The purpose of this assignment is to demonstrate the so called "small angle approximation": for small angles, sin(θ) ≅ θ, when the angle θ is expressed in radians. The question really translates to what does a small angle mean.
This really touches on the precision of a measurement. Precision is related to significant figures. The more precise your experiment, that is, the better your equipment, the more significant figures you would use to represent your result.
For example, this approximation comes up typically for the period of a pendulum. It is usually stated that the period of a pendulum is independent of the angle of release, for small angles. If you did an experiment where you plotted the period as a function of angle, the resulting data would be flat for small angles and then curve start to curve away from flat when the angle get larger. Experimentally, if you used the stop watch feature on your phone to do this experiment, you would not see a change in the curve until maybe 40 degrees. However, if you used some sort of electronic timer, like a vernier probe, the curve would start to change from flat at perhaps 5 degrees. The second experiment was done with more precision - more significant figures.
So, simple make a table to see this approximation in action.
θ sin(θ)
0.0000 .0000
0.0001 .0001
0.0010 .0010
0.0100 .0010
0.1000 .0998 To 2 significant figures, this is still the same - .0998 is .010 to 2 sig figs.
0.2000 .1987 To 2 significant figures, this is still the same - .1987 is .20 to 2 sig figs.
1.0000 .8414 Here, they agree to only 1 significant figure.
So you need to explore the region where the angle is between .20 and 1.0
It is tedious, but have fun!
