If $$\cos \theta + \cos 2\theta + \cos 3\theta = 0,$$      then the general value of $$\theta$$ is :

Solution :
$$\eqalign{ & {\text{Given, }}\cos \theta + \cos 2\theta + \cos 3\theta = 0 \cr & \Rightarrow \left( {\cos 3\theta + \cos \theta } \right) + \cos 2\theta = 0 \cr & \Rightarrow 2\cos 2\theta \cdot \cos \theta + \cos 2\theta = 0 \cr & \Rightarrow \cos 2\theta \cdot \left( {2\cos \theta + 1} \right) = 0 \cr & {\text{we have}},\cos \theta = \cos \alpha \cr & \Rightarrow \theta = 2n\pi \pm \alpha \cr & \therefore {\text{For general value of }}\,\theta ,\cos 2\theta = 0 \cr & \Rightarrow \cos 2\theta = \cos \frac{\pi }{2} \cr & \Rightarrow 2\theta = 2m\pi \pm \frac{\pi }{2} \cr & \Rightarrow \theta = m\pi \pm \frac{\pi }{4}{\text{ or }}\,2\cos \theta + 1 = 0; \cr & \Rightarrow \cos \theta = \frac{{ - 1}}{2} \cr & \Rightarrow \cos \theta = \cos \frac{{2\pi }}{3} \cr & {\text{So}},\theta = 2m\pi \pm \frac{{2\pi }}{3} \cr} $$