Why is the change in angle of precession equal to dL/L?

[trobinson41]

If a rapidly rotating gyroscope attached to a perch is released in a horizontal position -- that is, so that the axis of the gyroscope is horizontal -- the gyroscope will precess around it's perch. Let L represent its angular momentum. Let dL represent a small change in that momentum. Let dA represent the corresponding change in the angle of L. According to all the references I've found, dA = dL/L. I don't understand this. Shouldn't dL/L be the change in the sine of the angle, not the angle? See attached diagram. Thanks.

 

[RUber]

It looks like dL and dA are both measured in radial coordinates.
dL = L * dA will give you the appropriate measure to arrive at the point L(t + dt) in the illustration. If you were to use the sine, you would be headed to a point more than |L| away from the center.

 

[trobinson41]

Your mention of radial coordinates jogged my memory. The angle in radians is the arc length / radius. The radius in this case is L. Since the arc length approaches dL as the length of dL decreases, dL/L approaches dA. I think that's the reasoning that's being used. Thanks for your help.