Trigonometric expression - winding an unknown

• Mar 17th 2011, 11:29 AM
Wane
Trigonometric expression - winding an unknown
Hello everyone! I've been working on a trigonometric expression that led me nowhere. I have attached it, so that you kan take a look at it. What I am trying to express is Gama, but I am quite confused on how to do it. I tried several approaches, but I guess I lack the knowledge. Could someone help me, plese?
• Mar 17th 2011, 11:51 AM
topsquark
Quote:

Originally Posted by Wane
Hello everyone! I've been working on a trigonometric expression that led me nowhere. I have attached it, so that you kan take a look at it. What I am trying to express is Gama, but I am quite confused on how to do it. I tried several approaches, but I guess I lack the knowledge. Could someone help me, plese?

It's a monstrosity, no doubt, but there is a way. I'll outline it.

First I am assuming f is a constant, so I'll replace 1 + f with k for simplicity.

Get rid of the fraction. ie
$kR - R~cos(2 \beta - \gamma ) = k - cos( \gamma )$

Now expand the $cos(2 \beta - \gamma ) = cos(2 \beta )~cos( \gamma ) + sin(2 \brta )~sin( \gamma )$

Collect terms:
$k(R - 1) = (R~cos(2 \beta ) - 1)~cos( \gamma ) + sin(2 \beta )~sin( \gamma )$

Now isolate the $sin( \gamma )$ term and replace it with $\sqrt{1 - cos^2 ( \gamma )}$. Square both sides and collect like factors. You will now have a quadratic by which you can solve by the quadratic formula. Then take the inverse cosine of both sides.

I may/may not have some errors in there. But the basic method is still the same.

-Dan
• Mar 17th 2011, 12:50 PM
Wane
Thanks a lot! I'll have a go and I'll post my results here.