My textbook gives the following derivation of the angular momentum of gyroscopic precession:

> 
> The magnitude of the torque is $$\tau = r M g\sin\theta.$$
> Thus $$dL = r M g\sin\theta\, dt.$$
> The angle the top precesses through in time $dt$ is $$d\phi=\frac{dL}{L\sin\theta}
> =\frac{rMg\sin\theta}{L\sin\theta}dt=\frac{rMg}{L}dt.$$
> The precession angular velocity is $\omega\_P = \frac{d\phi}{dt}$ and from this equation we see that $$\omega\_P = \frac{rMg}{L}= \frac{rMg}{I \omega}.$$
> 
> 
> 

[(source)](https://openstax.org/books/university-physics-volume-1/pages/11-4-precession-of-a-gyroscope)

The derivation on Wikipedia is similar.

I am confused by the use of differentials here. I can see that in the first two lines, the authors have used the fact that $\tau = \frac{dL}{dt}$. But why, in the third line, is $d\phi = \frac{dL}{L\sin\theta}$?

There are many questions on this site seeking a conceptual understanding of precession. I think I understand the concept (how the angular momentum vector changes under the influence of gravity), but I am confused by this derivation.