Please see attached image.
Knowing only theta and x, find theta_2 and theta_3
(the green curve is an ellipse inscribed within the reddish unit circle)
These are very unexpected, interesting responses (to me), which is precisely what I hoped to get by posting this problem.
Opalg, I'm hesitant to say you're wrong, but your answer doesn't work when I graph it. Either it's wrong or I'm making some mistake when I graph it.
Simplependulum, I'm still trying to wrap my head around your response. I had never thought of expressing things in that way.
Here is my solution (obviously not the only one):
First, use the given info-- --to find the semi-minor axis of the ellipse, :
Then, the equations for and are, respectively:
You derive these equations by fooling around with (x, y) intercepts and pythagorus' theorem. I won't go into the details unless you really want.
Happily, these equations turn out to be reducible to: