Jtdylktuy

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 $frac{d(sin x)}{dx}=cos x$.

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.

The Maxwell equations that relate electric and magnetic fields to each other read (in vacuum, in SI units) as
begin{align}
nabla times mathbf E & = -frac{partialmathbf B}{partial t} \
nabla times mathbf B & = frac{1}{c^2} frac{partialmathbf E}{partial t},
end{align}
where the notation $nabla times{cdot}$ is a spatial derivative (the curl). This means that both sides have derivatives, and if you're applying them to a function like $cos(kx-omega t)$, then they will both change the cosine into a sine. This is what locks the phase of both waves to equal values.

This is one of those 'why' questions that physics can or cannot answer, depending on what you want from answer to 'why'.

If equations are a satisfactory explanation, then the Maxwells Equation in Emilios answer are a complete answer.

Unfortunately, not far beneath the surface of that answer is 'why do Maxwells Equations' fit reality?' or 'why do fields behave the way they do so that we can derive Maxwells Equations?'. Wigner along with many other physicists was similarly troubled by such questions.

It doesn't get any more intuitive if you go down further to QED to try to explain the classical behaviour.

At the lowest level, the answer is 'that's the way Nature behaves'.

The E and H fields in a time-harmonic EM wave are in phase in the time domain when the medium's polarization (electric and magnetic) are in phase with the corresponding fields. You can see that polarization fields inherently act as 'source' terms in Maxwell's equations, and hence, instantaneous polarization implies in-phase relationship. However, whenever there is dissipation (such as existence of conduction current, or out of phase polarization), the E and H fields are no longer in phase. In other words, the one phasor cannot respond instantaneously to the changes of the second one in time. Note that regardless of propagating or standing wave, E and H fields are in phase with each other in the time domain in a lossless medium (for a standing wave, they are 'out of phase' spatially).