Frequency Response Output
Hint:

I guess we may assume that the system is real-valued, i.e. its impulse response $h(n)$ is real-valued. This means that the response to the imaginary part of some complex input signal $\hat{x}(n)$ equals the imaginary part of the response to the complex signal $\hat{x}(n)$:

$$\mathcal{T}\{\Im\{\hat{x}(n)\}\}=\Im\{\mathcal{T}\{\hat{x}(n)\}\}$$

where $\mathcal{T}$ is the system operator.

Note that your input signal $x(n)$ is the imaginary part of

$$\hat{x}(n)=Ae^{j(\omega_0n+\theta)}\tag{1}$$

The frequency response is related to $h(n)$ by

$$H(\omega)=\sum_{n=-\infty}^{\infty}h(n)e^{-jn\omega}\tag{2}$$

The output $y(n)$ is given by the following convolution sum

$$y(n)=\sum_{k=-\infty}^{\infty}h(k)x(n-k)\tag{3}$$

Now use (3) with the complex input signal $\hat{x}(n)$ given by (1), make use of (2), and then take the imaginary part to obtain $y(n)$. The response of a system with frequency response $H(\omega)$ to a complex exponential $e^{j(\omega_0n+\theta)}$ is a fundamental result which you should remember.