Solution

For a plane sinusoidal wave, the general equation of the electromagnetic wave is given by:

\begin{equation} E(z,t)=E_0\sin(At+Bz+\varphi_0) \end{equation}

where $A$,$B$ and $\varphi_0$ are constants.

From the condition of the problem we know that:

\begin{equation} E(z,0)=E_0\sin(\frac{2\pi}{\lambda}z) \end{equation}

Using eq(1) we get:

\begin{equation} E_0\sin{(Bz+\varphi_0)}=E_0\sin(\frac{2\pi}{\lambda}z) \end{equation}

By comparing the phases we obtain that:

\begin{equation} B=\frac{2\pi}{\lambda};\varphi_0=0 \end{equation}

As the wave travels in the positive direction:

\begin{equation} \frac{dz}{dt}=c \end{equation}

Setting the phase($\varphi=At+Bz+\varphi_0$)to a constant value allows us to derive the phase velocity($c$):

\begin{equation} A+B\frac{dz}{dt}=\frac{d\varphi}{dt}=0 \end{equation}

So we get the constant $A$:

\begin{equation} A=-\frac{2\pi c}{\lambda} \end{equation}

Plugging all the constants into the eq(1) gives us:

\begin{equation} E(z,t)=E_0\sin(\frac{2\pi}{\lambda}(z-ct)) \end{equation}