# Help solving (complex) simultaneous equations

KFC

## Homework Statement

I am trying to solving the following complex equation for $$x$$ and $$\theta$$

$$a\sinh(2x) e^{-i\theta} + y\sinh^2x e^{-i2\theta} + y^*\cosh^2(x) = 0$$

where $$a$$ is real constant, $$x$$ and $$\theta$$ are also real parameter. $$y$$ is complex number, $$y^*$$ is the complex conjugate.

Solve for $$x$$ and $$\theta$$ (in terms of y and a)

2. The attempt at a solution
Let
$$y = |y| e^{i\varphi}$$

and multiply the equation with $$y$$

$$ay\sinh(2x) e^{-i\theta} + y^2\sinh^2x e^{-i2\theta} + |y|^2\cosh^2(x) = 0$$

Now let the real part and imaginary part equals ZERO.

$$\begin{cases} a\sinh(2x) |y|\cos(\theta-\varphi) + |y|^2\sinh^2(x)\cos(2\theta-2\varphi) + |y|^2\alpha^2 = 0, \\[3.8mm] a\sinh(2x) |y|\sin(\theta-\varphi) + |y|^2\sinh^2(x)\sin(2\theta-2\varphi) = 0 \end{cases}$$

I tryied to solve that two days ago, I tried many way to simpliy that but still find no way to get the soluton. Could anyone give me some hints?

Thanks

Last edited:

Mentor
This doesn't make sense to me. How can you solve one equation for two variables? This seems to me like asking someone to solve y = 2x for x and y. You can solve the equation for y in terms of x, or you can solve for x in terms of y, but you can't solve it simultaneously for both variables.

KFC
This doesn't make sense to me. How can you solve one equation for two variables? This seems to me like asking someone to solve y = 2x for x and y. You can solve the equation for y in terms of x, or you can solve for x in terms of y, but you can't solve it simultaneously for both variables.

How come. This is a equation for complex variable, the real part and imaginary part gives two equations.

Mentor
OK, I see.
$$a\sinh(2x) |y|\sin(\theta-\varphi) + |y|^2\sinh^2(x)\sin(2\theta-2\varphi) = 0$$
In your 2nd equation, I think the sine arguments should be the other way around. Also, you can expand $sin(2(\phi - \theta))$ as $2sin(\phi - \theta) cos(\phi - \theta)$. Then you'll have one factor the same in both terms, which hopefully leaves you with the other factor that you can do something with.