If $x\sin A+y\sin B+z\sin C=x^2\sin2A+y^2\sin2B+z^2\sin 2C=0$, show $x^3\sin3A+y^3\sin3B+z^3\sin 3C=0$ I'm having trouble with a question that came in one of my exam and is about complex numbers and trigonometry:

If $$x \sin A +y \sin B+z\sin C=0$$ and $$x^2 \sin 2A + y^2 \sin 2B + z^2 \sin 2C=0$$ then prove that $$x^3 \sin 3A+y^3 \sin 3B +z^3 \sin 3C=0$$ where $x$, $y$, $z$ belong to $\mathbb{R}$ and $A+B+C=\pi$.

What I tried: I tried maybe doing it as imaginary part of $xe^{iA} + ye^{iB} +ze^{iC} =0$ and similarly, but I'm reaching a dead end in that approach
 A: If $\sin{C}=0$, so $$x\sin{A}+y\sin(180^{\circ}k-A)=0,$$ where $k$ is an integer number.
Thus, easy to show (just consider two cases: $k$ is odd and $k$ is even) that $$x^3\sin3A+y^3\sin(540^{\circ}k-3A)=0.$$
Now, let $\prod\limits_{cyc}\sin{A}\neq0.$
Thus, the first condition gives $$z=-\frac{x\sin{A}+y\sin{B}}{\sin(A+B)}$$ and with the second we obtain:
$$(x^2\sin2A+y^2\sin2B)\sin(A+B)=2(x\sin{A}+y\sin{B})^2\cos(A+B)$$ or
$$(\sin{A}\cos{A}\sin(A+B)-\sin^2A\cos(A+B))x^2-2\sin{A}\sin{B}\cos(A+B)xy+$$
$$+(\sin{B}\cos{B}\sin(A+B)-\sin^2B\cos(A+B))=0$$ or
$$\sin{A}\sin{B}(x^2-2xy\cos(A+B)+y^2)=0$$ or
$$x^2-2xy\cos(A+B)+y^2=0,$$ which since $|\cos(A+B)|\neq1,$ gives $$x=y=0.$$
Can you end it now?
A: With $\alpha=xe^{iA}$, $\beta=ye^{iB}$ and $\gamma=ze^{iC}$ one has $\alpha\beta\gamma=-xyz$, which is a real number. Also, the first two equations of the problem imply that $\alpha+\beta+\gamma$ and $\alpha^2+\beta^2+\gamma^2$ are real numbers, so $\alpha\beta+\beta\gamma+\gamma\alpha= \frac 12((\alpha+\beta+\gamma)^2-(\alpha^2+\beta^2+\gamma^2))$ is real as well. Therefore, the polynomial $$(X-\alpha)(X-\beta)(X-\gamma)=X^3-(\alpha+\beta+\gamma)X^2+(\alpha\beta+\beta\gamma+\gamma\alpha)X-\alpha\beta\gamma$$ has real coefficients and either all its roots are real or there is one real root and the other two roots are each other's complex conjugate.
In both cases it follows for all positive $n$ that $\alpha^n+\beta^n+\gamma^n$ is real, which is equivalent with $x^n \sin nA+y^n \sin nB +z^n \sin nC=0$.
