Though its authorship is unknown, the Surya Siddhanta (c. 400) contains the roots of modern trigonometry. Some authors consider that it was written under the influence of Mesopotamia and Greece.
This ancient text uses the following as trigonometric functions for the first time:
- Sine (Jya).
- Cosine (Kojya).
- Inverse sine (Otkram jya).
It also contains the earliest uses of:
The Hindu cosmological time cycles explained in the text, which was copied from an earlier work, gives:
- The average length of the sidereal year as 365.2563795 days, which is only 1.4 seconds longer than the modern value of 365.2563627 days.
- The average length of the tropical year as 365.2421756 days, which is only 2 seconds shorter than the modern value of 365.2421988 days.
Aryabhata (476-550) wrote the Aryabhatiya. He described the important fundamental principles of mathematics in 332 shlokas. The treatise contained:
- Quadratic equations
- The value of π, correct to 4 decimal places.
Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata’s contributions include:
- Introduced the trigonometric functions.
- Defined the sine (jya) as the modern relationship between half an angle and half a chord.
- Defined the cosine (kojya).
- Defined the versine (utkrama-jya).
- Defined the inverse sine (otkram jya).
- Gave methods of calculating their approximate numerical values.
- Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals from 0° to 90°, to 4 decimal places of accuracy.
- Contains the trigonometric formula sin (n + 1) x – sin nx = sin nx – sin (n – 1) x – (1/225)sin nx.
- Spherical trigonometry.
- Continued fractions.
- Solutions of simultaneous quadratic equations.
- Whole number solutions of linear equations by a method equivalent to the modern method.
- General solution of the indeterminate linear equation .
- Proposed for the first time, a heliocentric solar system with the planets spinning on their axes and following an elliptical orbit around the Sun.
- Accurate calculations for astronomical constants, such as the:
- Solar eclipse.
- Lunar eclipse.
- The formula for the sum of the cubes, which was an important step in the development of integral calculus.
- In the course of developing a precise mapping of the lunar eclipse, Aryabhata was obliged to introduce the concept of infinitesimals (tatkalika gati) to designate the near instantaneous motion of the moon.
- Differential equations:
- He expressed the near instantaneous motion of the moon in the form of a basic differential equation.
- Exponential function:
- He used the exponential function e in his differential equation of the near instantaneous motion of the moon.
Bhāskara II (1114–1185) was a mathematician-astronomer who wrote a number of important treatises, namely the Siddhanta Shiromani, Lilavati, Bijaganita, Gola Addhaya, Griha Ganitam and Karan Kautoohal. A number of his contributions were later transmitted to the Middle East and Europe. His contributions include:
- Interest computation
- Arithmetical and geometrical progressions
- Plane geometry
- Solid geometry
- The shadow of the gnomon
- Solutions of combinations
- Gave a proof for division by zero being infinity.
- The recognition of a positive number having two square roots.
- Operations with products of several unknowns.
- The solutions of:
- Quadratic equations.
- Cubic equations.
- Quartic equations.
- Equations with more than one unknown.
- Quadratic equations with more than one unknown.
- The general form of Pell’s equation using the chakravala method.
- The general indeterminate quadratic equation using the chakravala method.
- Indeterminate cubic equations.
- Indeterminate quartic equations.
- Indeterminate higher-order polynomial equations.
- Gave a proof of the Pythagorean theorem.
- Conceived of differential calculus.
- Discovered the derivative.
- Discovered the differential coefficient.
- Developed differentiation.
- Stated Rolle’s theorem, a special case of the mean value theorem (one of the most important theorems of calculus and analysis).
- Derived the differential of the sine function.
- Computed π, correct to 5 decimal places.
- Calculated the length of the Earth’s revolution around the Sun to 9 decimal places.
- Developments of spherical trigonometry
- The trigonometric formulas: