Solve the following determinant equations: 3&-2& (3 ) -7&8& (2 ) … - Vidyadip

Question

Solve the following determinant equations:
$\begin{vmatrix}3&-2&\sin(3\theta)\\-7&8&\cos(2\theta)\\-11&14&2\end{vmatrix}=0$

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Answer

$\begin{vmatrix}3&-2&\sin3\theta\\-7&8&\cos2\theta\\-11&14&2\end{vmatrix}=0$
$\Rightarrow3(16-14\cos2\theta)+2(-14+11\cos2\theta)\\+\sin3\theta(-98+88)=0$
$\Rightarrow20(1-\cos2\theta)+10\sin3\theta=0$
$\Rightarrow20(2\sin^2\theta)+10(3\sin\theta-4\sin^3\theta)=0$
$\Rightarrow4\sin^2\theta+3\sin\theta-4\sin^3\theta=0$
$\Rightarrow4\sin^2\theta+3-4\sin^2\theta=0$
$\Rightarrow4\sin^2\theta-4\sin\theta-3=0$
$\Rightarrow(2\sin\theta+1)(2\sin\theta-3)=0$
$\Rightarrow\sin\theta=-\frac{1}{2}$ or $\sin\theta=\frac{3}{2}=1.5$
As $\sin\theta\in[-1,1]$
$\therefore\sin\theta=-\frac{1}{2}$
$\Rightarrow\theta=\text{n}\pi+(-1)^{\text{n}}\frac{\pi}{6},\text{n }\in\text{ z}$

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