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જો $4{\sin ^2}\theta + 2(\sqrt 3 + 1)\cos \theta = 4 + \sqrt 3 $ તો $\theta $ નો વ્યાપક ઉકેલ મેળવો.
A
$2n\pi \pm \frac{\pi }{3}$
B
$2n\pi + \frac{\pi }{4}$
C
$n\pi \pm \frac{\pi }{3}$
D
$n\pi - \frac{\pi }{3}$
Solution
(a) $4 – 4{\cos ^2}\theta + 2\,(\sqrt 3 + 1)\cos \theta = 4 + \sqrt 3 $
$ \Rightarrow $ $4{\cos ^2}\theta – 2\,(\sqrt 3 + 1)\cos \theta + \sqrt 3 = 0$
$ \Rightarrow $ $\cos \theta = \frac{{2(\sqrt 3 + 1) \pm \sqrt {4{{(\sqrt 3 + 1)}^2} – 16\sqrt 3 } }}{8}$
$ \Rightarrow $ $\cos \theta = \frac{{\sqrt 3 }}{2}{\rm{ or}}\,\,{\rm{1/2}}$
$\Rightarrow \theta = 2n\pi \pm \frac{\pi }{6}$ or $2n\pi \pm \pi /3$.
Standard 11
Mathematics
