Trigonometry

Hindu name for the science of Trigonometry is Jyotpatti-ganita or "the science of calculation for the construction of the sine". It is found as early as in the Brāhma-sphuṭa-siddhānta of Brahmagupta (628). In the recent years name appeared as Trikoṇamiti. The Hindus introduced and employed three trigonometrical functions namely jyā, koṭi-jyā , utkrama-jyā. They are functions of an arc of a circle, but not of an angle. If AP is an arc of a circle with centre at O, then its jyā=PM, koṭi-jyā=OM and utkrama-jyā=OA-OM =AM. Hence their relation with modern trigonometrical functions will be

jyā AP=Rsinθ, koṭi-jyā AP=Rcosθ, utkrama-jyā AP=R-Rcosθ=Rversinθ, where R is the radius of the circle and θ the angle subtended at the centre by the arc AP. Thus the values of the Hindu trigonometrical functions vary with the radius chosen. The earliest Hindu treatise in which the above trigonometrical functions are now found recorded is the Sūrya-siddhānta.

Jyā
Jyā in sanskrit means "a bow-string" and hence the " chord of an arc" for the arc is called " a bow" (dhanu, cāpa). The other names are

are jīvā, śiñjinī, guṇa , maurvī etc. This trigonometrical function is also called ardha-jyā (“half-chord") or jyārdha ("chord-half"). Thus Bhaskara II (1150) explicitly observes, "It should be known that ardha-jyā is here called "jyā".

Parameśvara (1430) remarks:

"A part of a circle is of the form of a bow, so it is called the "bow" (dhanu). The straight line joining its two extremities is the "bow-string"    (jīvā); it is really the "full-chord” (samasta-jyā). Half of it is here (called) the "half-chord" (ardha-jyā), and half that arc is called the "bow" of that half-chord. In fact the Rsine (jyā) and Rcosine (koṭi-jyā) of that bow are always half-chords."

Kamalakara (1658) is more explicit. "Having seen the brevity", says he, "the half-chords are called Jya by mathematicians in this (branch of) mathematics and are used accordingly." The function jya is sometimes distinguished as krama-jyā” or kramārdha-