Find the 4th root of the following complex number

Question

find all $z$ in polar form satisfying the following

$z^4 = -1 +i\sqrt{3}$

What I did was the following.

Let $z = r(\cos\phi\ +\ i\sin\phi)$

Therefore $z^4 = 2(\cos\frac{\pi}{3} + i\sin(\frac{\pi}{3}))= r^4(\cos4\phi\ +\ i\sin4\phi)$

Therefore by comparism, $r = 2^{1/4}, \ \phi = \frac{\pi}{12} + \frac{n\pi}{2}$ where $n \in \{0,1,-1,2\}$

But my answer seemed to be wrong and I was hoping if you guys can point out my mistake.

Any help or insight is deeply appreciated.

Answer 1

Therefore $z^4 = 2(\cos\frac{\pi}{3} + i\sin(\frac{\pi}{3}))= r^4(\cos4\phi\ +\ i\sin4\phi)$

You seem to use $r = 2$ (which is correct) and $\phi = \tfrac{\pi}{3}$, but clearly $-1+\sqrt{3}i$ with negative real part and positive imaginary part is not in the first quadrant, but in the second; so $\phi = \tfrac{2\pi}{3}$.