Solving $2\sin\left(2x\right)=3\left(1-\cos x\right)$ Background - this was part of a homework packet for students looking to skip HS pre-calc. There is a text book they use as well, but this particular problem was not in it.
$$2\sin\left(2x\right)=3\left(1-\cos\left(x\right)\right)$$
My first step was to eliminate the double angle.
$$4\sin\left(x\right)\cos\left(x\right)=3\left(1-\cos\left(x\right)\right)$$
and distribute on right side
$$4\sin\left(x\right)\cos\left(x\right)=3-3\cos\left(x\right)$$
And this is where I am stuck.
Now, I can see that $0$ and $2\pi$ are solutions, but with a graph of both sides, one more. The worksheet instructions do not say whether or not graphing is allowed, although the particular chapter in the book for this seems to rely heavily on calculator work.
My question

Can this be solved by manipulation, if so, how? If not, what is the hint to stop and go to the graph?



Rewrite $2\sin\left(2x\right)=3\left(1-\cos x\right)$ as
$$ 4\sin\frac x2\cos\frac x2\cos x= 3\sin^2\frac x2$$
Then, let $t= \tan\frac x2, \> \cos x = \frac{1-t^2}{1+t^2}$ and factorize
$$\sin\frac x2 \frac{4-3t-4t^2-3t^3}{1+t^2}=0
$$
The factor $\sin\frac x2=0 $ yields $\frac x2 =\pi n$ and $4-3t-4t^2-3t^3=0$ has one real root at
$$t= \frac19\left(-4+\sqrt[3]{584+9\sqrt{4227}}-\sqrt[3]{-584+9\sqrt{4227}}\right)
$$
Thus, the solutions are
$$x=2\pi n,\>\>\> 2\pi n + 2\arctan t$$
