Why is it impossible to estimate the inclination of an exoplanet with the Radial Velocity Method? The formula of the radial velocity is:

$$v_i= V - K(\sin(f_i+\omega)\ + e\sin\omega),$$ with
$$K = \frac{m_p}{m_s+m_p}\frac{na\sin i}{\sqrt{1-e^2}}.$$

Being $V$the systematic velocity of the system,
$m_p$ the mass of the planet, $m_s$ the mass of the star, $n = \frac{2\pi}{P}$ the mean motion and $P$ is orbital period of planet, $a$ the length of the semi-major axis of the planet, $i$ the inclination of the orbital plane with the ecliptic,
$e$ the eccentricity of the planet, $f_i$ the true anomaly at time $t_i$ and $\omega$ the longitude of periastron.

It's well known the usual procedure. You have data about the instances $t_i$, the radial velocities $v_i$ at time $t_i$ and then you also have the measurements uncertainties $\sigma_i$. With that data, you can estimate 6 parameters: $V$, $K$, $e$, $\omega$, $T$ (the periastron passage time) and $P$. After you estimate these parameters, you can calculate (if you know the mass of the star and you assume that $m_p&lt;&lt;m_s$), $m_p\sin i$, which is known as the minimum mass. So, as far as this way shows, you got no way to estimate the mass of the planet and the inclination separately. 

But what if instead of estimate this set of parameters, we introduce another one? Just one with all the parameters in the equation: $V$, $m_p$, $i$, $e$, $T$, $\omega$, etc.? Couldn't that be possible? But nonetheless, it's always assumed that it's impossible to know the inclination with the RV data. Is there some reason why?
Essentially the problem comes down to geometry: radial velocity provides a 1-dimensional view of a 3-dimensional system. See the diagram below:



The observer sees only the component of the velocity vectors along the line from the observer to the star. That means that they cannot distinguish between the two blue velocity vectors: they will see only the purple (radial) component. This means a larger velocity vector that is tilted further away from the line-of-sight can masquerade as a smaller vector in the line of sight.

No matter how you define your variables, you will always run into this issue because it is inherent to the geometry of the situation.

This does break down in the case of planet-planet interactions, which can break the degeneracy here (a good example of this is Gliese 876).