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Trigonometrical Equations
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સમીકરણ $cos^2\theta\, +\, sin\theta\, + 1\, =\, 0$ ના ઉકેલો ............ અંતરાલ આવેલ છે
A
$\left( { - \frac{\pi }{3}\,,\,\frac{\pi }{4}} \right)$
B
$\left( {\frac{\pi }{4}\,,\,\frac{3\pi }{4}} \right)$
C
$\left( {\frac{3\pi }{4}\,,\,\frac{5\pi }{4}} \right)$
D
$\left( {\frac{5\pi }{4}\,,\,\frac{7\pi }{4}} \right)$
Solution
$\cos ^{2} \theta+\sin \theta+1=0$
$\Rightarrow \quad 1-\sin ^{2} \theta+\sin \theta+1=0$
$\Rightarrow \quad \sin ^{2} \theta-\sin \theta-2=0$
$\Rightarrow \quad(\sin \theta+1)(\sin \theta-2)=0$
$\Rightarrow \quad \sin \theta+1=0$
$[\because \sin \theta-2=0 \text { is not possible }]$
$\Rightarrow \quad \sin \theta=-1$
$\Rightarrow \quad \theta=\frac{3 \pi}{2} \in\left(\frac{5 \pi}{4}, \frac{7 \pi}{4}\right)$
Standard 11
Mathematics
