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5Question: A philosopher of science ponders the rotational symmetry of a circular model representing fundamental particles, where the angle $ \theta $ satisfies $ \cos(3\theta) = \cos(2\theta) $ in the interval $ [0, 2\pi) $. Find the sum of all such angles $ \theta $.
Solution: We begin with the equation:
\cos(3\theta) = \cos(2\theta)
Using the identity $ \cos A = \cos B $ implies $ A = 2k\pi \pm B $ for some integer $ k $, we apply it here:
3\theta = 2k\pi \pm 2\theta
We consider two cases: