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I forgot about the laws of sine and cosine. They're easy to prove once you know them, though I don't know about discovering them from scratch.
I was reminded after the previous post that you can get another triangle with just a fair bit of algebra. sin 2*18 = cos 3*18, solve for sin 18; you'll get a quadratic, and a fairly simple solution (-1+sqrt(5))/2. (If you try sin 18 = cos 4*18, you get a quartic, and I don't know how to solve it analytically.)
I've been writing code to approximate log or powers via iterated square roots. It works, though takes a fair number of steps.
I was reminded after the previous post that you can get another triangle with just a fair bit of algebra. sin 2*18 = cos 3*18, solve for sin 18; you'll get a quadratic, and a fairly simple solution (-1+sqrt(5))/2. (If you try sin 18 = cos 4*18, you get a quartic, and I don't know how to solve it analytically.)
I've been writing code to approximate log or powers via iterated square roots. It works, though takes a fair number of steps.