Question

Solve $2{cos}^2\theta +3sin\theta =0$

Answer 1

$2{\cos }^2\theta +3\sin \theta =0$

$\Rightarrow 2(1-{\sin }^2\theta )+3\sin \theta =0$

$\Rightarrow 2-2{\sin }^2\theta +3\sin \theta =0$

$\Rightarrow 2{\sin }^2\theta -3\sin \theta -2=0$

$\Rightarrow 2{\sin }^2\theta -4\sin \theta +\sin \theta -2=0$

$\Rightarrow 2\sin \theta (\sin \theta -2)+(\sin \theta -2)=0$

$\Rightarrow (\sin \theta -2)(2\sin \theta +1)=0$

$\Rightarrow \sin \theta \ne 2$ since range of $\sin \theta \in [-1,1]$

$\therefore \sin \theta =\frac{-1}{2}$

$\therefore \theta =\pi +\frac{\pi }{6},2\pi -\frac{\pi }{6}$ in third and fourth quadrant.

$\therefore \theta =\frac{7\pi }{6},\frac{11\pi }{6}$