Complex Exponential Narrow Band Channel
The equations are a bit wrong. For a signal $x=cos(2\pi f_{c}t+n\pi), n\in \left\{0,1\right\}$ transmitted through a complex exponential channel $h=Ae^{i\theta}$, the received signal $y$ is not $hx$. $h$ is in the complex baseband notation and your signal $x$ is in the bandpass notation. 

To answer your actual question first, $\theta$ is the phase change that is incurred by the signal when it passes through the channel i.e. the delay. And the actual received signal is $y=Re\left\{x_{l}Ae^{i\theta}e^{i2\pi f_ct}\right\}$, where $x_{l}$ is the baseband representation of our signal $x$ which in our case would be something like $x_{l} = e^{i2\pi\frac{(n-1)}{2}}$, where $n\in \left\{0,1\right\}$.