Question

Which of the following is the general solution of the equation $2{\cos }^2\theta +3\sin \theta =0$ ?

Answer 1

The given equation is $2{\cos }^2\theta +3\sin \theta =0$
$\Rightarrow 2(1-{\sin }^2\theta )+3\sin \theta =0$
$\Rightarrow 2{\sin }^2\theta -3\sin \theta -2=0$
$\Rightarrow (\sin \theta -2)(2\sin \theta +1)=0$
$\Rightarrow 2\sin \theta +1=0$
$\Rightarrow \sin \theta =-\frac{1}{2}=\sin \left(\frac{-\pi }{6}\right)$
$\Rightarrow \theta =n\pi +(-1{)}^n(\frac{-\pi }{6});n\in Z$
$\Rightarrow \theta =n\pi +(-1{)}^{n+1}\frac{\pi }{6};n\in Z$