Homogeneous second-degree equation

Statement

Solve for $0^\circ\le\theta\le 360^\circ$ the homogeneous equation: $3\sin^2\theta-5\sin\theta\cos\theta+2\cos^2\theta=0.$

Solution

Since $\cos\theta=0$ is not a solution (it would give $3=0$), divide by $\cos^2\theta$: $3\tan^2\theta-5\tan\theta+2=0\Rightarrow \tan\theta=\dfrac{5\pm\sqrt{25-24}}{6}=\dfrac{5\pm 1}{6},$ i.e. $\tan\theta=1$ or $\tan\theta=\dfrac{2}{3}$.

$\tan\theta=1\Rightarrow \theta=45^\circ,\ 225^\circ$.

$\tan\theta=\dfrac23\Rightarrow \theta\approx 33{.}69^\circ,\ 213{.}69^\circ$.

$\boxed{\theta\approx 33{.}69^\circ,\ 45^\circ,\ 213{.}69^\circ,\ 225^\circ}$