PDE's - Finding a certain solution to Laplace's equation on a circle

Find a solution of Laplace's equation of the form which satisfies the boundary condition for all points on the circle, .

first, I found and

From and the above results, I can get .

Now, I plugged in the boundary condition:

I tried various trig substitutions here and couldn't seem to get anywhere. However, with this equation and the one above, I have two equations (but there are three unknowns). I am pretty sure I have to use the to write another equation so that I can solve for A, B, and C, but I do not know how to use the circle information.

Any help would be greatly appreciated.

Thanks in advance.