I have not had an oppurtunity to do any telltale experiments, but it struck me that some of my sessions show a very regular pattern where I can get a very good approximation of the wind direction, and therefore the angle of my runs.
I looked at a recent two last night. I printed them off and drew lines on them using a Protractor to try to estimate the wind direction. In both examples, where I got top speeds just over 40 knots, the angle worked out to 125 degrees off the wind.
Now that surprised me. I thought it would be more like 130 deg..
But I can think of a few areas where my wind direction calculations could have been misled, and also the wind strength was only high 20's to barely 30kts both times.
I have definitly found in the past that stronger winds lead to broader angles as well as faster speeds.
And I may have not been pointing as high on both tacks for other reasons. This could easily account for at least some of the expected difference.
If these estimates are somewhere close, and the gusts on those runs were, lets say 30 knots for simplicity, someone with mathematical abilities should be able to work out the Beta in this example.