 Solution

Let $\psi =kx-\omega t$ and use bold $\mathbf{B}$ for the magnetic flux density, to distinguish it from the scalar amplitude $B$ in $E_z$. For a plane electromagnetic wave travelling along $+x$, Faraday's law gives the wave-associated fieldHenceFor an explicit component check, $\nabla \times \mathbf{E}=(0,kB\sin \psi ,kA\cos \psi )$, while differentiating the displayed field gives $-\partial \mathbf{B}/\partial t$ equal to that vector. The field is transverse and also satisfies $\nabla \cdot \mathbf{B}=0$.

In vacuum, Ampère-Maxwell equation compatibility requires $\omega /k=c$, so each factor $k/\omega$ is $1/c$. More generally it is the inverse phase velocity on a suitable homogeneous-medium plane-wave branch. A time-independent background field could be added without changing the electric wave's Faraday equation; the requested propagating-wave field takes that independent background to be zero. Finally, $\mathbf{E}\times \mathbf{B}=(k/\omega )(E_y^2+E_z^2)\hat{\mathbf{x}}$ confirms the propagation direction and the relative signs.