1. Questions

Math/Physics

My question is this:
The potential at the surface of a sphere (radius R) is given by
V_0 = k cos 3theta
where k is constant. Find the potential inside and outside the sphere as well as the surface charge density (lower case sigma(theta)) on the sphere. (assume there is no charge inside or outside the sphere.

So that's my question but the problem is my teacher is really pickey in how students do it and I just want some clarfication so ok. First I did some trig. for the given problem V_0 =3k cos(theta) which I got 4k cos^3(theta) - 3k cos (theta). Then I went to the next step and used the Legendre polynomials which I only used P_1 and P_3 and changed the X's to cosines so P_1 (cos theta) = cos theta and
P_3 (cos theta) = [(5 cos^3 theta)/2 - (3cos theta)/2] which then I had to solve which I did but he did not like that I floated the k around.... since i replaced it as x and y and then in the end I added the k... 4kcos^theta - 3kcos thta = x[(5 cos^3 theta)/2 - (3cos theta)/2] + y cos theta which then I did (5x cos^3 theta)/2 + y - 3x/2 where then I did 4k= 5x/2 and got x = 8k/5 and for the other part I plugged the x to -3k = b- (3x/2) and got b = -3k/5. However he did not like the fact that I kept the k floating around and I do not understand that part....thats the only problem I have...he liked the rest of my work I have done so far but does anyone know what he means? Some clarfication would be a great help! since he did say the outcome (8k/5) P_3 (cos theta) - (3k/5) P_1 (cos theta) was fine but to re-do the way I found the 8k/5 and -3k/5. (Oh and I did find the inside and outside potential and currently finding the surface charge density so I don't need much help in that but any suggestions will be great in knowing which equation would be great to use to find surface potential) Thanks!