So I'm asked to find the square-root of i which has negative imaginary part. Thinking geometrically, I can see immediately that this has modulus 1 and angle 5pi/4.
However, I started answering this by equating the square of the solution to i, hence z^2 = i = e^2(i theta + r) = e^(i pi/2). I then reason that 2i theta + 2r = i pi/2 thus r = 0 (and hence modulus is 1, as expected) and theta = pi/4. Now this is obviously a solution, but why does it seem from my calculation that there is only one solution? At what step are there really two ways to solve for theta when I thought there was only one?