VEDIC MATHS

 

Name-Manav Nimish Kothari

PRN-23UF18402AC019

Trigonometry: The Jya and Cojya in Indian Mathematics — A Journey into Ancient Genius

 

Introduction — Where Angles Meet Eternity

Long before trigonometry became the heart of modern mathematics, ancient Indian scholars had already unlocked its mysteries.
They introduced concepts called Jya (Sine) and Cojya (Cosine) — terms rooted in Sanskrit and astronomy. These ideas bridged geometry, observation, and philosophy, revealing how India’s early mathematicians connected the cosmos with computation.

 

The Birth of Jya and Cojya

In modern trigonometry, the sine of an angle represents the ratio of the opposite side to the hypotenuse.
But in ancient India, mathematicians didn’t deal with triangles — they studied circles.

• Jya (or Ardha-jya) meant half the chord of an arc — the earliest form of sine.
• Cojya (Kojya or Kotijya) represented the complementary arc, the ancestor of the modern cosine.

The great scholar Aryabhata (5th century CE) first described these in Aryabhatiya. His ideas guided astronomers to predict eclipses, track planets, and measure time — proving that trigonometry was born not in a classroom, but under the open sky.

 

Computation of R-Sines

Ancient Indian mathematicians developed the concept of R-Sines, where “R” stood for the radius of a circle (commonly taken as 3438 units).
Using geometry and recursive techniques, they computed Jya values for various arcs without degrees or radians!

These R-Sine tables formed the backbone of early Indian astronomy and were used to:

• Predict planetary positions
• Measure celestial distances
• Design temples and observatories aligned with the stars

 

 

The precision achieved was remarkable — errors were just a few arc-minutes, showing their advanced mathematical understanding.

 

Madhava’s Marvel — The Infinite Sine and Cosine Series

In the 14th century, Madhava of Sangamagrama, a genius from Kerala, extended trigonometry into infinite series — laying the early foundation of calculus centuries before Newton and Leibniz.

He discovered that:

                   

This was a revolution — transforming trigonometry from a geometric tool into an analytical one.
Madhava’s series became the invisible thread connecting ancient India’s astronomy with modern science and engineering.

 

The Circle of Time

Picture this: an ancient Indian astronomer observing the sunset, his gnomon (sundial stick) casting a long shadow on a carved stone circle.
Every inch of that shadow told a story of angles, arcs, and Jya values.
The circle wasn’t just geometry — it was a cosmic clock, showing the rhythm of time, light, and mathematics working in perfect harmony.

 

Real-World Connection

Madhava’s infinite sine series mirrors the nature of human knowledge — expanding endlessly with every generation.
From predicting eclipses in ancient India to guiding satellites and drones today, trigonometry remains a bridge between earth and sky, past and future.

Just as sine waves describe sound, light, and motion, our own learning follows the same wave — rising, falling, and continuing infinitely.

 

 

Conclusion

From Aryabhata’s Jya to Madhava’s infinite series, India’s contribution to trigonometry is not just a mathematical legacy — it’s a philosophical masterpiece.
It teaches that when we combine curiosity with calculation, we touch infinity itself.The sine and cosine we study today are not merely ratios — they are echoes of the same rhythm that guided ancient minds to explore the universe.

 

Every angle tells a story, every curve hides a truth