Question

If $\cos x+\cos 2x+\cos 3x=0$, then the value of x will be

• A. $2n\pi \pm \frac{\pi }{3}$
• B. $2n\pi \pm \frac{2}{3}\pi$
• C. $n\pi \pm \frac{\pi }{3}$
• D. $n\pi +(-1{)}^n\frac{\pi }{3}$

Answer 1

The correct option is B $2n\pi \pm \frac{2}{3}\pi$

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

or, $(\cos x+\cos 3x)+\cos 2x=0$

or, $2\cos 2x\cos x+\cos 2x=0$

or, $\cos 2x(2\cos x+1)=0$. If $\cos 2x=0,$, then $2x=(2n+1)\frac{1}{2}\pi$

then $2x=(2n+1)\frac{1}{2}\pi$

or, $x=\frac{1}{4}(2n+1)\pi$ and if $2\cos x+1=0$

then, $\cos x=-\frac{1}{2}=\cos \left(\frac{2}{3}\pi \right)$ then $x=2n\pi \pm \frac{2}{3}\pi$