Output of an LTI system given its transfer function and input
You cannot solve this problem using the Laplace transform. The reason is that the Laplace transform of the input signal doesn't exist. You could use the Fourier transform, but in this case there's an even simpler way to determine the output signal. You need to know one important property of linear time-invariant (LTI) systems: their response to a sinusoidal input is a sinusoidal signal with the same frequency, but with its amplitude and phase altered according to the system's frequency response evaluated at the input frequency.

So for an input signal

$$x(t)=A\sin(\omega_0t+\phi)\tag{1}$$

the output is given by

$$y(t)=A\big|H(j\omega_0)\big|\sin\left(\omega_0t+\phi+\arg\big\{H(j\omega_0)\big\}\right)\tag{2}$$

where

$$H(j\omega)=\big|H(j\omega)\big|e^{j\arg\{H(j\omega_0)\}}\tag{3}$$

is the system's frequency response.