How are these parameters derived from readings of sensors in an aircraft: angle of attack, slip angle, and airspeed?

From what I've learned, the reading comes from

1. an angle of attack sensor and a slip angle sensor. Let's call the readings $$\hat\alpha$$ and $$\hat\beta$$ respectively. These are of either the weathervane type or the differential pressure type.
2. airspeed sensor, i.e. a pitot tube. This probe measures the total and static pressure $$\hat p_\text{total}$$ and $$\hat p_\text{static}$$, and the airspeed $$\hat u$$ is obtained from them.

Here's the catch: in real flight, the measured angles $$\hat\alpha$$ and $$\hat\beta$$ may depend on not only the true values $$\alpha$$ and $$\beta$$ but also other factors which include true airspeed $$u$$, and the measured airspeed $$\hat u$$ may depend on $$\alpha$$ and $$\beta$$ too. How is this problem solved? How are these sensors calibrated? Are they tested in a wind tunnel where every possible combination of $$(\alpha,\beta,u)$$ is given to produce $$(\hat\alpha,\hat\beta,\hat u)$$ and form a lookup table?