Question

What is the general value of $x$ if $\cos x+\cos 2x+\cos 3x=0$

• A. $2k\pi +\frac{2\pi }{3}$
• B. $2k\pi -\frac{2\pi }{3}$
• C. $k\pi +\frac{k}{4}$
• D. $k\pi -\frac{k}{4}$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $2k\pi +\frac{2\pi }{3}$
B $2k\pi -\frac{2\pi }{3}$
C $k\pi -\frac{k}{4}$
D $k\pi +\frac{k}{4}$

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

$\Rightarrow \cos x+2{\cos }^{2}x-1+4{\cos }^{3}x-3\cos x=0$

$\Rightarrow 4{\cos }^{3}x+2{\cos }^{2}x-2\cos x-1=0$

$\Rightarrow 2{\cos }^{2}x(2\cos x+1)-1(2\cos x+1)=0$

$\Rightarrow (2\cos x+1)(2{\cos }^{2}x-1)=0$

$\Rightarrow 2\cos x+1=0,2{\cos }^{2}x-1=0$

$\Rightarrow 2\cos x=-1,2{\cos }^{2}x=1$

$\Rightarrow 2\cos x=-1,{\cos }^{2}x=\frac{1}{2}$

$\Rightarrow x=2k\pi \pm \frac{2\pi }{3},x=k\pi \pm \frac{k}{4}$

$\therefore x=2k\pi \pm \frac{2\pi }{3},k\pi \pm \frac{k}{4}$ for $k\in Z$