A) \[\frac{\pi }{3}\]
B) \[\frac{\pi }{3},{{\cos }^{-1}}\left( \frac{3}{5} \right)\]
C) \[{{\cos }^{-1}}\left( \frac{3}{5} \right)\]
D) \[\frac{\pi }{3},\pi -{{\cos }^{-1}}\left( \frac{3}{5} \right)\]
Correct Answer: D
Solution :
Given, \[5\cos 2\theta +2{{\cos }^{2}}\frac{\theta }{2}+1=0,-\pi <\theta <\pi \] \[\Rightarrow \] \[5(2{{\cos }^{2}}\theta -1)+(1+\cos \theta )+1=0\] \[\Rightarrow \] \[10{{\cos }^{2}}\theta -5+1+\cos \theta +1=0\] \[\Rightarrow \] \[10{{\cos }^{2}}\theta +\cos \theta -3=0\] \[\Rightarrow \] \[10{{\cos }^{2}}\theta +6\cos \theta -5\cos \theta -3=0\] \[\Rightarrow \] \[2\cos \theta -1=0\]or\[5\cos \theta +3=0\] \[\Rightarrow \] \[\cos \theta =\frac{1}{2}\]or\[\cos \theta =-\frac{3}{5}\] \[\Rightarrow \] \[\theta =\frac{\pi }{3}\]or \[\theta ={{\cos }^{-1}}\left( \frac{-3}{5} \right)\] \[\Rightarrow \] \[\theta =\frac{\pi }{3}\] or \[\pi -{{\cos }^{-1}}\left( \frac{3}{5} \right)\]
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