$\frac{2\pi}{15},\frac{4\pi}{15},\frac{8\pi}{15},\frac{2\pi}{3},\frac{14\pi}{15}, \frac{16\pi}{15}, \frac{4\pi}{3},\frac{22\pi}{15}, \frac{26\pi}{15}, \frac{28\pi}{15}$

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

$cos5x=-\frac{1}{2}$

Solving for 5x:

$5x=\frac{2\pi}{3}+2n\pi$

$5x=\frac{4\pi}{3}+2n\pi$

The values for x will be

$x=\frac{\frac{2\pi}{3}+2n\pi}{5}$

$x=\frac{\frac{4\pi}{3}+2n\pi}{5}$

To find all the solutions, we'll give n values of 0, 1, 2,... until x stops belonging to the interval $(0,2\pi)$

For n=0

$x=\frac{\frac{2\pi}{3}}{5}=\frac{2\pi}{15}$

$x=\frac{\frac{4\pi}{3}}{5}=\frac{4\pi}{15}$

For n=1

$x=\frac{\frac{2\pi}{3}+2\pi}{5}=\frac{8\pi}{15}$

$x=\frac{\frac{4\pi}{3}+2\pi}{5}=\frac{2\pi}{3}$

For n=2

$x=\frac{\frac{2\pi}{3}+4\pi}{5}=\frac{14\pi}{15}$

$x=\frac{\frac{4\pi}{3}+4\pi}{5}=\frac{16\pi}{15}$

For n=3

$x=\frac{\frac{2\pi}{3}+6\pi}{5}=\frac{4\pi}{3}$

$x=\frac{\frac{4\pi}{3}+6\pi}{5}=\frac{22\pi}{15}$

For n=4

$x=\frac{\frac{2\pi}{3}+8\pi}{5}=\frac{26\pi}{15}$

$x=\frac{\frac{4\pi}{3}+8\pi}{5}=\frac{28\pi}{15}$

For n=5 we would find values such as

$x=\frac{\frac{2\pi}{3}+10\pi}{5}=\frac{32\pi}{15}$

$x=\frac{\frac{4\pi}{3}+10\pi}{5}=\frac{34\pi}{15}$

which don't lie in the interval $(0,2\pi)$

The whole set of results is

$\frac{2\pi}{15},\frac{4\pi}{15},\frac{8\pi}{15},\frac{2\pi}{3},\frac{14\pi}{15}, \frac{16\pi}{15}, \frac{4\pi}{3},\frac{22\pi}{15}, \frac{26\pi}{15}, \frac{28\pi}{15}$

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3 years ago