JyÃÂ, koá¹Âi-jyàand utkrama-jyàare three trigonometric functions introduced by Indian mathematicians 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. Jyàand koti-jyàare 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 jyàand koti-jyÃÂ.
Let 'arc AB' denote an arc 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 is R and the length of arc AB is s, the angle subtended by arc AB at O measured in radians is ø = 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 a dhanu or chÃÂpa 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 jyàwhich in Sanskrit means "a bow-string", presumably translating Hipparchus's with the same meaning. The word jëvá is also used as a synonym for jyàin 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. The half-chords were called ardha-jyÃÂs or jyÃÂ-ardhas. These terms were again shortened to jyàby omitting the qualifier ardha which meant "half of".
The Sanskrit word koá¹Âi has the meaning of "point, cusp", and specifically "the curved end of a bow". In trigonometry, it came to denote "the complement of an arc to 90ð". Thus koá¹Âi-jyàis "the jyàof the complementary arc". In Indian treatises, especially in commentaries, koá¹Âi-jyàis often abbreviated as kojyÃÂ. The term koá¹Âi also denotes "the side of a right angled triangle". Thus koá¹Âi-jyàcould also mean the other cathetus of a right triangle, the first cathetus being the jyÃÂ.
Utkrama means "inverted", thus utkrama-jyàmeans "inverted chord". The tabular values of utkrama-jyàare derived from the tabular values of jyàby 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 bÃÂá¹Âa, iá¹£u or à Âara all meaning "arrow".
An arc of a circle which subtends an angle of 90ð at the center is called a vritta-pÃÂda (a quadrat of a circle). Each zodiacal sign defines an arc of 30ð and three consecutive zodiacal signs defines a vritta-pÃÂda. The jyàof a vritta-pÃÂda is the radius of the circle. The Indian astronomers coined the term tri-jyàto denote the radius of the base circle, the term tri-jyàbeing indicative of "the jyàof three signs". The radius is also called vyÃÂsÃÂrdha, viá¹£kambhÃÂrdha, vistarÃÂrdha, etc., all meaning "semi-diameter".
According to one convention, the functions jyÃÂ and koti-jyÃÂ are respectively denoted by "Rsin" and "Rcos" treated as single words. Others denote jyÃÂ and koti-jyÃÂ 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).
The origins of the modern term sine have been traced to the Sanskrit word , or more specifically to its synonym . This term was adopted in medieval Islamic mathematics, transliterated in Arabic as (). Since Arabic is written without short vowels â and as a borrowing the long vowel is here denoted with yÃÂþ â this was interpreted as the homograph , (), which means "bosom". The text's 12th-century Latin translator used the Latin equivalent for "bosom", '. When became , it has been suggested that by analogy became co-sinus. However, in early medieval texts, the cosine is called the "sine of the complement", suggesting the similarity to is coincidental.