functional analysis – simple question about and unitary element and self-adjointness

In Murphy’s book P42, it says

If $A$ is a unital $C^*$– algebra and $a $ is self-adjoint in $A$, then $e^{ia}$ is unitary.

Why is the statement true?

I’ve been playing around with the involution. But I don’t see how we get $(e^{ia})^* = e^{-ia}$.

Thank you for your help!