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)#.
