 History

# The Islamic Roots of Trigonometry

Trigonometry is the branch of mathematics that studies trigonometric functions and their applications to geometry. It is this task – “measuring triangles” or “solving triangles”, determining all the elements of a triangle from three types of data – that has formed the basis of practical applications of trigonometry since ancient times.

## Why Did Muslims Start the Study of Trigonometry?

The reason for the emergence of trigonometry was astronomy, which Muslims diligently studied, especially because of its significance for determining the exact time of Prayers and to determine the position of Qibla . Before the Muslims, Greek astronomers used to calculate the sides and angles of certain triangles with indicators of known sides and angles in order to understand the movement of the Sun, the Moon, and the five planets known at that time.

Muslim scientists have made a great contribution to the development of trigonometry, in particular spherical. Their interest in this field was determined by the problems of astronomy and geodesy, the main of which were:

• accurate determination of the time of day;
• calculation of the future location of the heavenly bodies, the moments of Sunrise and Sunset, eclipses of the Sun and Moon;
• finding the geographical coordinates of the current location;
• calculating the distance between cities with known geographical coordinates; determining the direction to Mecca (Qibla) from a given location.

The treatise of Ptolemy came to the scientists of Europe thanks to the Muslims. They briefly translated the original Greek full name “The Great Mathematical Construction on Astronomy in 13 books” as “Al-Majisti”, which means “The Greatest”. This title reflected the deep respect that was widespread in Muslim academic circles for this book.

### Nasir al-Din al-Tusi and His Contribution to Trigonometry

The Muslim astronomer Nasir al-Din al-Tusi explains in his work “Intersecting Figures” how people can use this chord table was to solve problems about right-angled triangles. Nasir al-Din al-Tusi made an important observation, which established the connection between triangles and the arcs of circles. Accordingly, the sides of a triangle can be considered as chords, contracting arcs opposite to the angles of the triangle.

However, this table had two disadvantages. First, it required considerable work with the table and intermediate steps to calculate all the variations that might appear when searching for unknown lengths or angles of a right triangle. In contrast, trigonometry uses six familiar functions: sine, cosine, tangent, and their derivatives secant, cosecant, and cotangent. They are properties of modern techniques first developed and systematized by Muslim mathematicians. The second disadvantage of the table is the frequent need to double the angles in order to count the arc length.

### Al-Battani and His Contribution to Trigonometry

In fact, a number of Muslim scholars laid the foundation for trigonometry before the 10th century. Therefore, it allowed Nasir al-Din al-Tusi to collect, organize, and supplement their developments. One of the most influential figures in trigonometry was al-Battani, born in Harran (Turkey). He is one of the greatest Muslim astronomers, who passed away in 929 AD in Samarra. The driving force behind his groundbreaking development was observing the movement of the planets.

Al-Battani explained his mathematical operations and encouraged others to “continue observations and research” in order to refine and expand his work. Like al-Battani, Ibn Yunus, and Ibn al-Haytham developed spherical trigonometry and used its laws to solve problems in astronomy. Al-Battani was the first to use the terms “sine” and “cosine”. He defined them as length rather than proportion, as we do today. The scientist referred to the “tangent” as the “elongated shadow”. It means the shadow of an imaginary horizontal rod mounted on a wall. Al-Biruni defined the trigonometric functions of tangent and cotangent, which were theoretically improvised on the knowledge of Ancient Civilizations.

However, it is worth noting that the Arabic word geb angle is translated as “pocket”. In Arabic it means sinus (in the context of anatomy), the meaning of which has passed into the Latin sinus.

### Summary

The invention of trigonometric functions and their application in mathematics was revolutionary. The technologies, sciences, and mathematics on which industrial societies depend are based on trigonometry. Nasir al-Din al-Tusi and his fellow Muslim mathematicians and scientists could not have imagined how their work would ever be applied. But their discoveries are an integral part of our modern society.