find all the solutions in there interval (0,2pi) for cos5x=-1/2

Answer:[tex]\frac{2\pi}{15},\frac{4\pi}{15},\frac{8\pi}{15},\frac{2\pi}{3},\frac{14\pi}{15}, \frac{16\pi}{15}, \frac{4\pi}{3},\frac{22\pi}{15}, \frac{26\pi}{15}, \frac{28\pi}{15}[/tex]Step-by-step explanation:Solving trigonometric equations. We are given a condition and we must find all angles who meet it in the provided interval. Our equation is [tex]cos5x=-\frac{1}{2}[/tex]Solving for 5x: [tex]5x=\frac{2\pi}{3}+2n\pi[/tex][tex]5x=\frac{4\pi}{3}+2n\pi[/tex]The values for x will be [tex]x=\frac{\frac{2\pi}{3}+2n\pi}{5}[/tex][tex]x=\frac{\frac{4\pi}{3}+2n\pi}{5}[/tex]To find all the solutions, we'll give n values of 0, 1, 2,... until x stops belonging to the interval [tex](0,2\pi)[/tex]For n=0 [tex]x=\frac{\frac{2\pi}{3}}{5}=\frac{2\pi}{15}[/tex][tex]x=\frac{\frac{4\pi}{3}}{5}=\frac{4\pi}{15}[/tex]For n=1 [tex]x=\frac{\frac{2\pi}{3}+2\pi}{5}=\frac{8\pi}{15}[/tex][tex]x=\frac{\frac{4\pi}{3}+2\pi}{5}=\frac{2\pi}{3}[/tex]For n=2 [tex]x=\frac{\frac{2\pi}{3}+4\pi}{5}=\frac{14\pi}{15}[/tex][tex]x=\frac{\frac{4\pi}{3}+4\pi}{5}=\frac{16\pi}{15}[/tex]For n=3 [tex]x=\frac{\frac{2\pi}{3}+6\pi}{5}=\frac{4\pi}{3}[/tex][tex]x=\frac{\frac{4\pi}{3}+6\pi}{5}=\frac{22\pi}{15}[/tex]For n=4 [tex]x=\frac{\frac{2\pi}{3}+8\pi}{5}=\frac{26\pi}{15}[/tex][tex]x=\frac{\frac{4\pi}{3}+8\pi}{5}=\frac{28\pi}{15}[/tex]For n=5 we would find values such as  [tex]x=\frac{\frac{2\pi}{3}+10\pi}{5}=\frac{32\pi}{15}[/tex][tex]x=\frac{\frac{4\pi}{3}+10\pi}{5}=\frac{34\pi}{15}[/tex]which don't lie in the interval [tex](0,2\pi)[/tex]The whole set of results is [tex]\frac{2\pi}{15},\frac{4\pi}{15},\frac{8\pi}{15},\frac{2\pi}{3},\frac{14\pi}{15}, \frac{16\pi}{15}, \frac{4\pi}{3},\frac{22\pi}{15}, \frac{26\pi}{15}, \frac{28\pi}{15}[/tex]