# Introduction

*Trigonometry* is a branch of mathematics concerned with angles and ratios. The word itself comes from the Greek words *trigon* meaning *triangle*, and *metron* meaning *a measure*. Literally translated, therefore, it means the *measurement of triangles*. More specifically it relates to the angles and sides of triangles, and the relationships between them. Trigonometry provides us with a means of performing precise calculations to find unknown values such as the size of an angle or the length of a side. Trigonometry is also vital to the study of cyclic phenomena such as waveforms and oscillations. Although the relationship between triangles and circles is perhaps not immediately obvious, its importance was recognised by the ancient Greeks and became the basis of their trigonometric calculations.

The evolution of trigonometry began long before the ancient Greeks, however. From the dawn of civilisation, mankind has had an abiding fascination with the heavens. Probably one of the earliest tools used in the study of astronomy was a rod called a *shadow stick*. The rod was placed upright in the ground, and the shadow cast by the rod could be used to find the position of the sun in the sky. A number of ancient civilisations, including those of *Babylon*, *Mesopotamia*, *Egypt*, *China* and *India*, used such devices to observe the movement of the sun across the sky, establish the *cardinal directions* (*north*, *south*, *east* and *west*), and mark the passage of time. Other simple devices were used by the Babylonians and the Egyptians to track the movement of the objects visible in the night sky, and to record their positions relative to one another.

The movement of the stars and planets across the sky could be used to measure the passage of time during the hours of darkness, and to determine the seasons as the position of the constellations changed throughout the year. It is thought that the concept of dividing a circle into three hundred and sixty degrees may have originated with the Babylonians, who discovered that the position of the sun with respect to the celestial sphere changes by approximately one *degree of arc* (to give it its full title) each day. The Babylonians used a *sexagesimal* number system (i.e. a number system based on the number sixty), and may have chosen the number three hundred and sixty because it was more convenient to work with than three hundred and sixty five.

In any event, the Babylonians recorded a vast amount of information relating to the movement of the stars and planets, the cycles of the sun and the moon, and the timing of lunar and solar eclipses. Although they mistakenly believed that the sun, the stars and the planets revolved around the earth, the Babylonians recognised that the events unfolding in the night sky throughout the year were cyclic. The data they collected would be invaluable to the astronomers and mathematicians of future civilisations.

Both the Babylonians and the Egyptians studied the relationships between the sides and angles of triangles. Indeed, the Egyptians used such knowledge in the construction of the pyramids, but the emergence of trigonometry as we know it today really began with the ancient Greeks, who discovered that the size of the angle subtended by an arc of a circle is a function of the length of the chord subtending the arc and the diameter of the circle. This discovery eventually led to the development of many theorems and trigonometric identities that have modern equivalents. The Greeks, of course, did not have calculators. In order to obtain values for the chord function of an angle they had to perform complex and time consuming calculations. The results were recorded in tables, the first of which is believed to have been compiled by the Greek astronomer and mathematician *Hipparchus* in the second century BCE.

Many of the writings of Greek astronomers and mathematicians found their way first to India, during the fourth century CE, and by the end of the first millennium CE to China, having been refined and expanded upon by Indian scholars. By that time, the extent and influence of the Islamic empire was at its height. Islamic astronomers and mathematicians translated many of the ancient writings of the Babylonians, Egyptians and Greeks into Arabic, and collaborated closely with their Chinese and Indian counterparts. Most significantly, they developed trigonometry into a branch of mathematics in its own right and established a body of knowledge that would prove essential for the advancement of the study of astronomy, the creation of accurate maps and charts, and the navigation of the world's oceans. By the end of the thirteenth century CE Islamic influence had begun to decline, but the work of Islamic scholars had already found its way to Europe, where it was translated into Latin.

By the eighteenth century trigonometry had essentially evolved into the form we know today. *Johannes Müller von Königsberg* (better known as *Regiomontanus*), a German mathematician, astronomer, and Catholic bishop, among other things, had written his famous work *De Triangulis Omnimodus* (*On Triangles of Every Kind*), which was published in 1533. It was the first book written by a European that dealt solely with trigonometry. The work was essentially a formalisation of everything that was known up to that point about plane and spherical trigonometry, and influenced the work of *Nicolaus Copernicus*. Copernicus was the German astronomer who revolutionised astronomy by proposing a heliocentric cosmology, in which the sun, and not the earth, was at the centre of the universe. The work of Copernicus, published over a three year period shortly before his death in 1543, included detailed coverage of the trigonometric theorems underpinning the study of astronomy.

The contribution of Swiss mathematician *Leonard Euler* in the eighteenth century was to show how trigonometric functions could be used to solve many of the integral and differential equations of *calculus*, a new branch of mathematics that emerged during the latter half of the seventeenth century, and which is variously attributed to both the English physicist, mathematician and astronomer *Isaac Newton*, and to the German mathematician and philosopher *Gottfried Wilhelm Leibniz*. Today, a working knowledge of trigonometry is vital to many fields of endeavor, including mechanical and electronic engineering, architectural design, surveying, computer game design - the list is virtually endless. And of course it is crucial to navigation and astronomy. In fact, trigonometry probably influences every sphere of our lives in one way or another.