<s>[INST] If $\sin x +\sin 2x + \sin 3x = \sin y\:$ and $\:\cos x + \cos 2x + \cos 3x =\cos y$, If  $\sin x +\sin 2x + \sin 3x = \sin y\:$ and $\:\cos x + \cos 2x + \cos 3x =\cos y$, then $x$ is equal to
(a) $y$
(b) $y/2$
(c) $2y$
(d) $y/6$
I expanded the first equation to reach $2\sin x(2+\cos x-2\sin x)= \sin y$, but I doubt it leads me anywhere. A little hint would be appreciated. Thanks!
 [/INST] If $z = e^{ix} = \cos x + i \sin x$
Then we have $z + z^2 + z^3 = e^{iy} = w$ (say)
Divide by $z^2$ to see that
$$z + \frac{1}{z} + 1 = \frac{w}{z^2}$$
The left side is real and thus
$$w = az^2$$
Since $|w| = |z| = 1$ we must have that $|a| = 1$
Thus $$w = \pm z^2$$
This gives rise to two equations:
$$z + z^3 = 0$$
and
$$z + 2z^2 + z^3 = 0$$
I will leave the rest to you. 
And as others said, be careful that $e^{ix}$ is periodic and for the question to make sense you might need to put bounds on $x,y$.
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