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(i) Solve the equation:
\sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0.
(ii) Supposing the solutions are in the form of arcs AB with one end at the point A, the beginning of the arcs of the trigonometric circle, and P a regular polygon inscribed in the circle with one vertex in A, find:

1) The subsets of arcs having the other end in B in one of the vertices of the regular dodecagon.
2) Prove that no solution can have the end B in one of the vertices of polygon P whose number of sides is prime or having factors other than 2 or 3.

Slični zadaci

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