Question

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

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

Answer 1

Answer: (D)

$\cos 7\theta =\cos \theta -\sin 4\theta$
$p\sin 4\theta =\cos \theta -\cos 7\theta$
$p\sin 4\theta =2\sin 4\theta \sin 3\theta$
$p\sin 4\theta (1-2\sin 3\theta )=0$
$\therefore \sin 4\theta =0$ or $\sin 3\theta =\frac{1}{2}$
$\Rightarrow 4\theta =n\pi$ or $3\theta =n\pi +(-1{)}^n\frac{\pi }{6}$
$\Rightarrow \theta =\frac{n\pi }{4}$ or $\frac{n\pi }{3}+(-1{)}^n\frac{\pi }{18}$