A) \[\theta =2\,m\pi \pm 2\pi /3\]
B) \[\theta =2m\pi +\pi /4\]
C) \[\theta =m\pi +{{(-1)}^{n}}\,2\pi /3\]
D) \[\theta =m\pi +{{(-1)}^{n}}\,\pi /3\]
Correct Answer: A
Solution :
Given \[\cos \theta + cos 2\theta + cos3\theta = 0\] \[\Rightarrow \,\,\,\,\left( cos 3\theta + cos \theta \right) + cos 2\theta =\theta \] \[\Rightarrow \,\,\,2 cos 2\theta . cos \theta + cos 2\theta =\theta \] \[\Rightarrow \,\,\,\cos 2\,\theta \left( 2cos\theta +1 \right)=\theta \] we have, \[\cos \theta = cos\,\alpha \Rightarrow \,\, \theta = 2n\pi \pm \alpha \] \[\therefore \,\,\,For general value of \theta , cos 2\,\theta = \theta \] \[\Rightarrow \,\,\,cos2\,\theta =cos\frac{\pi }{2}\,\,\,\,\Rightarrow \,\,\,2\,\theta =\,\,2m\pi \,\,\pm \,\,\frac{\pi }{2}\] \[\Rightarrow \,\,\,\theta =m\pi +\frac{\pi }{4}\,\,\,or\,\,2\cos \theta +1=0\] \[\Rightarrow \,\,\cos \theta =\frac{-1}{2}\,\,\Rightarrow \,\,\cos \,\theta =\cos \,\frac{2\pi }{3}\] So, \[\theta =2m\pi \pm \frac{2\pi }{3}\]
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