History of trigonometry

Trigonometry is the study of triangles and the relationships between their angles and sides. Its history goes back thousands of years, beginning with early civilizations like the Egyptian mathematics and Babylonian mathematics. These ancient people used simple triangle measurements to solve practical problems, such as calculating distances and building structures.

Later, during the time of Hellenistic mathematics, scholars began studying triangles more systematically. This knowledge traveled to India, where it grew even more advanced. A famous mathematician named Aryabhata lived in the sixth century AD and discovered important functions like the sine function, cosine function, and versine function, which are still used today.

During the Middle Ages, Islamic mathematics helped trigonometry become a separate area of study. Great thinkers like al-Khwarizmi and Abu al-Wafa expanded on this knowledge. Eventually, trigonometry reached Europe through translations of Arabic and Greek texts during the Renaissance.

In more recent times, especially during the Age of Enlightenment, trigonometry became even more important. Famous mathematicians such as Isaac Newton, James Stirling, and Leonhard Euler helped shape the modern form of trigonometry that we use in schools and science today.

Etymology

The word "trigonometry" comes from ancient Greek words for "triangle" and "measure." The terms "sine" and "cosine" began with a Latin word that came from an Arabic idea, thanks to a mistranslation.

Other important words like "tangent" and "secant" also have Latin roots. "Tangent" means "touching," while "secant" means "cutting." The prefix "co-" in words like "cosine" was first used by a mathematician named Edmund Gunter in the year 1620 to describe angles that complete a right angle. The words "minute" and "second" for small parts of an angle also come from Latin.

Main article: Sine and cosine § Etymology Main article: Pythagorean identities

Ancient

The ancient Egyptians and Babylonians studied triangles for many years. They looked at the sides of triangles but did not yet understand angles.

Ancient Greek and Hellenistic mathematicians used something called a chord. This is a line that connects two points on a circle. They used chords to understand angles and circles better. Later, Claudius Ptolemy made tables to show how chords change with different angles. These tables helped astronomers predict where stars and planets would be.

In India, mathematicians like Aryabhata made big steps in trigonometry. They were the first to define the sine and cosine functions, which help us understand angles in circles. They made tables to show the values of sine for different angles.

Medieval

Previous works from India and Greece were later translated and expanded in the medieval Islamic world by Muslim mathematicians of mostly Persian and Arab descent. These mathematicians created many new theorems that made trigonometry easier to use without relying on old methods.

Important advances came from mathematicians like Muhammad ibn Mūsā al-Khwārizmī, who made accurate tables for sine and cosine values. Later, Abū al-Wafā' al-Būzjānī used all six main trigonometric functions and discovered important formulas that helped simplify calculations. These ideas made trigonometry more practical for tasks like navigation and astronomy.

Modern

The book Trigonometria (1595) by Bartholomaeus Pitiscus was the first to use the term “trigonometry.” He discovered important relationships between angles and sides of triangles.

Later, Leonhard Euler helped make trigonometry easier to understand by creating new ways to express these relationships using special numbers. His work set the foundation for how we study trigonometry today.