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]
