Finding roots for complex numbers polynomial

Question

For finding the solutions for $z^3−5=−12i$,

I calculated $\theta$ as $-67.3^\circ$. I can then write $z=r^{(1/3)}e^{i\theta/3}$ where I can substitute $e^{(i\theta/3)}$ as $\operatorname{cis}(2\pi k/3+θ/3)$ where I took $k$ as $0,\,1,\,2$. Is this right?

Answer 1

Since you don't say what $r$ is, it's hard to say. Anyway, if you take $r=\sqrt{5^2+(-12)^2}=13$, yes, it is correct. A better approximation for $\theta\left(=-\arccos\left(\frac5{13}\right)\right)$ would be $-67.4^\circ$.