$x={A} \sin \omega {t}+{B} \sin \omega {t}$

${v} ={A} \sin \omega {t}+{Bcos} \omega {t}$

${{dt}}={A} \omega \cos \omega {t}-{B} \omega \sin \omega {t}$

${At} {t}=0, {x}(0)={B}$

${v}(0)={A} \omega$

${x}={A} \sin \omega {t}+{B} \sin \left(\omega {t}+90^{\circ}\right)$

$A_{\text {net }}=\sqrt{A^{2}+B^{2}}$

$\tan \alpha=\frac{B}{A} \Rightarrow \cot \alpha=\frac{A}{B}$

$\Rightarrow \quad x=\sqrt{A^{2}+B^{2}} \sin (\omega t+\alpha)$

$\Rightarrow \quad x=\sqrt{A^{2}+B^{2}} \cos (\omega t-(90-\alpha))$

$x=C \cos (\omega t-\phi)$

$\Rightarrow C=\sqrt{A^{2}+B^{2}}$

$C= \sqrt{\frac{[v(0)]^{2}}{\omega^{2}}+[x(0)]^{2}}$

$\phi= 90-\alpha$

$\tan \alpha=\cos \alpha=\frac{A}{B}$

$\Rightarrow \tan \phi=\frac{v(0)}{x(0) \cdot \omega}$

$\phi= \tan ^{-1}\left(\frac{v(0)}{x(0) \omega}\right)$