You can find the angles using the unit circle. You can also find the angle using algebra.
I want to show you the algebra way.
Since cos(x) = cos(2•pi - x), this leads to two equations.
The two equations are as follows:
cos (x) = -1/2...the given problem.
We also have cos (2•pi - x) = -1/2.
To isolate x for both trigonometric equations, use the inverse trig function idea.
Equation 1
arccos(cos x) = arccos (-1/2)
x = 2•pi/3
Equation 2
arccos(cos (2•pi - x)) = arccos (-1/2).
Note: 2•pi - x = 2•pi/3. We are not done with this equation. We must isolate x. In other words, like any other equation, solving for x is needed.
When we subtract 2•pi from both sides, the answer for x in the second equation is 4•pi/3.
As you know, 2•pi/3 and 4•pi/3 lie between 0 and 2•pi on the unit circle.
Answer: 2•pi/3, 4•pi/3
