The set of equations 

$$\begin{gathered} \lambda x-y+\left(\cos \theta \right)z=0, \\ 3x+y+2z=0, \\ \left(\cos \theta \right)x+y+2z=0 \end{gathered}$$

$0\le \theta <2\pi$,has non-trivial solution(s):

Let  $D=\left|\begin{array}{ccc}\lambda & -1& \cos \theta \\ 3& 1& 2\\ \cos \theta & 1& 2\end{array}\right|$
= $\lambda (2-2)+1(6-2\cos \theta )+\cos \theta (3-\cos \theta )$
 $$\begin{gathered} =6-{\cos }^2\theta +\cos \theta \\ =6+\frac{1}{4}-({\cos }^2\theta -\cos \theta +\frac{1}{4}) \\ =\frac{25}{4}-{(\cos \theta -\frac{1}{2})}^2 \end{gathered}$$
 Minimum value of D is  $\frac{25}{4}-{(-1-\frac{1}{2})}^2=4>0$
i.e.  $D\ne 0$

Given system has only trivial solutions