Trigonometrical Equations
medium

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

A

$\theta = 2m\pi \pm \frac{{2\pi }}{3}$

B

$\theta = 2m\pi \pm \frac{\pi }{4}$

C

$\theta = m\pi \pm {( - 1)^m}\frac{{2\pi }}{3}$

D

$(a)$ and $(b)$ both

Solution

$\cos \theta + \cos 2\theta + \cos 3\theta = 0$ 

==> $(\cos \theta + \cos 3\theta ) + \cos 2\theta = 0$

==> $2\cos 2\theta \cos \theta + \cos 2\theta = 0$

==> $\cos 2\theta (2\cos \theta + 1) = 0$ 

==> $\cos 2\theta = 0 = \cos \frac{\pi }{2}$

==> $\theta = \frac{\pi }{4}$

==> $\theta = 2m\pi \pm \frac{\pi }{4}$

or $\cos \theta = \frac{{ – 1}}{2} = \cos \frac{{2\pi }}{3}$ 

==> $\theta = 2m\pi \pm \frac{{2\pi }}{3}$.

Standard 11
Mathematics

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