finding the total differential in a special orthogonal coordinate system

Question

Solving one important applied physics problem I faced the need to isolate the total differential using some new orthogonal coordinate system. Another words, I have the following expression in spherical coordinate system: $\dfrac{\sin 2\theta {\rm d}\theta}{r}$. It is necessary to choose another curvilinear coordinates $\alpha, \beta, \gamma$ so that $$\dfrac{\sin 2\theta {\rm d}\theta}{r}= {\rm d}f (\alpha, \beta, \gamma) $$. The function $f$ should be as simple as possible (it can be just one new coordinate)