Question

Consider the system of linear equations in $x,y,z$

$\left(\sin 3\theta \right)x-y+z=0$
$\left(\cos 2\theta \right)x+4y+3z=0$
$2x+7y+7z=0$
Find the value of $\theta$ for which this system has non-trivial solutions.

Answer 1

The system will have a non-trivial solution if

$\left[\begin{array}{ccc}\sin 3\theta & -1& 1\\ \cos 2\theta & 4& 3\\ 2& 7& 7\end{array}\right]$

or $(28-21)sin3\theta -(-7-7)cos2\theta +2(-3-4)=0$

or $7sin3\theta +14cos2\theta -14=0$

or $sin3\theta +2cos2\theta -2=0$

$\therefore sin\theta =0$ which gives $\theta =n\pi$

or $sin\theta =\frac{1}{2}$ which gives $\theta =n\pi +(-1{)}^n,\frac{\pi }{6}$,
where n is an integer.