7.35 is nothing more than a definition of spherical coordinates.

⎧    x = r sinθ cosϕ
⎪
⎨    y = r sinθ sinϕ
⎪
⎩    z = r cosθ

Some differential forms may come in handy.

∂/∂θ:
⎧    ∂x = r cosϕ cosθ ∂θ
⎪
⎨    ∂y = r sinϕ cosθ ∂θ
⎪
⎩    ∂z = - r sinθ ∂θ

∂/∂ϕ:
⎧    ∂x = - r sinθ sinϕ ∂ϕ
⎪
⎨    ∂y = r sinθ cosϕ ∂ϕ
⎪
⎩    ∂z = 0 ∂ϕ


7.47 is the set of algebraic conditions expressed by the vector definition 𝐋 = 𝐫 × 𝐩.

⎧    L̂𝓍 = yp𝓏 - zp𝓎 = -ιħ (y ∂͟_  - z ∂͟_ )
⎪                           ∂z      ∂y
⎪
⎨    L̂𝓎 = zp𝓍 - xp𝓏 = -ιħ (z ∂͟_  - x ∂͟_ )
⎪                           ∂x      ∂z
⎪
⎪    L̂𝓏 = xp𝓎 - yp𝓍 = -ιħ (x ∂͟_  - y ∂͟_ )
⎩                           ∂y      ∂x

Substituting 7.35 into 7.47,

⎧    L̂𝓍 = -ιħ (r sinθ sinϕ ∂͟_  - r cosθ ∂͟_ )
⎪                          ∂z           ∂y
⎪
⎨    L̂𝓎 = -ιħ (r cosθ ∂͟_  - r sinθ cosϕ ∂͟_ )
⎪                     ∂x                ∂z
⎪
⎪    L̂𝓏 = -ιħ (r sinθ cosϕ ∂͟_ - r sinθ sinϕ ∂͟_ )
⎩                          ∂y               ∂x



Geometry is shown on the attached notes page.
    
For L̂𝓍:

    ∂͟z͟ = -r sinθ
    ∂θ
    ∂͟y͟ = r sinθ cosϕ 
    ∂ϕ

    L̂𝓍 = -ιħ ( r sinθ sinϕ ∂͟θ͟ ∂͟_  - r cosθ ∂͟ϕ͟ ∂͟_ )
                           ∂θ ∂z           ∂ϕ ∂y    

    L̂𝓍 = ιħ ( -r sinθ sinϕ ∂͟θ͟ ∂͟_  + r cosθ ∂͟ϕ͟ ∂͟_ )
                           ∂z ∂θ           ∂y ∂ϕ

    L̂𝓍 = ιħ ( −͟r͟ s͟i͟n͟θ͟ sinϕ ∂͟_ +   c͟o͟s͟θ͟    ∂͟_ )
              -r sinθ      ∂θ  sinθ cosϕ  ∂ϕ

    L̂𝓍 = ιħ ( sinϕ ∂͟_ + c͟o͟t͟θ͟ ∂͟_ )
                   ∂θ   cosϕ ∂ϕ                   

For L̂𝓎:

    ∂x = r cosϕ cosθ ∂θ
    ∂z = 0 ∂ϕ

    L̂𝓎 = -ιħ (r cosθ ∂͟_  - r sinθ cosϕ ∂͟_ )
                     ∂x                ∂z

    L̂𝓎 = ιħ (-r cosθ ∂͟θ͟ ∂͟_  + r sinθ cosϕ ∂͟ϕ͟ ∂͟_ )
                     ∂θ ∂x                ∂ϕ ∂z

    L̂𝓎 = ιħ (-r cosθ ∂͟θ͟ ∂͟_  + r sinθ cosϕ ∂͟ϕ͟ ∂͟_ )
                     ∂x ∂θ                ∂z ∂ϕ

    L̂𝓎 = ιħ (- _͟1͟  cosθ ∂͟_  + r sinθ cosϕ 0 ∂͟_ )
              cosϕ cosθ ∂θ                  ∂ϕ

    L̂𝓎 = ιħ (- _͟1͟  ∂͟_ )
              cosϕ ∂θ

    L̂𝓎 = -ιħ _͟1͟  ∂͟_ 
            cosϕ ∂θ

For L̂𝓏:

    ∂x = -r sinθ sinϕ ∂ϕ
    ∂y = r sinθ cosϕ ∂ϕ

    L̂𝓏 = -ιħ (r sinθ cosϕ ∂͟͟ϕ͟ ∂͟_ - r sinθ sinϕ ∂͟͟ϕ͟ ∂͟_ )
                          ∂ϕ ∂y               ∂ϕ ∂x

    L̂𝓏 = -ιħ (r͟ s͟i͟n͟θ͟ c͟o͟s͟ϕ͟ ∂͟_ + r͟ s͟i͟n͟θ͟ s͟i͟n͟ϕ͟ ∂͟_ )
              r sinθ cosϕ ∂ϕ   r sinθ sinϕ ∂ϕ 

    L̂𝓏 = -ιħ ∂͟_ ( 1 + 1 )
             ∂ϕ       

    L̂𝓏 = -2ιħ ∂͟_ 
              ∂ϕ

So, according to my calculus, the final solutions should be the set

⎧    L̂𝓍 = ιħ ( sinϕ ∂͟_ + c͟o͟t͟θ͟ ∂͟_ )
⎪                   ∂θ   cosϕ ∂ϕ          
⎪
⎨    L̂𝓎 = -ιħ _͟1͟  ∂͟_ 
⎪            cosϕ ∂θ
⎪
⎪    L̂𝓏 = -2ιħ ∂͟_ 
⎩              ∂ϕ
   

The spherical representation, i.e. ending place, is the set

⎧    L̂𝓍 = ιħ (sinϕ ∂͟_  + cosϕ cotθ ∂͟_ )
⎪                  ∂θ              ∂ϕ
⎪
⎨    L̂𝓎 = ιħ (-cosϕ ∂͟_ + sinϕ cotθ ∂͟_ )
⎪                   ∂θ             ∂ϕ
⎪
⎪    L̂𝓏 = -ιħ ∂͟_ 
⎩             ∂ϕ


My set DOES NOT match this. I must be going about this the wrong way. I have to give it more thought. Perhaps a purely geometric approach will improve my answers: I'll try that over the weekend.