We have: cos(x) = frac(1)(2); 0 le x le 2 pi
Let the reference angle be cos(x) = frac(1)(2):
Applying arccos to both sides of the equation:
Rightarrow arccos(cos(x)) = arccos(frac(1)(2))
Rightarrow x = frac(pi)(3)
So, the reference angle is x = frac(pi)(3)
Now, the interval is given as 0 le x le 2 pi, covering all four quadrants.
The value of cos(x) is positive, i.e. + frac(1)(2).
So, we need to find the values of x in the first and fourth quadrants (where values of cos(x) are positive).
Rightarrow x = frac(pi)(3), 2 pi - frac(pi)(3)
Rightarrow x = frac(pi)(3), frac(6 pi)(3) - frac(pi)(3)
therefore x = frac(pi)(3), frac(5 pi)(3)
Therefore, the solutions to the equation are x = frac(pi)(3) and x = frac(5 pi)(3).