Question

Let $S=\{\theta \in [-2\pi ,2\pi ]:2{\cos }^2\theta +3\sin \theta =0\}$. Then the sum of the elements of S is?

• A. $\frac{13\pi }{6}$
• B. $\pi$
• C. $2\pi$
• D. $\frac{5\pi }{3}$

Answer 1

The correct option is C $2\pi$

$2(1-{\sin }^2\theta )+3\sin \theta =0$
$\Rightarrow 2{\sin }^2\theta -3\sin \theta -2=0$
$\Rightarrow (2\sin \theta +1)(\sin \theta -2)=0$
$\Rightarrow \sin \theta =-\frac{1}{2};\sin \theta =2$ (reject)
Roots: $\pi +\frac{\pi }{6},2\pi -\frac{\pi }{6},-\frac{\pi }{6},-\pi +\frac{\pi }{6}$
$\Rightarrow$ sum of values $=2\pi$.