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EASIER THAN YOU THINK...

Trigonometry

White Sphere Labelled with the Word Trigonometry and All Kinds of Trigonometric Symbols, Graphs and EquationsHow 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.

The Right-Angled Triangle

Trigonometry and Non-Right-Angled Triangles

The Unit Circle and Trigonometric Identities

On Solving Trigonometric Equations

How to Sketch Trigonometric Functions

Brilliant!!! This helps so much!!! So much easier than anything else I have found. This is now simple.
Thanks again for showing the simple way of doing this. I look forward to the next video!! Absolutely brilliant.
Once again, excellent! I’ve watched many others, that have not helped nearly as much as these. Thanks for making it so easy to understand. When things get complicated, it is easy to make a mistake, but your method of writing down the structure helps to prevent mistakes.
Thanks a million!!! I’ve watched many videos and read tutorials, but still could not get two in a row correct on Khan Academy. Your explanation was so clear and simple that it finally made sense to me and now I can get all of the problems correct!!! Thank you very much.
It all makes total sense now! Thanks very much for all the derivative videos!!! Very helpful.
MemorizeAndLearn (on four different CCM YouTube videos about Differentiation)

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