Find $\lambda$ and $\theta$ such that it validates this matricial equation Find $\lambda$ and $\theta$  such that it validates the matricial equation
$$ \left( \begin{array}{cc}
1 & 2 \\
2 & 3
\end{array} \right)
%
\left( \begin{array}{cc}
\cos \theta \\
\sin \theta 
\end{array} \right) = \lambda\left( \begin{array}{cc}
\cos \theta \\
\sin \theta 
\end{array} \right)
$$

What I have tried:
$$\left\{ \begin{array}{ll}
         \cos \theta + 2\sin \theta  &= \lambda \cos \theta\\
        2\cos \theta + 3\sin \theta  &=  \lambda \sin \theta.\end{array} \right.  $$
$$\left\{ \begin{array}{ll}
         \ -2\cos \theta -4 \sin \theta  &= -2\lambda \cos \theta\\
        2\cos \theta + 3\sin \theta  &=  \lambda \sin \theta.\end{array} \right.  $$
$$-\sin \theta = -2\lambda \cos \theta + \lambda \sin \theta$$
$$ (\lambda + 1)\sin \theta = 2 \lambda \cos \theta$$
I can´t find $\lambda$ and $\theta$

Here are the steps:
1- Find the eigenvalues of the coefficient matrix which are $2\pm \sqrt{5}$.
2- Find the associated eigenvectors with the values.
3- equalling the vectors with your sine and cosine vector, find the $\theta$.
