But how can an angle be wrong? If it were so wrong why would we still call it an angle? I caught a glimpse of these stygian depths pondering how to prove that cos²x+sin²x =1.
If x is a real number, as any holy and pure angle ought to be, then we can factorise
cos²x +sin²x =(cosx+isinx)(cos x - isin x) = eix e-ix = 1
People seem to find this insufficiently macho and want to prove how manly they are by multiplying out the power series. I tried to warn against doing that, but lydianrain insisted on doing it regardless.
These manipulations are valid in any nicely normed field. So the result is proved for complex angles. For example cos 1/2+i/3 = 0.9268-0.1628i and sin 1/2+i/3 = 0.5063+0.298i, from which you can soon verify that cos² 1/2+i/3 + sin² 1/2+i/3 = 1.
Complex angles are obviously wrong. They are incomprehensible. What could turning through 1/2+i/3 radians possibly mean? On the other hand, one might just shrug and comment sarcastically "Mathematics is incomprehensible, who knew?". Perhaps this is merely ordinary incomprehensibility and not the excrescence of evil.
Notice though that proving the theorem by multiplying out power series in a single variable has darker implications, far worse than I have adumbrated. Any nicely normed power associative algebra will do and the Cayley-Dickson construction spawns an infinite tower of 2n dimensional hypercomplex numbers that serve as increasingly wrong angles growing exponentially more disturbing as they voyage further and further beyond human comprehension.
We could start with the quaterions. For example cos(1/2+2i/3-j/4+3k/5) = 1.286 - 0.368i + 0.138j - 0.331k and sin(1/2+2i/3-j/4+3k/5) = 0.703 + 0.673i -0.252j + 0.606k. You can check for yourself that the squares add up to one.
The octonians works just as well. With x = 0.1-0.25i+0.3j-0.7k+0.2l+0.5il+0.4jl+0.6kl we get cos x = 1.809+0.031i-0.038j+0.088k-0.025l-0.063il-0.050jl-0.075kl and also sin x = 0.182-0.313i+0.376j-0.878k+0.251l+0.627il+0.502jl+0.752kl leading to the sum of squares being one as before.
The darkness just keeps on deepening for the sedenions expand the horror into 16 dimensions. For example if x = 0.1+0.2e1+0.3e2+0.4e3 -0.25e4+0.5e5-0.7e6+0.6e7 -0.3e8+0.3e9-0.1e10-0.6e11+0.23e12 +0.85e13+0.12e14+0.11e15 it remains possible to compute the cosine, which turns about to be 2.748-0.031e1-0.046e2-0.061e3 +0.038e4-0.077e5+0.107e6-0.092e7 +0.046e8-0.046e9+0.015e10+0.092e11 -0.035e12-0.130e13-0.018e14-0.017e15. Torturing myself with another computation just as foul yields the sine which is 0.276+0.306e1+0.459e2+0.612e3 -0.382e4+0.765e5-1.071e6+0.918e7 -0.459e8+0.459e9-0.153e10-0.918e11 +0.352e12+1.300e13+0.184e14+0.168e15. Yet again the sum of squares is 1.
Is their no escape? No, it is an infinite tower, power associative at every step and offering a climb that ends only in insanity, fall, and death. Respect Lovecraft's insights and flee from angles that are wrong.
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