Jyā, koti-jyā and utkrama-jyā
Jyā, koti-jyā and utkrama-jyā are three trigonometric functions introduced by Indian astronomers and mathematicians. 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. Jyā and koti-jyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of "sine" and "cosine" has been traced back to the Sanskrit words jyā and koti-jyā.
An arc of a circle is like a bow and so is called a dhanu or cāpa 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". The word jīvā (literally "life", cognate with quick, especially in its archaic sense as in "the quick and the dead".) is also used as a synonym for jyā in geometrical literature (Suryasiddhanta 2.57).
At some point in the historical development of trigonometry in India, the exact date of which could not be determined, 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-jyās or jyā-ardhas. These terms were shortened to jyā by omitting the qualifier ardha which meant "half of".
The Sanskrit word koṭi means, amongst others, "the curved end of a bow" or "the end or extremity in general". Hence 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 rightangled triangle one of whose sides is the jyā.
Utkrama means "reversed", "going out" or "exceeding". Utkrama-jyā means "reversed sine". 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".
Technical meaning of jyā
Let 'arc AB' denote an 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 ) )
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".
Modern terms for jyā and koti-jyā
According to one convention, the functions jyā and koti-jyā are respectively denoted by "Rsin" and "Rcos" treated as single words. 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).
From jyā to sine
The origins of the modern term sine have been traced to the Sanskrit word jyā. It has already been noted that the Sanskrit word jīvā had been used as a synonym for jyā. This term was adopted by early Arab mathematicians and they pronounced it as jība since "b" was the closest voiced consonant they had to "v". In Arabic jība is spelled "jiyba" with the consonants j, y and b written out, leaving vowels "i" and "a" as just a dash below and above the line. In fact, except for holy scripture like the Qur'ān, vowels are rarely written at all. Early Latin translators of Arabic works, unaware of an earlier Sanskrit borrowing, naturally read "jyb" as the purely Arabic word "jayb" and translated it into Latin as sinus having the meaning of "bosom" or "bay". Following suit with foreign borrowing, the Latin word sinus was borrowed into English as sine this time with all vowels intact. When jyā became sinus, by analogy kojyā became ko-sinus or with the Latin preference for "c" spelled co-sinus and in English cosine. This word history for "sine" is interesting because it follows the path of trigonometry in Sanskrit from India, through the Arabic language from Baghdad through Spain, into western Europe in the Latin language, and then to modern languages such as English."
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