radius of the circle. The Indian astronomers coined the termtri-jyā to denote the radius of the base circle, the termtri-jyā being indicative of "thejyā of three signs". The radius is also calledvyāsārdha,viṣkambhārdha,vistarārdha, etc., all meaning "semi-diameter".[1]
According to one convention, the functionsjyā andkoti-jyā are respectively denoted by "Rsin" and "Rcos" treated as single words.[1] Others denotejyā andkoti-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).[3]
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-centuryLatin translator used the Latin equivalent for "bosom",sinus.[6] Whenjyā becamesinus, it has been suggested that by analogykojyā becameco-sinus. However, in early medieval texts, the cosine is called thecomplementi sinus "sine of the complement", suggesting the similarity tokojyā is coincidental.[7]