Let us consider $$f(z):=\sum\limits\_{j=1}^{j=n}a\_j\sin(\lambda\_jz) $$
where all $a\_j$ and $\lambda\_j$ (of course, $\lambda\_j$ are distinct) are real numbers and $n \in \mathbb{Z},\, n \ge 3$. The question is: are all the zeroes of $f(z)$ real?



Zeroes of linear combination of sines
Note that $\sinh$ is a strictly increasing positive unbounded function on $(0,\infty)$ so for say $n \ge 3$, the equation $\sinh 1+ \sinh 2+...\sinh (n-1) =\sinh x$ has a unique positive solution $x\_n> n-1$. 


Using that $\sin {iy}=i\sinh y$ for real $y$, the above means that $i$ is a root of the equation $\sin z+\sin 2z +..\sin {(n-1)z}-\sin x\_nz=0$


For $n=2$ we can adapt this and use that $2\sinh 1=\sinh x\_2$ for a unique $x\_2>1$ and get that $i$ is a root of $2\sin z-\sin x\_2z$


