# How do you deal with a possibly imaginary or not imaginary term in the polar form of a complex number?

Sorry about the confusing title but I'll try to illustrate what I mean here.

I have the equation $z=e^{-i(t+z)}$ where $z$ can be of the form $x+iy$, $x$, or $iy$ (for $x,y,t \in \mathbb{R}$). How do I convert this into rectangular form without splitting into cases? Or is this a necessity?

Edit:

I think I just simplified my question too much to put it up here. What I am trying to separate is this complex number:

z = exp$\left[-i(\omega t-k\left(\sin({\theta_{1}}) x + \sqrt{\sin^{2}{\theta_{2}}-\sin^2{\theta_{1}}}\right) \right]$

where I need to consider both $\theta_{1} < \theta_{2}$ and $\theta_{2} < \theta_{1}$ and $\omega t$ and $k\sin{\theta_{1}}$ are both real.

• Strictly speaking $e^{-i(t+z)}$ is not an equation – Henry Sep 16 '15 at 23:52

Well if $z$ can be complex then:
$$e^{-i(x+iy+t)}=e^{-ix+y-it}=e^{-ix}e^{y}e^{-it}$$
You know $x,y,t$ are all real. Just use Euler's formula $$e^{i\theta}=\cos\theta+i\sin\theta$$ and you'll get it split into cartesian form.
• Ah I see what you mean, I think I just simplified my question too much to put it up here. What I am trying to separate is this complex number: exp$\left[-i(\omega t-k\left(\sin({\theta_{1}}) x + \sqrt{\sin^{2}{\theta_{2}}-\sin^2{\theta_{1}}}\right) \right]$ where I need to consider both $\theta_{1} < \theta_{2}$ and $\theta_{2} < \theta_{1}$. Does your answer still apply? – Daniel Ward Sep 16 '15 at 23:52
• Also $\omega t$ and $k\sin{\theta_{1}}$ are both real. – Daniel Ward Sep 16 '15 at 23:55