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Inverse Function Identities

Hello,

I've been working on a problem I gave myself to work on over the holiday to curb my boredom, and I think I have almost solved half of it. I have this system of equations:

$\displaystyle Xsin(a) + Ysin(b) = d$

$\displaystyle Xcos(a) - Y cos(b) = h$

$\displaystyle phi = a + b$

The a's and b's are actually alpha's and beta's, respectively, in my notebook, I just don't know how type them here. $\displaystyle X$ and $\displaystyle Y$ are constants, $\displaystyle h$ and $\displaystyle d$ are given, and $\displaystyle d^2 = x^2 + y^2$.

Anyway I want to solve the system for phi. I tried this in a couple of ways, but my only successful attempt was solving the top two for alpha and beta and adding the two results together, yielding the following:

Attachment 25878

As you can see I already substituted for d since my answer had only d squared terms.

Now what I would like to do is simplify that further if at all possible. Any help would be much appreciated.

Re: Inverse Function Identities

Square each of the top two equations and add them.

Simplify and you finish up with

$\displaystyle \cos(a+b)=\frac{X^{2}+Y^{2}-d^{2}-h^{2}}{2XY}.$