How can we navigate on a sphere? How can we navigate on and measure flat ground? How do we survey the land that we own, use, and live on? How do we find and study the stars and planets? How can we measure the height of mountains?

Mathematicians first started asking these questions during the days of the Sumerian and Babylonian Empires. Their astronomers dividied the sky into 360 sections (roughly one for each day of the year and the Earth orbited the Sun and revealed different parts of the night sky) and gave us our measure of degrees (a full revolution can be divided into 360 degrees).

Greek scholars during the 3rd Century BC, such as Euclid (Alexandria, Egypt) and Archimedes (Syracuse, Sicily), studied triangles in great depth geometrically.

During the 2nd Century AD, the Greek astronomer Ptolemy (Alexandria, Egypt) printed detailed trigonometric tables in his famous Almagest and this remained the basic text and reference for trigonometric tables in Europe for the next 1200 years! For those who are curious, I have found a complete copy of Ptolemy’s Almagest on the Internet. Unfortunately, it is in Greek, but on pages 134-141 and 174-187 you can view some tables that use Greek numeration (Greeks used letters of their alphabet to represent numbers) to provide details about the 12 regions of the zodiac. Pages 210-215 and 282-293 appear to contain trigonometric tables.

It is thanks to Indian mathematicians and astronomers such as Aryabhata (476–550) that the trigonometric ratios that we recognise were developed. By the 10th Century, Islamic scholars, who had combined their studies of Greek and Indian mathematics, were using all six trigonometric ratios that we recognise. They had constructed tables for them and were using them to solve problems in spherical geometry.

Although such material was increasingly known in Europe, it was not until 1464 that the German mathematician, Regiomontanus, published his *De Triangulis* (I can only find a full Portugese translation on the Internet) that presented the current state of trigonometry. Even so, by the time Nicolaus Copernicus wrote his *De revolutionibus orbium coelestium* in 1543, he had to devote a couple of chapters to explaining the basic concepts of trigonometry to his readers (I can only find an English translation of Book 1 for you).

After this period, the discipline of trigonometry developed very rapidly … driven largely by the need for accurate navigation created by voyages of discovery/mapping and the expansions of empires.

You will quickly learn that although trigonometry began as a study of ratios within triangles, the definitions were eventually broadened to distances based on the unit circle (and angles of any size) and, finally, the science of analysis applied the ratios to series, waves, complex numbers and calculus.

Trigonometry and Non-Right-Angled Triangles

The Unit Circle and Trigonometric Identities

On Solving Trigonometric Equations

How to Sketch Trigonometric Functions

Graeme was a fantastic tutor who substantially improved my marks in all areas. He’s a great person to learn from and has a good sense of humour, too.

Blake H (student, 2013-2015)

See all Testimonials