I understand that the period for the graph 4 + 2sin(2Θ/3) is 3π. The way I got that was by 2π/b. b is 2/3. Therefore I did 2π/(2/3) and got 3π for the period. Can someone guide me to solve part b? I need to get the domain values of Θ to get the whole graph without repeating any part of the graph. I posted a picture of the graph below and part b in case I didn't explain it clearly.
For the graph to repeat, $r$ needs to repeat, as well as the angle
The angle repeats when $\theta$ changes by $2\pi$
$r$ repeats when $\theta$ changes by $3\pi$ as you have found. So you need the $\theta$ that is the smallest multiple of both $2\pi$ and $3\pi$, which is $6\pi$. So $6\pi$ is the answer.
You can also see that the graph makes 3 loops around the origin hinting that it repeats after $3\times2\pi$