What is $\Re(e^i)$?

Question

What is $\Re(e^i)$?

My book says answer is $e$. Can that be right?

I know $e^{i \pi} = \cos \pi + i \sin \pi = -1$

So that would mean $e^i = \cos 1 + i \sin 1$ and the real part would be $\cos 1$. That's not the book's answer.

Answer 1

You're right, $e^i=\cos 1+i \sin 1$. So the answer is $\cos 1$.