YYZ,
You will notice in the Bristol notes, that they provide a table of values for the various angles in the progress tests. If you use those then you will always get the same answer as those listed, else you may need to do a bit of rounding.
Correct me if I am wrong somebody but...
...As for how this Sin, Cos and Tan numbers are calculated. They are the various ratios of the opposite, adjacent and hypotenuse (spelling?).
Sin A = Opp/Hyp
Cos A = Adj/Hyp
Tan A = Opp/Adj
Just remeber SOHCAHTOA.
Also:
Tan A = Sin A / Cos A (and the transpositions thereof)
So...
If in the northen hemisphere you were to choose a point along the GC track and then draw line perpendicular to it to intersect with a line drawn along the Rhumb line track and measure the distance from the start point to the right angle and from the start point to the point of intersection, divide those distances (the GC track is the Opp and the RL track is the Hyp) you would get the Sin of the angle of convergence. Obviously this is utterly impractical so they porivide the tables or you provide the calculator with the right buttons.
Or something like that!!