Sideslip vs Freestream Azimuth Angle Why is the sideslip angle, $\beta$, defined as 

$$\beta=sin^{-1}\left(\frac{v}{|{\bf v}|} \right)$$

and not the same as freestream azimuth angle $\psi$

$$\psi=tan^{-1}\left(\frac{v}{u} \right)$$

where ${\bf v}=u \hat {\bf b}_x+v \hat {\bf b}_y+w \hat {\bf b}_z$ is the airspeed of the aircraft in body coordinates, $\hat {\bf b}$, and $|\cdot|$ is euclidean norm of $\bf v$. $\psi$ can be interpreted as the angle between $\hat {\bf b}_x$ and the projection of $\bf v$ onto the bodies $x$-$y$ plane. What is the physical interpretation of $\beta$?

Thanks in advance!
Dispensing with the math, because it makes my head hurt...

Azimuth refers to geographic direction relative to a compass reference point, so azimuth angle refers to the deviation between the body's longitudinal axis and its horizontal path parallel to the earth's surface (heading vs track along the ground).

Sideslip angle refers to the deviation, referenced along the lateral axis of the body (parallel to the wings in other words), between the body's longitudinal axis and the airflow, independently of direction relative to the earth.