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!

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