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).
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.
Our daughter Angelina was home schooled and from the start did not like maths. Over the years I have battled through, trying 5 different programs along the way. Angelina was surviving but not enjoying the journey but, when it came to algebra, the future looked dim. A friend recommended Graeme to me as her three sons had been tutored by him. She could not recommend him more highly. My daughter has just finished year 12 maths and did very well, thanks to Graeme (that would be an understatement). Graeme has a love for his subject and a genuine interest in his students. Graeme seems to meets his students where they are and tailors the lessons to meet their individual needs and interests. Graeme not only explains concepts clearly, but we all found him an interesting, knowledgeable and humble man. We are extremely happy that Graeme was recommended to us. Now our daughter is looking forward to uni with a grateful heart. We also could not recommend Graeme more highly.
Angela K (parent, 2013)
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