Trigonometry

a Boy with a book in front of a blackboard with Trigonometry Formula written on it

Indians have always had a noteworthy relationship with mathematics. Therefore, it’s no surprise that trigonometry was borne in India. Trigonometry developed in ancient India to solve astronomy problems. The earliest reference to trigonometry in India occurs in the text “Suryasidhanta” (c. 400). Important works from the 4th & the 5th century AD, like the Siddhantas, first defined the sine as the relationship between half an angle & half a chord, while also defining the cosine & inverse sine.

Shortly thereafter the Indian mathematician and astronomer Aryabhata collected and expanded upon the developments of the Siddhantas in a path-breaking work, the Aryabhatiya. The Siddhantas & the Aryabhatiya contain the earliest surviving tables of sine & versine values, in intervals from 0 to 90, to an accuracy of 4 decimal places. These texts used the words “jya” for sine, “kojya” for cosine, “utkrama-jya” for versine and “otkram jya” for inverse sine. The words “jya” and “kojya” eventually became sine and cosine after a mistranslation.

An interesting note is that, in Sanskrit, “jya-ardha” means chord-half, and this was frequently abbreviated to just “jya”. Indian works were eventually translated to Arabic, and when an Arabic work was translated to Latin, the term was mistranslated to mean “bosom or breast”, and thus was written as the Latin word “sinus”, and this is how the term “sine” was coined.

The first sine table can be traced to the early 5th century, in the text “Paitamahasiddhanta”. The first well-preserved text containing a somewhat accurate sine table appeared in Aryabhata’s aforementioned “Aryabhatiya”, written in the year 499 AD.

One of India’s greatest mathematicians, Madhava is sometimes called the greatest mathematician-astronomer of medieval India. Madhava came from a town called Sangamagrama, in Kerala, near the southern tip of India. He founded the Kerala School of Astronomy and Mathematics in the late 14th Century.

While Madhava’s predecessors were nervous about the concept of infinity, Madhava was more than happy to play around with infinity, particularly with the infinite series. He showed how, although one can be approximated by adding a half plus a quarter plus an eighth plus a sixteenth, etc., (as even the ancient Egyptians and Greeks had known), the exact total of one can be achieved only by adding up infinitely many fractions.

Madhava went even further and linked the idea of an infinite series with geometry and trigonometry. He realised that, by successively adding and subtracting different odd number fractions to infinity, he could hone in on an exact formula for π (this was two centuries before the German polymath Gottfried Wilhelm Leibniz came to the same conclusion in Europe). Today, trigonometry has a variety of applications including, but not limited to, astronomy, construction, physics, flight engineering and criminology.

Another example of the application of trigonometry is in music. Sound travels in waves and this pattern though not as regular as a sine or cosine function, is still useful in developing music. A computer cannot obviously listen to and comprehend music as we do, so computers represent music mathematically by its constituent sound waves. And this means sound engineers need to know at least the basics of trigonometry.

Fun Fact

Aryabhata approximated the value of π equivalent to 3.1416, correct to four decimal places. He used this to estimate the circumference of the Earth, arriving at a figure of 24,835 miles, only 70 miles off its true value.