Find exactly all solutions to the equation \( 2\cos ^2 \theta = 3 \sin \theta +3\) , \( 0 \leq \theta \leq 2 \pi \)

5th Ed: #54

First, we can use the Pythagorean Identity to get an equation of \( \sin \theta \)
\( 2 \left ( 1 - \sin ^2 \theta \right ) = 3 \sin \theta +3 \) , \( 0 \leq \theta \leq 2 \pi \)
\( 0 = 2 \sin ^2 \theta + 3\sin \theta + 1 \), \( 0 \leq \theta \leq 2 \pi \)
\( 0 = (2\sin \theta +1 )(\sin \theta +1 \), \( 0 \leq \theta \leq 2 \pi \)
\( \sin \theta = -\frac{1}{2} \) , \( \sin \theta = -1 \), \( 0 \leq \theta \leq 2 \pi \)
\( \theta = \frac{7\pi}{6},\frac{11\pi}{6},\frac{3\pi}{2} \)