1. ## Find real solutions

Find real solutions of the system

sin x + 2 sin(x+y+z) = 0,
sin y + 3 sin(x+y+z) = 0,
sin z + 4 sin(x+y+z) = 0.

2. We have $\displaystyle \sin(x+y+z)=-\frac{\sin x}{2}=-\frac{\sin y}{3}=-\frac{\sin z}{4}$.
Adding the first two equations and substracting the third, we have:
$\sin x+\sin y-\sin z+\sin(x+y+z)=0$
$\displaystyle 2\sin\frac{x+y}{2}\cos\frac{x-y}{2}+2\sin\frac{x+y}{2}\cos\frac{x+y+2z}{2}=0$
$\displaystyle 2\sin\frac{x+y}{2}\left(\cos\frac{x-y}{2}+\cos\frac{x+y+2z}{2}\right)=0$
$\displaystyle4\sin\frac{x+y}{2}\cos\frac{x+z}{2}\c os\frac{y+z}{2}=0$ (1)

If $\displaystyle\sin\frac{x+y}{2}=0\Rightarrow x+y=2k\pi\Rightarrow y=2k\pi-x$
Then $\displaystyle\frac{\sin x}{2}=-\frac{\sin x}{3}\Rightarrow\sin x=0\Rightarrow x=l\pi\Rightarrow y=m\pi\Rightarrow z=n\pi$

In the same way if one of the others two factors from (1) is 0.