Help with project - Freudenstein equation

[8point1]

Homework Statement

Its easier to see if I link the problem:
http://faculty.olympic.edu/jjbrown/documents/ENGR%20111/ENGR%20111%20Project%201%202011.pdf

The Attempt at a Solution

I don't have a problem finding the answers, but the other steps we need to show are giving me a hard time. My hang up is the sentence: "By isolating in terms of β, then adding the squares of those equations one obtains the Freudenstein equation (you are to show this)."

I tried working the above equation to solve for β, but I'm just not seeing it. Any help would be greatly appreciated. Thanks

 

[cepheid]

Just follow the steps that were given. Isolating the terms in beta means moving them to one side and everything else to the other side. Hence the equations become:$$b\sin\beta = c\sin\phi- a\sin\theta$$$$b\cos\beta = c\cos\phi - a\cos\theta$$The sum of the squares of the left-hand sides is just $b^2(\sin^2\theta + \cos^2\theta) = b^2$

So, the sum of the squares of the right-hand sides has to be equal to that. Some trig identities are definitely going to come into play here.

 

[8point1]

Thanks for the reply.

cool, I see how to get $b^2$ using trig ID's.

But adding the sums of the right hand I'm not seeing the algebra.

and seeing how this equation:

$b^2=c^2sin^2\theta+c^2cos^2\theta-a^2sin^2\phi-a^2cos^2\phi$

gets to this equation:

$R^1cos\theta-R^2cos\phi+R^3-cos(\theta-\phi)$

 

[cepheid]

Thanks for the reply.

cool, I see how to get $b^2$ using trig ID's.

But adding the sums of the right hand I'm not seeing the algebra.

and seeing how this equation:

$b^2=c^2sin^2\theta+c^2cos^2\theta-a^2sin^2\phi-a^2cos^2\phi$

gets to this equation:

$R^1cos\theta-R^2cos\phi+R^3-cos(\theta-\phi)$

First of all, I don't think you're really squaring the right-hand sides correctly:

$$(c\sin\phi- a\sin\theta)^2 = (c\sin\phi- a\sin\theta)(c\sin\phi- a\sin\theta)$$

$$= c^2\sin^2\phi -2ac\sin\phi\sin\theta - a^2\sin^2\theta$$

 

[8point1]

You're right, I didn't foil the right side. gets confusing with all the sines and cosines.

Should that last term be +?

 

[cepheid]

You're right, I didn't foil the right side. gets confusing with all the sines and cosines.

Should that last term be +?

Yes.

 

[8point1]

Okay, thanks for your help!