I know it can be explicitly shown that for (i) $u(r,\theta)=c$, and for (ii) $u(r,\theta)=r\sin(\theta)$ by separating variables and finding the Fourier coefficients, but is there any mathematical reasoning behind this without having to tediously compute the coefficients? Again I also know that if the B.C. was $u(1,\theta)=\sin^2(\theta)$, then $u(r,\theta)=\frac{1}{2}(1-r^2\cos(2\theta))$. These solutions seem very intuitive and I wonder if it can be argued that these are the solutions without explicit computation of Fourier coefficients?
Also, how can it be shown that in each case the solutions are unique? I.e. for (i) I'm aware that both the real and imaginary parts of an analytic function are harmonic, hence since $f(z)=c$ is analytic then $\text{Re}(f(z))=c$ solves Laplace's equation and satisfies the boundary condition, but how do we know that this is the only solution? This is as far as my background goes on this, I'm not aware of any additional theorems etc.