# Solving a system of non-linear (trig) equations:

I am having trouble trying to solve the following equations:

$\sin(\alpha)+\sin(\beta)=\dfrac {1000} A$

$\sin(\alpha)+\sin(\gamma)=\dfrac {800} A$

$\dfrac {20(1+\cos(\alpha-\beta))} {\cos(\beta)} -\dfrac {20(1+\cos(\alpha-\gamma))} {\cos(\gamma)}-.225=0$

I tried plugging into MatLab using fsolve() and it gave me values that do not solve the all three of the equations. Trying to expand in a taylor series gets really ugly really fast. So does anyone have any advice on how to solve these. Thanks

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What does your third equation equal? Are your unknowns the three greeks or is it the three greeks and A? If its four unknowns then you have to use a trig identity to get the fourth equation. –  riotburn Jul 8 '13 at 14:45
Whoops... Added it. The A is an arbitary constant based on another equation (for simplicity, its roughly 1000). –  yankeefan11 Jul 8 '13 at 14:57
One naive way to start is : replace $\sin\beta,\sin\gamma$ in $(3)$ with $\sin\alpha$ using $(1),(2)$. Then replace $\cos\beta,\cos\gamma$ in $(3)$ with $\sin\alpha$ or $\cos\alpha$ if need be –  lab bhattacharjee Jul 8 '13 at 15:06
FYI: With a brute-force first pass, I got a degree-$10$ polynomial in $\sin\gamma$. –  Blue Jul 8 '13 at 15:10

Its likely that the fsolve isn't working in matlab because you're initial guess isn't close enough, causing the optimization software to get stuck. If you try starting with some values that approximate the answer you want, you might get an exact answer from matlab.

Some values that worked for me were

$A = 4000$

$\alpha = 13^\circ$

$\beta = -3^\circ$

$\gamma = -3^\circ$

hope this helps.

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I used the values that you suggested as an initial guess. I got results that work very well with my modal. Any idea as to how to approach it from a mathematical standpoint and solve the system of equations analytically? –  yankeefan11 Jul 9 '13 at 20:39