a
\({{\rm{y}}_1} = {\rm{a}}\sin (\omega {\rm{t}} + {\rm{kx}} + 0.57)\)

\(\therefore\) Phase, \(\phi_{1}=(\omega t+k x+0.57)\)

\(y_{2}=\operatorname{acos}(\omega t+k x)=a \sin \left(\omega t+k x+\frac{\pi}{2}\right)\)

\(\therefore\) Phase, \(\phi_{2}=\omega t+k x+\frac{\pi}{2}\)

Phase difference, \(\Delta \phi=\phi_{2}-\phi_{1}\)

\(=\left(\omega t+k x+\frac{\pi}{2}\right)-(\omega t+k x+0.57)\)

\(=\frac{\pi}{2}-0.57=1.57-0.57=1 \mathrm{\,radian}\)