Question

If $\left|\begin{array}{ccc}\sin 3\theta & -1& 1\\ \cos 2\theta & 4& 3\\ 2& 7& 7\end{array}\right|=0$, then find the number of values of $\theta$ in $[0,2\pi ]$.

Answer 1

$\left|\begin{array}{ccc}\sin 3\theta & -1& 1\\ \cos 2\theta & 4& 3\\ 2& 7& 7\end{array}\right|C_2\to C_2+C_3$

$\Rightarrow \left|\begin{array}{ccc}\sin 3\theta & 0& 1\\ \cos 2\theta & 7& 3\\ 2& 14& 7\end{array}\right|C_2\to C_2/7$

$7\left|\begin{array}{ccc}\sin 3\theta & 0& 1\\ \cos 2\theta & 1& 3\\ 2& 2& 7\end{array}\right|$

$\Rightarrow 7\left(\sin 3\theta \right(7-6\left)\right)+1(2\cos 2\theta -2)$

$\Rightarrow 7\left(\sin 3\theta \right(1\left)\right)+2(\cos 2\theta -1)2\left(2\right({\cos }^2\theta -1\left)\right)$

$7(3\sin \theta -7{\sin }^3\theta )+4(-{\sin }^2\theta )$

$\Rightarrow -28{\sin }^3\theta -4{\sin }^2\theta +21\sin \theta$

$=-\sin \theta [28{\sin }^2\theta +4\sin \theta -x]=0$

$\sin \theta =0$

or $28{\sin }^2\theta +4\sin \theta -21=0$

$\sin \theta =\frac{2\sqrt{37}\pm 1}{14}$

or $\theta =0$

and $\theta ={\sin }^{-1}\left(\frac{2\sqrt{37}\pm 1}{14}\right)$