# Trigonometry with the Ancient Indian Mathematicians

The most significant development of trigonometry in ancient times was in India. Influential works from the $4^{th}$ & the $5^{th}$ century AD, known as Siddhantas first defined the sine as the relationship between half an angle & half a chord, while also defining the cosine, versine ($1-$ cosine) & inverse sine.

Soon afterwards, another Indian mathematician & astronomer Aryabhata collected & expanded upon the developments of the Siddhantas in a path breaking work, the Aryabhatiya. The Siddhantas & the Aryabhatiya contain the earliest survivng tables of sine & versine values, in $3.75^\circ$ intervals from $0^\circ$ to $90^\circ$, to an accuracy of $4$ decimal places. They used the words jya for sine, kojya for cosine, utkrama-jya for versine & otkram jya for inverse sine. The words jya & kojya eventually became sine & cosine after a mistranslation.

Aryabhata was the first one. Others after him expanded on these works of trigonometry. In the $6^{th}$ century AD, Varahamihira discovered the identity $sin^2x + cos^2x = 1$. He improved Aryabhata's sine table & discovered an early version of the Pascal's Triangle.

Such was the ingenuity of the ancient Indian mathematicians!

Note by Ameya Salankar
7 years, 3 months ago

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