Snell's Law Variation

[BFM_Edison]

You probably know what Snell's Law is if you're looking at this thread. As a refresher, here is the wikipedia page on it: http://en.wikipedia.org/wiki/Snell's_Law

I'd rather you not give me a direct answer to this question (assuming there is one). We'll use this general form for Snell's Law: n₁ sin(theta₁) = n₂ sin(theta₂) where the two thetas are the angles with respect to the normal. What I'm trying to know is if there is a way to define cos(theta₂) in terms of n₁, n₂, and cos(theta₁). If not, give me a hint as to how to show why this is the case. Then I can decide whether to go into reflection off of surfaces in 3-space and possibly n-space.

[Goalie]

Let's say that x²=y².  Couldn't you square both sides and then use Pythagorean identities to solve for cos(theta₁)?  That's what I'd do, anyway, even if it is a bit messy.

[BFM_Edison]

Not sure what x and y are referring to. Anyways, I've done some proof similar to the normal derivation, but there's definitely something wrong with it and for some reason I'm not seeing it. Hay, I has idea. Nvm. But it can't involve using the normal equation and squaring it. The final solution can't have any absolute values, squares, or square roots. Need to keep the sign correct. Or will it work. Hmmmm. Will have to test it out vector-wise. No, doesn't look like it will work.

Now tell me what's wrong with this proof here please. Line number only is fine.

http://img516.imageshack.us/img516/238/failproofri1.jpg

[Goalie]

The only thing that I see is that you made a comment next to line 5 about what looks like the derivitive of (l-x)².  But I could not find this applied to the equation anywhere.

That's the only thing I see.  I could bring it into my Calc III class tommorrow and see what my professor says.

[BFM_Edison]

That was just a little test to make sure that it went negative and had a two since I couldn't figure the error out.

And I'll be bringing it to my Physics teacher today.