Origin of Trigonometric Functions & Zero

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Trigonometric functions:

Astronomy is the oldest of all the natural sciences and in India it dates back to the period of Indus Valley Civilization during 3rd millennium BCE, when it was used to create calendars. So the use of trigonometric functions arises from the early relation between Mathematics and Astronomy. However, the first work on trigonometric functions related to chords of a circle and the first known table of chords was produced by the Greek mathematician Hipparchus in about 140 BC. Although these tables have not survived, but it is claimed that twelve books of tables of chords were written by Hipparchus.

The next Greek mathematician to produce a table of chords was Menelaus in about 100 AD. Menelaus worked in Rome producing six books of tables of chords which have been lost but his work on spherics has survived and is the earliest known work on spherical trigonometry.

But the first actual appearance of the sine of an angle appears in the work of the Hindus. Sine and versine were first described in detail by Aryabhata in the late 5th century, but were likely developed earlier in the Siddhantas, astronomical treatises of the 3rd or 4th century. Aryabhata, in about 500 AD, gave tables of half chords which now really are sine tables and used jya for sin. This same table was reproduced in the work of Brahmagupta (in 628) and detailed method for constructing a table of sines for any angle were give by Bhaskara in 1150.

The Hindu word jya for the sine, was adopted by the Arabs who called it jiba, a meaningless word with the same sound as jya. Now jiba became jaib in later Arab writings  which simply means a ‘fold’. When European authors translated the Arabic mathematical works into Latin they translated jaib into the word sinus meaning fold in Latin.  Later on, Fibonacci’s use of the term sinus in his books soon encouraged the universal use of sine. So in this way Aryabhata’s Jya became sin or sine.

Later, the 6th-century astronomer Varahamihira made some important mathematical discoveries.  He gave his sine tables  with more accurate values where he improved those of  given by Aryabhata. Accuracy was very important for these Indian mathematicians since they were computing sine tables for applications to astronomy and astrology. There are certain trigonometric formula’s which we see in our schools and get scared of –

sin x = cos(π/2 – x),   sin2x + cos2x = 1, and (1 – cos 2x)/2 = sin2x.

Zero:

Indians were the first to use the zero as a symbol in arithmetic operations, although Babylonians used zero to signify the ‘absent’ in their work. In those earlier times a blank space was used to denote zero, later when it created confusion, a dot was used to denote zero (could be found in Bakhshali manuscript). In 500 AD, Aryabhata again gave a new symbol for zero i.e “0”. The concept of zero as a number and not merely a symbol or an empty space for separation was first given by Aryabhata and by the 9th century AD, practical calculations were carried out using zero. In 498 AD, Aryabhata stated that “sthānāt sthānaṁ daśaguņaṁ syāt” i.e. “from place to place each is ten times the preceding,” which is the origin of the modern decimal-based place value notation.