Sharp values of wavefunction in polar coordinates

[ParoxysmX]

Homework Statement

Consider the function in polar coordinates

ψ(r,θ,\phi) = R(r)sinθe^{i\phi}

Show by direct calculation that ψ returns sharp values of the magnitude and z-component of the orbital angular momentum for any radial function R(r). What are these sharp values?

The Attempt at a Solution

I -think- for L_{z} to be sharp, you have to impose the eigenfunction condition

-i\hbar \frac{dψ}{d\phi} = L_{z}ψ

which means that the radial function R(r) would cancel with itself if you were to rearrange the above for L_{z}. However I could have completely the wrong idea here.

 

[DrClaude]

How about calculating the expectation values corresponding to "the magnitude and z-component of the orbital angular momentum "?