Hello Everyone! Today we are asked to prove that both this expressions are true: $\displaystyle \left | e^{\theta i } \right |=1$ and $\displaystyle \overline{e^{\theta i}}=e^{-\theta i}$. To be quite honest, I am not very sure as to how I can prove these, but my attempt for the first one, I was able to prove that $\displaystyle \sin^2(\theta)+cos^2(\theta)=1$ but was unsure as to how I would be able to turn it into $\displaystyle \sin(\theta)+cos(\theta)=1$ which is what I need to prove the first equality.

As for the second one I did something like this:

$\displaystyle e^{\theta i}=\cos(\theta)+i\sin(\theta)$

So that:

$\displaystyle \overline{e^{\theta i}}=\cos(\theta)-i\sin(\theta)$

and by properties of odd and even function:

$\displaystyle \overline{e^{\theta i}}=\cos(-\theta)+i\sin(-\theta)$

Thus, it should be true to say that :

$\displaystyle \overline{e^{\theta i}}=e^{-\theta i}$

The proof above, for me, is quite sloppy, but I think I could live with that. But for the first one, I really have no idea as to how to prove it. Anyone got some ideas as to how I should prove both equivalence? Thanks everyone in advance!