Sine of complex numbers.

Question

It is stated that the system, whose displacement is defined by sin[√(A²-1) + X ], is at rest when A is greater than 0 and smaller than 1.

How can this be shown?

Answer 1

Use the identity $\sin(a+ib)=\sin(a)\cosh(b)+i\cos(a)\sinh(b)$

$\implies \sin(\sqrt{A^2-1}+X)=\sin(i\sqrt{1-A^2}+X)=\sin(X)\cosh(\sqrt{1-A^2})+i\cos(X)\sinh(\sqrt{1-A^2})$

Here, $\sinh(x), \cosh(x)$ are the hyperbolic sine and cosine functions.