Then we get to the part they couldn't apply successfully, so we broke it down. First I asked them to find (say) 200 degrees on their graph. This seemed a nice challenge for them and it was nice to hear their thought process.
Then I asked them to find sin(200deg) on their graph. Also, enlightening.
Then I asked them to reason through finding arcsin(sin(200deg)), and I had them talk out their reasoning as to why it was placed where it was.
And finally, we took it to the unit circle, so they could summarize and wrap it up.
I think I may start with this technique the next time through because it shows everything on one page and they can work through the visuals of WHY even through you MAY have an angle in the first quadrant, it still is not valid for an arcsin function because the angle is more than 360 degrees (for example).