Jyā, Koti-jyā And Utkrama-jyā

Jyā, koṭi-jyā and utkrama-jyā 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. 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" have been traced back to the Sanskrit words jyā and koti-jyā.

Definition

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'' ) ) = ''R'' versin ( ''s'' / ''R'' )

Terminology

An arc of a circle is like a bow and so is called a ''dhanu'' or ''chāpa'' 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", presumably translating Hipparchus's with the same meaning.
The word ''jīvá'' is also used as a synonym for ''jyā'' in 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-jyā''s or ''jyā-ardha''s. These terms were again shortened to ''jyā'' by omitting the qualifier ''ardha'' which meant "half of".

The Sanskrit word ''koṭi'' 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 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 other cathetus of a right triangle, the first cathetus being the ''jyā''.

''Utkrama'' means "inverted", thus ''utkrama-jyā'' means "inverted chord".
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".

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".

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 ,
or more specifically to its synonym .
This term was adopted in medieval Islamic mathematics, transliterated in Arabic as ( جيب). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with ''yāʾ'' – this was interpreted as the homograph , ( جيب), which means "bosom". The text's 12th-century Latin translator used the Latin equivalent for "bosom", '' sinus''. 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.

See also

*Versine

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