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. 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 ) 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 χορδή 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-jyās or jyā-ardhas. 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°".