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