Question

If $\cos\theta + \cos 7\theta + \cos 3\theta + \cos 5\theta = 0,$ then $\theta =$

• A. $\frac{n\pi}{4}$
• B. $\frac{n\pi}{2}$
• C. $\frac{n\pi}{8}$
• D. None of these

Answer 1

Answer: (C)

$\frac{n\pi}{8}$

Answer 2

Combining $\theta$ and $7\theta,3\theta\text{and }5\theta,$ we get

$\mathbf{2}\mathbf{\cos}\mathbf{4}\mathbf{\theta(}\mathbf{\cos}\mathbf{3}\mathbf{\theta +}\mathbf{\cos}\mathbf{\theta}\mathbf{) = 0}$

$\therefore$ $4\cos 4\theta.\cos 2\theta.\cos\theta = 0 \Rightarrow$ $4\frac{1}{2^{3}\sin\theta}(\sin 2^{3}\theta) = 0$

$\Rightarrow$ $\sin 8\theta = 0.$ Hence $\mathbf{\theta =}\frac{\mathbf{n\pi}}{\mathbf{8}}$