Legs, if I'm following (and I'm not convinced I am) - are you saying the difference between the sine and the tangent is that the sine is the distance of the tarmac and the tangent is the distance of the ground beneath the tarmac? Ie. because the tarmac gains altitude, 100m going uphill will not travel as horizontally far as 100m flat? Or the other way round? Or both wrong?
Anyway, to Mr Vandal: percentages kind of work in reverse to the fractions that 515mm describes - 7% means that over 100m you'd gain 7m. That's why 1 in 5 = 20% - if over 5m you gain 1m, then over 100m you'd gain 20m.
As for climb categorisations; it varies from race to race, and they aren't always internally consistent. In the Tour de France you'd be looking at something like this as a base guide (these are very rough and don't take into account fluctuations in the slope, but as a broad rule of thumb they'd be reasonable I think):
Cat. 4 - A 12% average gradient over 0.5 km; down to a 3% average gradient over 2.0-5.0km. Any gradients in between would have to be of a distance somewhere in between.
Cat. 3 - 12% would probably need to be the average over 0.5-1km; and a 3% average would need to be 5-8km.
Cat. 2 - 12% over 1.5-4km; 3% over 8-20km.
Cat. 1 - 12% over 4-6km; 3% over 20-40km.
Cat. H - 12% over 6km+; 3% over 40km+.
You could tone all of these figures down immensely for the Tour of Britain. Category 1 climbs there are often the equivalent of Category 3 climbs in the Tour de France (this is down to the organisers being rubbish, there are some decent climbs in this country as most here will well know).
Like I said, undulations in the slope can distort figures, which is why those are only rough. The
Coll de Pradell is a good example of a climb which going on the figures alone you'd say was first category, but take a look at the profile and you can see it would be an hors category climb.