Complex numbers in Euler's rule

Question

I know that $\sin(ix) = \sinh(x)$ based on manipulation of Euler's rule. However if I introduce a constant $a$, is this relation correct?

$\sin(iax)=i\sin(ax)$

Answer 1

Yes. Given that: $\sinh(x) = \frac{(e^x - e^{-x})}2$ ,
$$\sinh(iax) = \frac{e^{iax} - e^{-iax}}2$$ And therefore:

$$\sinh(iax)= \frac{1}{2}(\cos(ax) + i\sin(ax) $$ $$- \cos(ax) + i\sin(ax))$$ $$ = 2i\frac{\sin(ax)}2 = i\sin(ax)$$

Answer 2

When we say that $$ \sin(ix) = i\sinh(x) $$ in this context that means that the two sides are equal no matter what complex number we put instead of $x$. When you write $ax$ instead of $x$, that is just another number, and equality still holds.