We started by making an envelope for their notebooks.
Then we SLOWLY with lots of back and forth questioning created this sheet for sine:
We had to refresh our memory on Inverse Functions in general from previous knowledge. Then we flipped the paper over (after we discussed the "x and y switch roles") by keeping our thumb (positive x) and index finger (positive y) attached to the paper:
And we had a discussion on what part we should restrict the domain of the sine graph to to make this inverse function a function. THEN we shaded the right graph part with a sharpie.
We did the same with the cosine graph the next day. Then because of a discussion with some students AND the fact that even though we stress that ONLY ONE output is possible, they still are confused with a problem such as:
Without a calculator, find arccos ( cos ( 200 °) ).
So after the patty paper, I had them explore with this sheet:
* Funny note: when we were making the graphs and free-style drawing, a student said, "hey, they should make patty paper lined with a graph/grid". I flipped the patty paper box over and showed her that it really was paper meant to be put between meat patties or some such food.