How to write $a^{ix}$ in terms of $\sin(x)$ and $\cos(x)$?

Question

We know that $e^{ix} = \cos(x) + i\sin(x)$ and the plot of $2^{ix}$ seems to have sinusoidal behavior.

http://goo.gl/Xfg2wp

Can we claim that we can write $a^{ix}$ in terms of $\sin(x)$ and $\cos(x)$?

Answer 1

Hint:If $a>0$ then $a=e^{\ln a}$

Answer 2

$a=e^{\ln a}, a>0$

$$(e^{\ln a})^{ix}=e^{ix \ln a}=\cos(x \ln a)+i\sin(x\ln a)$$