Does there exist a complex function that takes one complex number to another?

Question

Is there any meaningful relationship between the values $$z_1 = a \cos(x) + i b \sin(x), z_2 = b \cos(x) - i a \sin(x) $$ where $a,b,x$ are real numbers? Basically, $z_1$ lies on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ while $z_2$ lies on the corresponding point of that ellipse turned 90 degrees. However, I don't know how to come up with a bijective function that turns $z_1$ in $z_2$.

Answer 1

Thre is no such relationship, at least not if $a, b, x$ are all allowed to vary simultaneously through all real values.

For example, \begin{gather*} \text{if } (a, b, x) = \left(1, 1, \frac{\pi}{4}\right), \text{ then } z_1 = \frac{1+i}{\sqrt{2}} \text{ and } z_2 = \frac{1-i}{\sqrt{2}}, \text{ but} \\ \text{if } (a, b, x) = \left(1, -1, -\frac{\pi}{4}\right), \text{ then } z_1 = \frac{1+i}{\sqrt{2}} \text{ and } z_2 = \frac{-1+i}{\sqrt{2}}, \end{gather*} so there would have to be two different values of $z_2$ corresponding to the same value of $z_1$.

I'll see if I can add a counterexample where $a$ and $b$ are both strictly positive, which I imagine is what you had in mind.

I think this is OK: \begin{gather*} \text{if } (a, b, x) = \left(2, 1, \cos^{-1}\left(\frac{1}{\sqrt{5}}\right)\right), \text{ then } z_1 = \frac{2+2i}{\sqrt{5}} \text{ and } z_2 = \frac{1-4i}{\sqrt{5}}, \text{ but} \\ \text{if } (a, b, x) = \left(1, 2, \cos^{-1}\left(\frac{2}{\sqrt{5}}\right)\right), \text{ then } z_1 = \frac{2+2i}{\sqrt{5}} \text{ and } z_2 = \frac{4-i}{\sqrt{5}}. \end{gather*}