**Introduce Trigonometry by Aryabhata (Ind.)**

In *Ganitapada* 6, Aryabhata (Ind) gives the area of a triangle as:

Tribhujasya phalashriram samadalakoti bhujardhasamvargah

that translates to: “for a triangle, the result of a perpendicular with the half-side is the area.

Aryabhata (Ind) discussed the concept of sine in his work by the name of ‘ardha-jay’ , which literally means “half-chord”. For simplicity, people started calling it jya, When Arabic writers translated his works from Sanskrit into Arabic, they referred it as *jiba*. However, in Arabic writings, vowels are omitted, and it was abbreviated as *jb*. Later writers substituted it with *jaib*, meaning “pocket” or “fold (in a garment)”. (In Arabic, *jiba* is a meaningless word.) Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic *jaib* with its Latin counterpart, *sinus*, which means “cove” or “bay”; thence comes the English *sine*. Alphabetic code has been used by him to define a set of increments. If we use Aryabhata’s table and calculate the value of sin(30) (corresponding to hasjha) which is 1719/3438 = 0.5; the value is correct. His alphabetic code is commonly known as the Aryabhata cipher.

### Indeterminate equations

A problem of great interest to Indian Mathematician group since ancient times has been to find integer solutions to equations that have the form ax + by = c, a topic that has come to be known as Diophantine equation. This is an example from mathematician Bhaskara (Ind) commentary on Aryabhatiya:

**Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7**

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as this, can be notoriously difficult. They were discussed extensively in ancient Vedic text ‘sulabha sutras’ Sulba , whose more ancient parts might date to 800 BCE. Aryabhata’s method of solving such problems is called the *ku**ṭṭ**aka * method. *Kuttaka* means “pulverizing” or “breaking into small pieces”, and the method involves a recursive algorithm for writing the original factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantine equations and is often referred to as the Aryabhata algorithms The diophantine equations are of interest in cryptology

To be continue ………3