Why do electromagnetic waves have the magnetic and electric field intensities in the same phase?

My question is: in electromagnetic waves, if we consider the electric field as a sine function, the magnetic field will be also a sine function, but I am confused why that is this way.

If I look at Maxwell's equation, the changing magnetic field generates the electric field and the changing electric field generates the magnetic field, so according to my opinion if the accelerating electron generates a sine electric field change, then its magnetic field should be a cosine function because $\frac{d(\sin x)}{dx}=\cos x$.

Answer

E and B are in phase for a running plane wave, but are out of phase for a standing wave. This can be easily seen by considering the vector potential, $A(t, x) $. Using $E = \partial_t A$ and $B=\partial_x A$. For $A=sin(\omega t - kx) $ you find that E and B are in phase. For $A=sin(\omega t) sin(kx) $, a standing wave, E and B are out of phase.