how to solve nonlinear system of equations

Question

I have $A, B, C, D, E, F.$ I want to calculate a and b from the following system of equations:

I know I should solve this system using $3$ equations and $3$ unknowns, but it is not linear.

can any one show me how to solve it?

Edit: We know that $a>=b>=0$ and $tetta$ is unknown.

Answer 1

Observe that $A+C=a^2+b^2$, that $A-C=\left(b^2-a^2\right)\,\cos(2\Theta)$, and that $B=\left(b^2-a^2\right)\,\sin(2\Theta)$. Hence, $\left(b^2-a^2\right)^2=(A-C)^2+B^2$. You only need to solve for $a^2$ and $b^2$ from $a^2+b^2=A+C$ and $a^2-b^2=\pm\sqrt{(A-C)^2+B^2}$.

If $\Theta$ is an unknown, then there are $8$ possible values of $(a,b)$ (due to signs of $a$, of $b$, and of $a^2-b^2$). If $\Theta$ is known, then as mathreadler said, there are $4$ possible values of $(a,b)$.

Answer 2

You can find $a^2$ and $b^2$ from line 1 and 3. Then you just need to find the sign of a and b, only 4 combinations to try. Of course you need to check that the other equations are not being violated by the solution you find.