Part of the problem is that since I've never had a trig class, I don't know what I am suppose to memorize and what I am suppose to be able to derive, which is only important since tests are timed, but I've made a point to memorize the reference angles and make sure I can derive them. But anyway, after looking at this for a few days a few things have made sense, but a part of it is still confusing.
I know that the reference angle for sin 1/2 and cos 1/2 is 30 degrees and 60 degrees respectively and tan 1 is 45 degrees. I am able to get the answer pi/6 since 30 degrees equals pi/6. I can also get the answer 13pi/6 because I know that sin has a period of 2pi so pi/6 + 2pi = 13pi/6.
By the same logic the answer 5pi/6 + 2pi = 17pi/6, but I don't know why we can use 5pi/6 as one of the x values on the interval [0, 4pi]. Looking at the picture, I can see that there is an intersection value right before pi, but I don't know why we use 5pi/6. The actual value of that angle is 150 degrees - the reference angle is 30 degrees, but why do we use the reference angle instead of the the 150? Do we use it because it's at the same height (which you mentioned) or is there another reason?
I know I've complicated a simple problem, but it's a years worth of trig inside of a few weeks so every bit of understanding helps
Thanks for all the help so far though.
Also, and this is not really important, but I think we don't use 7pi/6 or 11pi/6 since those would be negative sine values right?