Steady-State Output from Transfer Function
You may know that one important property of linear time-invariant (LTI) systems is that the complex exponential $e^{j\omega_0}$ is an eigenfunction, and the corresponding eigenvalue is given by the system's frequency response evaluated at $\omega_0$.

So the response to an input signal $$x[n]=e^{jn\omega_0}$$ is given by $$y[n]=H(e^{j\omega_0})e^{jn\omega_0}$$

Now try to show that from this it follows that the response to a sinusoidal input signal $$x[n]=\sin(\omega_0n)$$ is given by $$y[n]=\big|H(e^{j\omega_0})\big|\sin\left[\omega_0n+\angle H(e^{j\omega_0})\right]$$ (if the system is real-valued). Now should you have all the information to compute the reconstructed output signal.