Question

If $\cos 7\theta =\cos \theta -\sin 4\theta$, then the general value of $\theta$ is

• A. $n\pi /4,n\pi /4+\pi /18$
• B. $n\pi /3,n\pi /3+{(-1)}^n\pi /18$
• C. $n\pi /4,n\pi /3+{(-1)}^n\pi /18$
• D. $n\pi /6,n\pi /3+{(-1)}^n\pi /18$

Answer 1

The correct option is C

$n\pi /4,n\pi /3+{(-1)}^n\pi /18$

Finding the general value of $\theta$:

Given,

$$\begin{gathered} \cos 7\theta =\cos \theta –\sin 4\theta \\ \sin 4\theta =\cos \theta –\cos 7\theta \end{gathered}$$

Using the formula

$\sin 4\theta =2\sin \frac{(\theta +7\theta )}{2}\sin \frac{(7\theta –\theta )}{2}$ $\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{x}\mathbf{}\mathbf{–}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{y}\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\frac{\mathbf{(}\mathbf{x}\mathbf{}\mathbf{+}\mathbf{}\mathbf{y}\mathbf{)}}{\mathbf{2}}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\frac{\mathbf{(}\mathbf{y}\mathbf{}\mathbf{–}\mathbf{}\mathbf{x}\mathbf{)}}{\mathbf{2}}$

$\Rightarrow$ $\sin 4\theta =2\sin 4\theta \sin 3\theta$

$\Rightarrow$$2\sin 4\theta \sin 3\theta –\sin 4\theta =0$

$\Rightarrow$ $\sin 4\theta (2\sin 3\theta –1)=0$

$\Rightarrow$ $\sin 4\theta =0,\sin 3\theta =1/2$

$\Rightarrow$ $4\theta =n\pi ,\sin 3\theta =\sin \pi /6$

$\Rightarrow$ $\theta =n\pi /4,3\theta =n\pi +{(-1)}^n\pi /6$

$\mathbf{\therefore }\mathit{\theta }\mathbf{}\mathbf{=}\mathbf{}\mathit{n}\mathit{\pi }\mathbf{/}\mathbf{4}\mathbf{,}\mathbf{}\mathit{n}\mathit{\pi }\mathbf{/}\mathbf{3}\mathbf{}\mathbf{+}\mathbf{}{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{n}}\mathbf{}\mathit{\pi }\mathbf{/}\mathbf{18}$

Hence, option ‘C’ is correct.