Impulse Response of a LTI system
There's no need for the explicit use of the Fourier or Laplace transform. I think the most straightforward way is to rewrite your input signal as a sum of cosine functions:

$$x(t)=3(\cos(2t)+\cos(4t))$$

Since you (should) know that your filter is an ideal low pass filter with cut-off frequency \$\omega_c=3\$ you know immediately that the term \$\cos(4t)\$ will be completely suppressed, whereas the term \$\cos(2t)\$ will appear at the output just with a scaling. I'm sure you can determine that scaling yourself. So your output signal is simply

$$y(t)=A\cos(2t)$$

where \$A\$ is determined by the original scaling of the input signal and by the scaling of the low pass filter.