(*^

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:[font = special1; inactive; preserveAspect; ]
                                                                        Spherical Coordinates

The cell below shows how Mathematica  can be used to convert from spherical to Cartesian coordinates.
;[s]
7:0,0;71,1;93,0;120,2;131,0;156,1;183,0;197,-1;
3:4,13,9,Times,0,12,0,0,0;2,13,9,Times,1,12,0,0,0;1,13,9,Times,2,12,0,0,0;
:[font = input; preserveAspect; ]
x[rho_, phi_, theta_] := rho Sin[phi] Cos[theta]
y[rho_, phi_, theta_] := rho Sin[phi] Sin[theta]
z[rho_, phi_, theta_] := rho Cos[phi]

Cart[rho_, phi_, theta_] := 
   {x[rho, phi, theta], y[rho, phi, theta], z[rho, phi, theta]}
   
Cart[5, Pi/2, Pi/4]
:[font = special1; inactive; preserveAspect; ]
The cell below shows how Mathematica  can be used to convert from Cartesian to spherical coordinates.
;[s]
5:0,0;25,1;36,0;61,2;88,0;102,-1;
3:3,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;1,13,9,Times,1,12,0,0,0;
:[font = input; preserveAspect; ]
Rho[x_, y_, z_]   := Sqrt[x^2 + y^2 + z^2]
S[x_, y_, z_]     := Sqrt[x^2 + y^2]
Phi[x_, y_, z_]    := ArcCos[z/Rho[x, y, z]]
Theta[x_, y_, z_] := If[x >= 0, ArcSin[y/S[x, y, z]],
                                Pi - ArcSin[y/S[x, y, z]]]
                                
Sphere[x_, y_, z_] := 
   N[{Rho[x, y, z], Phi[x, y, z], Theta[x, y, z]}]
   
Sphere[5/Sqrt[2], 5/Sqrt[2], 0]
^*)