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 son Joshua is currently in year 11 preparing for the HSC. Josh was doing reasonably well in mathematics and yet we felt, and his teachers agreed, that if he really applied himself he could be achieving much better results. We asked around for a maths tutor and Graeme Henderson came highly recommended. Joshua began to see Mr Henderson one hour per week and from the very first week Joshua’s motivation escalated. His maths improved to the point where he is now often achieving 100% in his exams, he applies himself to do hours of homework, something he had rarely done in the past, but most of all Josh has found a love of mathematics. Maths is no longer a chore. I believe this all stems from Mr Henderson’s gifted ability not only as a teacher, also as an encourager and motivator. I get motivated just listening to them! Josh told us this week that he wishes he could be tutored every day after school. When the results come through from Joshua’s exams, he is delighted and this has inspired him to become motivated to spend much more time applying himself to his other subjects. Josh is working hard to gain a great result in his HSC. I can personally recommend Mr Henderson as a truly wonderful tutor.
Deborah B (Parent, 2012)
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