Equilateral Triangles "Inscribed in" a Cosine Curve

Author:
Roy Wright
A question from Twitter: Is there an equilateral triangle with all three vertices on a cosine curve? The amplitude of the cosine curve can be changed using the red slider below. If the amplitude is high enough, it is possible to find one, two, or more equilateral triangles with a vertex along the -axis. (Having GeoGebra find more general triangles, as done with parabolas before, is harder here.)
  1. Can you find the exact coordinates for an equilateral triangle whose vertices all lie along the graph of ? If not, can you find them to, say, five decimal places?
  2. Can you find the amplitude required for such a triangle to exist, to five decimal places?