Negative imaginary exponents I was reading this question earlier: Understanding imaginary exponents
In the answer, the answerer says

$A^i=x+iy$
Furthermore, we can write $A^{−i}=x−iy$ for the same $x$ and $y$.

Can someone explain to me why this is true? It kind of makes sense to me but how can we make the leap to just negating the imaginary component?

If $A>0$ is real, we define $A^z=e^{z\ln A}$. So 
$A^i=e^{i\ln A}=\cos(\ln A)+i\sin(\ln A)$
$A^{-i}=e^{-i\ln A}=\cos(-\ln A)+i\sin(-\ln A)=\cos(-\ln A)-i\sin(\ln A)$
