Question

Find the general solution of $2co{s}^{2}x+3sinx=0$.

Answer 1

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

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

$\Rightarrow 2\left({\sin }^{2}x\right)-3\sin x-2=0$

$\Rightarrow 2\left({\sin }^{2}x\right)-4\sin x+\sin x-2=0$

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

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

$\Rightarrow \sin x=2,\sin x=-\frac{1}{2}$

(Not possible)

$\sin x=-\frac{1}{2}=-\sin \frac{\pi }{6}=\sin (\pi +\frac{\pi }{6})$

$=\sin \frac{7\pi }{6}$

$\Rightarrow \sin x=\sin \frac{7\pi }{6}$

$\Rightarrow x=\{n\pi +{(-1)}^n\frac{7\pi }{6}\}wheren\in I$

Hence, the general solution is:

$x=\{n\pi +{(-1)}^n\frac{7\pi }{6}\},n\in I$