Hello:

I am trying to solve this system:

A*cos(x) + B*sin(y) = C
D*sin(x) + E*cos(y) = F

I have tried several substitutions, but it will always lead to a 4th degree equation. I habe also used identities, for example sin(y) = sqrt(1-cos(y)^2), sin(x)=(exp(i*x)-exp(i*x))/2*i and cos(x)=(exp(i*x)+exp(i*x))/2, and so on...

For instance:

A*cos(x) + B*sin(y) = C => sin(y)^2 = ((C-A*cos(x))/B)^2
D*sin(x) + E*cos(y) = F => cos(y)^2 = ((F-D*sin(x))/E)^2 = 1- sin(y)^2

So ((F-D*sin(x))/E)=^2 = 1 - ((C-A*cos(x))/B)^2

When replacing sin(x) = sqrt(1-cos(x)^2 and expanding, I will get a 4th degree equation.

HOWEVER: I remember that in highschool I used to solve this without having to solve a 4th degree equation. I do know the solution for a 4th degree equation, but as you may know, it's a BIG formula.

The solution for the system is:

y1= arcsin((M*G+K)/(G^2+H^2))
y2= arcsin((M*G-K)/(G^2+H^2))

x1= arccos((F-E*y1)/d)
x2= arccos((F-E*y2)/d)

Where M=C^2+B^2+F^2-A^2, G=2*F*E, H=-2*B*C

I got it like 10 years ago! but now I can't remember the trick. I need to document a program I wrote for solving equations.

Thanks!