What is the conjugate of $[i + e^{iπt}]$?

Question

What is the conjugate of the following complex number?

$$ Z(t) = i + e^{iπt} $$

Is it $Z(t) = i - e^{iπt}$ or $Z(t) = i - e^{-iπt}$?
$t\in [0,1]$

Answer 1

$$Z(t) = i + \cos(\pi t) + i\sin(\pi t)$$ $$Z(t) = \cos(\pi t) + i(1 + \sin(\pi t))$$

$$conjugate (Z(t)) = \bar{Z}(t) = \cos(\pi t) - i(1 + \sin(\pi t))$$

$$\bar{Z}(t) = -i + \cos(-\pi t) + \sin(-\pi t))$$ $$\bar{Z}(t) = -i + e^{-i\pi t}$$

Answer 2

I suppose that $t$ is real.

If $I$ is real,then

$ \overline{z(t)}=I+e^{-i \pi t}.$

If $I$ is not real,then

$ \overline{z(t)}=\overline{I}+e^{-i \pi t}.$