Question

Find the general solution for the trigonometric equation $\cos 2x+\cos x=0.$

• A. $x\in \{2n\pi \pm \frac{\pi }{3}\}\cup \{2n\pi \pm \frac{\pi }{2}\},n\in Z$
• B. $x\in \{2n\pi \pm \frac{\pi }{3}\}\cup \{2n\pi \pm \pi \},n\in Z$
• C. $x\in \{2n\pi +\frac{\pi }{3}\}\cup \{2n\pi +\pi \},n\in Z$
• D. None of the above.

Answer 1

The correct option is B $x\in \{2n\pi \pm \frac{\pi }{3}\}\cup \{2n\pi \pm \pi \},n\in Z$

Given: $\cos 2x+\cos x=0$

We know $\cos 2x=2{\cos }^{2}x-1$
Substituting this in the above equation we get,
$2{\cos }^{2}x-1+\cos x=0$
$\Rightarrow 2{\cos }^{2}x+\cos x-1=0$

Let $\cos x=t$
$\Rightarrow 2t^2+t-1=0$
$\Rightarrow 2t^2+(2-1)t-1=0$
$\Rightarrow 2t^2+2t-1t-1=0$
$\Rightarrow 2t(t+1)-1(t+1)=0$
$\Rightarrow (2t-1)(t+1)=0$

Here, either $2t-1=0$ or $t+1=0.$
$\Rightarrow t=\frac{1}{2}$ or $t=-1$
$\Rightarrow \cos x=\frac{1}{2}$ or $\cos x=-1$

For $\cos x=\frac{1}{2},$ i.e., $\cos x=\cos \frac{\pi }{3}$
General solution is $x=2n\pi \pm \frac{\pi }{3},n\in Z$
Represent the above solution set as $A.$

For $\cos x=-1,$ i.e., $\cos x=\cos \pi$
General solution is $x=2n\pi \pm \pi ,n\in Z$
Represent the above solution set as $B.$

The complete general solution is $A\cup B,$ i.e., $x\in \{2n\pi \pm \frac{\pi }{3}\}\cup \{2n\pi \pm \pi \},n\in Z$