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."

Kamalākara (1658) is more explicit. "Having seen the brevity", says he, "the half-chords are called Jyā by mathematicians in this (branch of) mathematics and are used accordingly. The function jyā is sometimes distinguished as krama-jyā or kramārdha-jyā from krama, "regular" or "direct" meaning "direct sine" or "direct half-chord".

The modern term sine is derived from the Hindu name. The sanskrit term jīvā was adopted by the early Arab mathematicians but was pronounced as jiba. Later corrupted in their tongue into jaib. The latter word was confused by the early Latin translators of the Arabic works such as Gherardo of Cremona (c. 1150 A.D.) with a pure Arabic word of alike phonetism but meaning differently "bosom" or "bay" and was rendered as sinus, which also signifies "bosom" or "bay".

The degeneration and variations of the term krama-jyā are still more interesting. In the Arabic tongue it was corrupted into karaja or kardaja. According to Fihrist, the title of a work of Ya'kūb ibn Ṭārik (c. 770 A.D.) is "On the table of kardaja.” This table was copied from the Brāhma-sphuṭa-siddhānta of Brahmagupta. In the same connexion, al-Khowārizmi (825) used the variant karaja. In the Latin translations of the term we find several variants such as kardaga, karkaya, gardaga or cardaga. These terms had in foreign lands also the restricted uses for the arc of 3° 45', some- times of 15°.

Koti-Jyā
The Sanskrit word koți means, amongst others "the curved end of a bow" or "the end or extremity in general"; hence in Trigonometry it denoted "the complement of an arc to 90°. Hence Koti-Jyā is " the Jyā of the complimentary arc”. The modern term cosine appears to be connected with koṭijyā, for in Hindu works, particularly in the commentaries koṭijyā is often abbreviated into kojyā. When jyā became sinus, kojyā naturally became ko-sinus or co-sinus.

Utkrama-Jyā
Utkrama means "reversed", "going out" or "exceeding". Hence the term utkrama-jyā literally means "reversed sine". This function is so called in contradis- tinction to krama-jyā, for it is, rather its tabular values are, derived from the tabular values of the latter by subtracting the elements from the radius in the reversed order. Or in other words it is the exceeding portion of the krama-jyā taken into consideration in the reversed order.

Tangent and Secant
The Hindus approached very near the tangent and secant functions and actually employed them in astronomical calculations, though they did not expressly recognise them as separate functions. The Sūrya-siddhānta gives the following rule for calculating the equinoctial midday shadow of the gnomon at a station:

"The sine of the latitude (of the station) multiplied by 12 and divided by the cosine of the latitude gives the equinoctial mid-day shadow."

Here 12 is the usual height of a Hindu gnomon. So that

S = (jyā ø X h) / (kojyā ø)

where ø denotes the latitude of the place, S denotes equinoctial mid-day shadow and h denotes gnomon. This is equivalent to

S = h tanθ

To find the mid-day shadow (s) of the gnomon (h) and the hypotenuse (d), having known the meridian zenith distance (z) of the sun, we have the rules:

s=h tan z, d=h sec z.

Similar rules occur in other astronomical works also. In the Gaṇita-sāra-saṅgraha of Mahāvīra (850) by the term "shadow” of a gnomon is sometimes meant the ratio of the actual shadow to the height of the gnomon. This ratio, as has been just stated, is equal to the tangent of the zenith distance of the sun.