Question

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. $\theta = m\pi + ( - 1)^{m}\frac{2\pi}{3}$
• D. $\theta = m\pi + ( - 1)^{m}\frac{\pi}{3}$

Answer 1

Answer: (A)

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

Answer 2

$\Rightarrow \cos\theta + \cos 2\theta + \cos 3\theta = 0 \Rightarrow (\cos\theta + \cos 3\theta) + \cos 2\theta = 0 \Rightarrow 2\cos 2\theta.\cos\theta + \cos 2\theta = 0 \Rightarrow \cos 2\theta(2\cos\theta + 1) = 0$

$\Rightarrow \cos 2\theta = 0 = \cos\frac{\pi}{2}$⇒ $2\theta = 2m\pi \pm \pi/2$ $\Rightarrow \theta = m\pi \pm \frac{\pi}{4}$ or

$\cos\theta = - \frac{1}{2} = \cos\frac{2\pi}{3}$ $\Rightarrow \theta = 2m\pi \pm \frac{2\pi}{3}.$