Find all solutions to the equation in the interval [0, 2Ο€). (3 points) cos 4x - cos 2x = 0 0, two pi divided by three. , four pi divided by three. pi divided by six , pi divided by two , five pi divided by six , seven pi divided by six , three pi divided by two , eleven pi divided by six 0, pi divided by three. , two pi divided by three. , Ο€, four pi divided by three. , five pi divided by three. No solution

Accepted Solution

Answer:[tex]x=0,x=\pi,x=\frac{\pi}{3},x=\frac{2\pi}{3},x=\frac{4\pi}{3},x=\frac{5\pi}{3}[/tex]Step-by-step explanation:This is a trigonometric equation where we need to use the cosine of the double-angle formula [tex]cos4x=2cos^22x-1[/tex]Replacing in the equation [tex]cos4x - cos2x = 0[/tex]We have [tex]2cos^22x-1 - cos 2x = 0[/tex]Rearranging [tex]2cos^22x - cos 2x-1 = 0[/tex]A second-degree equation in cos2x. The solutions are: [tex]cos2x=1,cos2x=-\frac{1}{2}[/tex]For the first solution cos2x=1 we find two solutions (so x belongs to [0,2\pi)) [tex]2x=0, 2x=2\pi[/tex]Which give us [tex]x=0,x=\pi[/tex]For the second solution [tex]cos2x=-\frac{1}{2}[/tex]We find four more solutions [tex]2x=\frac{2\pi}{3},2x=\frac{4\pi}{3},2x=\frac{8\pi}{3},2x=\frac{10\pi}{3}[/tex]Which give us [tex]x=\frac{\pi}{3},x=\frac{2\pi}{3},x=\frac{4\pi}{3},x=\frac{5\pi}{3}[/tex]All the solutions lie in the interval [tex][0,2\pi)[/tex]Summarizing. The six solutions are [tex]x=0,x=\pi,x=\frac{\pi}{3},x=\frac{2\pi}{3},x=\frac{4\pi}{3},x=\frac{5\pi}{3}[/tex]