Those of you who hate math: sorry if I'm alienating you.
I was pleasantly amazed to find that somebody had designed a new model of trigonometry, one that is more fundamental, and in many ways simpler, than traditional trigonometry. This "new trig", called Rational Trigonometry, involves no angles, no sin(), cos() or tan(), and shuns circles. Instead, calculations are done with ordinary algebra and involve "spreads" instead of "angles" and, rather than distance, "quadrance" (the square of distance) is emphasized. With R. Trig, many answers can be found by hand that require a calculator in traditional trig. I quite like this new trig, and I'm inclined to think it should be taught in high schools instead of traditional trig. The first chapter of a textbook about it is available on the web.
It's astounding, given how far math has come, that something so fundamental has taken this long to be developed. Basically, it seems like it just never occurred to anyone before. I certainly would never have thought of it, and I got 95% in grade 12 math. I guess it's a mindset you get into, when you believe it's a "solved problem": when you believe that, there's no chance you'll go looking for any other solution.
Incidentally, this form of trigonometry looks like it would be very useful for computer algorithms, particularly on portable hardware. Fast cosines (and other trig functions) normally require a floating-point unit, which PDAs, cell phones and portable game consoles often do not have. Rational Trigonometry replaces those trig functions with multiplication, division and the occasional square root, which less powerful hardware can handle much more easily.