The experiment was a success! Students walked into class today asking to know what the inverse trig functions do (they phrased it as, “What does the negative-one mean?”), two students had reasonable theories to explain it, and one hit it right on the nose.
Before going over the worksheet, I teamed up the students for an activity where one student would write a sine or cosine value on the board, indicating whether it were a sine or a cosine; the next student would name one or several angles with that sine/cosine value and draw the angle/s on the unit circle. They were performing the inverse sine and cosine functions without knowing it. Later when a student attempted to explain his theory for the meaning of sin−1, he suddenly came out with, “I think it’s what we were doing on the board!”