, and are 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. and 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 and .
Definition
Let '' denote an arc whose two extremities are and of a circle with center . If a perpendicular is dropped from to , then:
of of of
If the radius of the circle is and the length of is , the angle subtended by at measured in radians is . The three Indian functions are related to modern trigonometric functions as follows:
Terminology
An arc of a circle is like a bow and so is called a or 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 which in Sanskrit means "a bow-string", presumably translating Hipparchus's with the same meaning. The word is also used as a synonym for in geometrical literature.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" (cognate with English quick) 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 s or s. These terms were again shortened to by omitting the qualifier which meant "half of".
The Sanskrit word 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 ". Thus is "the of the complementary arc". In Indian treatises, especially in commentaries, is often abbreviated as . The term also denotes "the side of a right angled triangle". Thus could also mean the other cathetus of a right triangle, the first cathetus being the .
means "inverted", thus means "inverted chord". The tabular values of are derived from the tabular values of 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 , or all meaning "arrow".
An arc of a circle which subtends an angle of at the center is called a (a quadrat of a circle). Each zodiacal sign defines an arc of and three consecutive zodiacal signs defines a . The of a is the radius of the circle. The Indian astronomers coined the term to denote the radius of the base circle, the term being indicative of "the of three signs". The radius is also called , , , etc., all meaning "semi-diameter".
According to one convention, the functions and are respectively denoted by "Rsin" and "Rcos" treated as single words. Others denote and 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 , 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 – and as a borrowing the long vowel is here denoted with – this was interpreted as the homograph , (), which means "bosom". The text's 12th-century Latin translator used the Latin equivalent for "bosom", .Various sources credit the first use of to either: Plato Tiburtinus's 1116 translation of the Astronomy of Al-Battani Gerard of Cremona's c. 1150 translation of the Algebra of al-Khwārizmī Robert of Chester's 1145 translation of the tables of al-Khwārizmī See Merlet, A Note on the History of the Trigonometric Functions in Ceccarelli (ed.), International Symposium on History of Machines and Mechanisms, Springer, 2004See Maor (1998), chapter 3, for an earlier etymology crediting Gerard.See When became , it has been suggested that by analogy became . However, in early medieval texts, the cosine is called the "sine of the complement", suggesting the similarity to is coincidental.
See also Versine
References