Find all solutions of the equation

Question

Find all solutions of the equation 2cos⁡3x=1 in the interval [0,π). 

The answer is x1= ,  x2=   and x3=   with x1<x2<x3. 

Answer 1

2cos⁡(3x) = 1

cos(3x) = 1/2

Let θ = 3x so the problem becomes cos(θ) = 1/2. Using the unit circle this will occur at θ = π/3 and θ = 5π/3. It also will occur (continuing around the unit circle) at θ = 7π/3

So we have:

θ = π/3 meaning 3x = π/3 or x = π/9

θ = 5π/3 meaning 3x = 5π/3 or x = 5π/9

θ = 7π/3 meaning 3x = 7π/3 or x = 7π/9

You can now see why we needed to keep going around the unit circle to get all the answers for "x" up to π