I learnt using sin (mathematics) expressions in after effects a while ago. I think of sharing it here since i find it pretty cool.These are mainly used to flip the wings. They are used in the butterfly animated below, i think the animation looks pretty smooth.
I am just trying to show the use of this expression as used to loop abutterfly.
The result:- Side view:-
Top view:-
Actually, this mathematical sin() function has a period of 2 angle with maxima of 1 at /2 and 3/2. It's symmetric across the x-axis across equal and different parts in its cycle.This function is pretty well suited to representbutterfly wing wiggle.
Not my image:-
Maybe this information is helpful to someone of you.