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Trigonometrical Equations
hard
Let $S\, = \,\left\{ {\theta \, \in \,[ - \,2\,\pi ,\,\,2\,\pi ]\, :\,2\,{{\cos }^2}\,\theta \, + \,3\,\sin \,\theta \, = \,0} \right\}$. Then the sum of the elements of $S$ is
A
$\frac{{13\,\pi }}{6}$
B
$2\pi $
C
$\pi $
D
$\frac{{5\,\pi }}{3}$
(JEE MAIN-2019)
Solution
$2\,(1 – {\sin ^2}\theta )\, + \,3\,\sin \,\theta \, = \,0$
$ \Rightarrow {\kern 1pt} 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 off values $=\,2\pi $
Standard 11
Mathematics
