The trigonometric functions can be thought of as distances on a triangle inside/outside of the unit circle. One imagines that this is closest to the way ancient Greek mathematicians thought of the trigonometric functions. I saw an interesting youtube video wherein Professor Zvezdelina Stankova described the circle inversion proof of Ptolemy's theorem, and I thought it would be useful to expand a little on how the use of polar coordinates and a spatial view of trigonometric functions makes this proof and circle inversion in general more clear. See the following pdf paper describing these ideas. Also, go to the math fun page and look at the Excel sheet files below the button that links to this paper ("Inversion through a circle"; coffee cup button, the Excel sheets are the four smaller cyan buttons with inverted teardrops below the coffee cup link) for a better visualization.
Comments