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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
2 years, 6 months ago

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Great Indians great mathematics Raghu Ram · 8 months ago

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