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  4. If cos θ + cos 2θ +cos 3θ = 0, then the general value of θ is :

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

Trigonometric Functions

Solution:

Given $cos \,\theta + cos\, 2\theta +cos\, 3\theta = 0$
$\Rightarrow \quad\left(cos\,3\theta + cos \,\theta \right) + cos \,2\theta = 0$
$\Rightarrow \quad 2cos \,2\theta . cos\, \theta +cos\, 2\theta = 0$
$\Rightarrow \quad cos \,2\theta . \left(2cos\, \theta + 1\right)=0$
we have, $cos\, \theta = cos\, \alpha \Rightarrow \theta = 2n\pi \pm \alpha$
$\therefore \quad$ For general value of $ \theta, \,cos\, 2 \theta = 0$
$\Rightarrow \,cos \,2\theta = cos \frac{\pi}{2} \quad\Rightarrow \quad2\theta = 2m\pi \pm \frac{\pi }{2}$
$\Rightarrow \quad\theta=\, m\pi \pm \frac{\pi }{4}$ or $2 \,cos\, \theta + 1 = 0 ;$
$\Rightarrow \quad cos \,\theta = \frac{-1}{2} \,\Rightarrow \,cos \,\theta = cos \frac{2\pi }{3}$
So, $\theta = 2m\pm \frac{2\pi }{3}$