The displacement of a particle executing simple harmonic motion is given by $y=A_0+A\sin \omega t+B\cos \omega t$. Then the amplitude of its oscillation is given by:

The term $A_0$ just represents a shift in the mean position and does not affect the amplitude of the oscillation.The oscillating part is $y^{\prime }=A\sin \omega t+B\cos \omega t$.This can be written as $y^{\prime }=R\sin (\omega t+\varphi )$, where the resultant amplitude $R$ is:
$$R=\sqrt{A^2+B^2+2AB\cos (90^{\circ })}=\sqrt{A^2+B^2}$$
(Since $\sin \omega t$ and $\cos \omega t$ have a phase difference of $90^{\circ }$).