Question 45
(ROM 4) (a) Solve the equation
$$\sin^3x+\sin^3\left(\frac{2\pi}{3}+x\right)+\sin^3\left(\frac{4\pi}{3}+x\right)+\frac34\cos2x=0.$$
(b) Suppose the solutions are in the form of arcs $AB$ of the trigonometric circle (where $A$ is the beginning of arcs of the trigonometric circle), and $P$ is a regular $n$-gon inscribed in the circle with one vertex at $A$.
(1) Find the subset of arcs with the endpoint $B$ at a vertex of the regular dodecagon.
(2) Prove that the endpoint $B$ cannot be at a vertex of $P$ if $2,3\nmid n$ or $n$ is prime.
$$\sin^3x+\sin^3\left(\frac{2\pi}{3}+x\right)+\sin^3\left(\frac{4\pi}{3}+x\right)+\frac34\cos2x=0.$$
(b) Suppose the solutions are in the form of arcs $AB$ of the trigonometric circle (where $A$ is the beginning of arcs of the trigonometric circle), and $P$ is a regular $n$-gon inscribed in the circle with one vertex at $A$.
(1) Find the subset of arcs with the endpoint $B$ at a vertex of the regular dodecagon.
(2) Prove that the endpoint $B$ cannot be at a vertex of $P$ if $2,3\nmid n$ or $n$ is prime.