### Origin of Trigonometric Functions & Zero

### 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), sin^{2}x + cos^{2}x = 1, and (1 – cos 2x)/2 = sin^{2}x.**

### 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**.