A) \[\left( \frac{-\pi }{4},\frac{\pi }{4} \right)\]
B) \[\left( \frac{\pi }{4},\frac{3\pi }{4} \right)\]
C) \[\left( \frac{3\pi }{4},\frac{5\pi }{4} \right)\]
D) \[\left( \frac{5\pi }{4},\frac{7\pi }{4} \right)\]
Correct Answer: D
Solution :
We have, \[{{\cos }^{2}}\theta +\sin \theta +1=0\] \[\Rightarrow \]\[1-{{\sin }^{2}}\theta +\sin \theta +1=0\] \[\Rightarrow \]\[{{\sin }^{2}}\theta -\sin \theta -2=0\] \[\Rightarrow \]\[(sin\theta +1)(sin\theta -2)=0\] \[\Rightarrow \]\[\sin \theta +1=0\] \[\Rightarrow \]\[\sin \theta =-1\] \[[\because \,\sin \theta \ne 2]\] \[\Rightarrow \]\[\theta =\frac{3\pi }{2}\in \left( \frac{5\pi }{4},\frac{7\pi }{4} \right)\]
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