Question 11
For real numbers $x$, let
$$
P(x) = 1 + \cos(x) + i \sin(x) - \cos(2x) - i \sin(2x) + \cos(3x) + i \sin(3x),
$$
where $i = \sqrt{-1}$. For how many values of $x$ with $0 \le x \lt 2\pi$ does $P(x) = 0$?
$$
P(x) = 1 + \cos(x) + i \sin(x) - \cos(2x) - i \sin(2x) + \cos(3x) + i \sin(3x),
$$
where $i = \sqrt{-1}$. For how many values of $x$ with $0 \le x \lt 2\pi$ does $P(x) = 0$?