Madhava of Sangamagrama is widely recognized for his pioneering work in calculus, particularly his development of the concept of the derivative and the use of infinite series to approximate functions.

The Pythagorean theorem is a well-known concept in geometry, but it is not directly related to the mathematical work of Madhava of Sangamagrama, who focused on calculus and trigonometry.

Madhava's work on infinite series had multiple significant implications. It provided a method for approximating functions using polynomials, leading to the development of the concept of the derivative. Additionally, it enabled the calculation of areas and volumes of complex shapes.

Madhava used the sine function to approximate the value of pi. He developed an infinite series expansion for the sine function, which allowed him to calculate its values for different angles.

Madhava's mathematical investigations were primarily driven by his desire to solve practical problems in astronomy and engineering. He sought to develop mathematical techniques that could be applied to real-world problems, such as calculating the positions of celestial bodies and designing structures.

The Newton-Raphson method is a numerical method for finding the roots of a function. It was developed by Isaac Newton and Joseph Raphson in the 17th century, and is not associated with Madhava of Sangamagrama.

The Madhava-Gregory series is an infinite series expansion for the sine function. It was developed by Madhava of Sangamagrama and later rediscovered by James Gregory in the 17th century.

Madhava's work on calculus had a profound influence on the development of mathematics in later centuries. It laid the foundation for the development of differential and integral calculus, leading to the discovery of new mathematical theorems and formulas. Additionally, it inspired other mathematicians to explore new areas of mathematics.

Madhava used his work on infinite series to calculate the area of a circle. He developed an infinite series expansion for the area of a circle, which allowed him to approximate its value.

Madhava of Sangamagrama primarily used Sanskrit to record his mathematical work. Sanskrit was the dominant language of scholarship and intellectual discourse in India during his time.

The concept of the logarithm is not directly associated with Madhava of Sangamagrama. He primarily focused on the development of calculus and trigonometry.

Madhava's development of infinite series expansions for trigonometric functions was motivated by multiple factors. He sought to simplify the calculation of trigonometric values, approximate the values of pi and other constants, and solve practical problems in astronomy and engineering.

The use of arithmetic series is not directly associated with Madhava of Sangamagrama. He primarily focused on the use of power series, geometric series, and harmonic series in his mathematical investigations.

The Wallis formula is a formula for calculating the area of a circle. It was discovered by John Wallis in the 17th century, and is related to the work of Madhava of Sangamagrama on infinite series.

The concept of the logarithm is not directly associated with Madhava of Sangamagrama. He primarily focused on the development of calculus and trigonometry.