Indian mathematicians made profound contributions to trigonometry, shifting from the ancient Greek system of chords (used by Ptolemy and others) to the more modern sine and cosine functions, which simplified calculations and theorems related to triangles.

In his Aryabhatiya (499 CE), Aryabhata introduced the first known sine table in India.

Instead of the Greek chord function, he used the concept of ardha-jya (half-chord), which later became the sine function (jya or jiva in Sanskrit).

His sine table consisted of values for sin⁡(A)\sin(A) at 3.75° intervals, using a circle with a radius of 3438 units (since π≈3.1416\pi \approx 3.1416 was approximated as 3.1416 × 360/2π = 3438). This was an essential step in the development of modern trigonometry.

In his Pañcasiddhāntikā, Varāhamihira provided explicit trigonometric identities, including:

• sin⁡2A+cos⁡2A=1\sin^2 A + \cos^2 A = 1 (one of the first clear formulations of this identity)
• The double-angle formula: sin⁡2A=2sin⁡Acos⁡A\sin 2A = 2 \sin A \cos A

His work also included calculations for eclipse predictions and astronomical models using trigonometric principles.

Later Indian mathematicians like Bhaskara I (7th century), Brahmagupta (7th century), and Bhaskara II (12th century) extended these ideas and refined sine and cosine calculations.

The Indian “jiva” (sine) was transliterated into Arabic as “jiba” and later misunderstood as “jaib” (meaning “fold” or “bay”) by European translators, leading to the modern term “sine” in Latin.

The Indian sine function was adopted by Islamic mathematicians, such as Al-Battani and Al-Khwarizmi, and later reached Europe through translations during the Renaissance.