Deep Dive · Vedic Mathematics
Vedic Origins of Trigonometry
From Baudhayana's diagonal theorem to Madhava's infinite series — how India invented trigonometry and named the sine function.
Jya (Sine)
The Sanskrit term 'jya' (ज्या) meaning 'bowstring' became the root of the modern word 'sine' through Arabic translation (jiba) and Latin mistranslation (sinus). India literally named trigonometry.
Kotijya (Cosine)
Kotijya (कोटिज्या) — the 'complement bowstring' — is the cosine function. Indian mathematicians recognized sine and cosine as complementary functions and tabulated both.
Utkrama-jya (Versine)
Utkrama-jya (उत्क्रम-ज्या) is the versine (1 - cos). Indian astronomers used this extensively for computing planetary positions, as it avoids negative numbers.
Trijya (Radius)
Trijya (त्रिज्या) means 'triple bowstring' and refers to the radius of the reference circle. Aryabhata used a radius of 3438 (minutes of arc in a radian), making the sine table directly usable for astronomical calculations.
Baudhayana's Sulba Sutra
~800 BCEDescription
States the diagonal theorem (Pythagorean theorem) centuries before Pythagoras. Provides geometric constructions for altars requiring precise angular calculations.
Source: Baudhayana Sulba Sutra 1.48
Original Sanskrit
दीर्घचतुरश्रस्याक्ष्णया रज्जुः पार्श्वमानी तिर्यक्मानी च यत्पृथग्भूते कुरुतस्तदुभयं करोति
Translation
“The diagonal of a rectangle produces both areas which its length and breadth produce separately.”
Aryabhata's Sine Table
499 CEDescription
Aryabhatiya presents a table of 24 sine differences (jya values) for angles from 0 to 90 degrees in steps of 3.75 degrees. This predates European sine tables by nearly a millennium.
Source: Aryabhatiya, Dasagitika Sutra, Verse 12
Original Sanskrit
मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व
Translation
“The encoded sine differences: 225, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164...”
Brahmagupta's Interpolation
628 CEDescription
Brahmasphutasiddhanta introduces second-order interpolation formulas for computing sine values between table entries, a precursor to modern numerical methods.
Source: Brahmasphutasiddhanta, Chapter 25
Original Sanskrit
गतभोग्यखण्डकान्तरदलं गतभुक्तशोध्यमथवा धनम्
Translation
“Half the difference of the tabular differences is to be subtracted from or added to the tabular difference.”
Madhava's Infinite Series
~1350 CEDescription
Madhava of Sangamagrama discovers the infinite series for sine, cosine, and arctangent — the "Taylor series" — nearly 300 years before Taylor and Gregory.
Source: Tantrasangraha (Nilakantha, recording Madhava)
Original Sanskrit
निहत्य चापवर्गेण चापं तत्तत्फलानि च। हरेत् समूलयुग्वर्गैः त्रिज्यावर्गाहतैः क्रमात्॥
Translation
“Multiply the arc by the square of the arc, and take the results one after another, dividing by squares of successive even numbers multiplied by the square of the radius.”
Kerala School Refinements
~1400 CEDescription
Nilakantha Somayaji, Jyeshthadeva, and others in the Kerala School extend Madhava's work. Jyeshthadeva's Yuktibhasa (1530) provides the first known rigorous proofs of these series expansions.
Source: Yuktibhasa by Jyeshthadeva (~1530 CE)
Aryabhata encoded 24 sine differences using an ingenious alphabetic notation in just one verse. The verse “makhi-bhakhi-phakhi...” encodes the values: 225, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154, 143, 131, 119, 106, 93, 79, 65, 51, 37, 22, 7.
These differences allow reconstruction of a full sine table for every 3.75 degrees. The radius (trijya) is 3438, equal to the number of arc-minutes in a radian. This means the jya values are directly usable in astronomical calculations without conversion.
The maximum error in Aryabhata's table compared to modern values is less than 1.5%, and for most entries the error is under 0.5%. European sine tables of comparable accuracy did not appear until the 15th century — nearly 1,000 years later.
Sine Series
Madhava (~1350 CE) discovered that sin(x) can be expressed as an infinite series:
This is known in the West as the “Taylor series for sine,” attributed to Brook Taylor (1715) — over 350 years after Madhava.
Arctangent Series (pi)
Madhava also discovered the infinite series for pi:
Known in Europe as the “Leibniz formula” (1674) — over 300 years after Madhava used it to calculate pi to 11 decimal places.
दीर्घचतुरश्रस्याक्ष्णया रज्जुः पार्श्वमानी तिर्यक्मानी च यत्पृथग्भूते कुरुतस्तदुभयं करोति
“The diagonal of a rectangle produces both areas which its length and breadth produce separately.”
This is the Pythagorean theorem (a² + b² = c²), stated by Baudhayana in his Sulba Sutra around 800 BCE — approximately 300 years before Pythagoras of Samos (570-495 BCE).
Baudhayana also provides the value of √2 as 1.4142156..., accurate to five decimal places. The modern value is 1.4142135..., giving an error of only 0.0014%.
Explore More
Dive into hands-on projects or explore other Vedic science topics.
Note on Sources
All references cite specific texts, chapters, and verses. Translations are based on standard academic editions. We encourage readers to verify all sources independently.