I’ve run into three topics related to trigonometry lately.
First, as I’d mentioned a few days ago, I ran into the identity on Mathematics Diary that for the interior angles of a triangle, the sum of the tangents equals the product of the tangents. I mentioned a converse to this identity here.
Next, a discussion came up on StackOverflow regarding how trigonometric functions are calculated. I happened to know something about this because of a consulting job I did a few years ago where I helped design the transcendental function algorithms for a microchip. Calculus instructors speculate, or even assert, that computers use Taylor series to compute trig functions. Maybe some do, but it’s not the most efficient or most common way.
Finally, this morning Brent Yorgey at The Math Less Traveled discusses a neglected aspect of the law of sines. The way I learned this theorem, and apparently the way the Brent learned it as well, was
a/sin(a) = b/sin(b) = c/sin(c)
where a, b, and c are interior angles of a triangle. But there’s more. Not only do the three ratios equal each other, they also equal something interesting: the diameter of the circle that circumscribes the triangle.
See Brent Yorgey’s post for a proof.