4cos(x)+5 = 34cos(x)+5=3
=> 4cos(x) = -2⇒4cos(x)=−2
=> cos(x) = -1/2⇒cos(x)=−12
From a unit circle, we can tell that cos(x) = -1/2cos(x)=−12 at x = +-pi/3+2pin, n in ZZ. Note that we add the 2pin as cos(x) is periodic with a period of 2pi.
=> x = pi/3 + 2pin or x = -pi/3 + 2pin
Now we can look to see what values for n leave x within our desired interval of [0, 2pi]
For x = pi/3+2pin, only n=0 works.
For x = -pi/3+2pin, only n=1 works.
Thus, we get our final solution set
x = pi/3+2pi*0 = pi/3
or
x = -pi/3 + 2pi*1 = (5pi)/3
:. x in {pi/3, (5pi)/3}
