Wolfram Alpha gives following results for $\cos(i x)$ and $\sin(i x)$ ,where $x\in\mathbb{R}$:
$\cos(ix)=\cosh(x)$
$\sin(ix)=i\sinh(x)$
What is a reason why the first number is real while the second is complex ?
Definition of the $\cosh(x)$ and $\sinh(x)$ may be found here.