We have the following:
are the complex number
[tex]\begin{gathered} z_1=9cis\frac{5\pi}{6}_{} \\ z_2=3\text{cis}\frac{\pi}{3} \\ \frac{z_1}{z_2} \end{gathered}[/tex]
So magnitudes are r₁ = 9, and r₂ = 3 and arguments are ∅₁ = 5π/6, and ∅₂ = π/3
[tex]\frac{z_1}{z_1}=\frac{r_1}{r_2}\cdot\text{cis(}\emptyset_1\cdot\emptyset_{2})[/tex]
replacing:
[tex]\begin{gathered} \frac{z_1}{z_2}=\frac{9}{3}\cdot\text{cis}(\frac{5\pi}{6}-\frac{\pi}{3}) \\ \frac{z_1}{z_2}=3\cdot\text{cis}(\frac{5\pi}{6}-\frac{2\pi}{6}) \\ \frac{z_1}{z_2}=3\cdot\text{cis}(\frac{3\pi}{6}) \\ \frac{z_1}{z_2}=3\cdot\text{cis}(\frac{\pi}{2})\rightarrow\text{cis}(\frac{\pi}{2})=\cos \mleft(\frac{\pi}{2}\mright)+3i\sin \mleft(\frac{\pi}{2}\mright) \\ \frac{z_1}{z_2}=3\cdot\lbrack\cos (\frac{\pi}{2})+i\sin (\frac{\pi}{2})\rbrack \\ \frac{z_1}{z_2}=3\cdot\lbrack0+i\cdot1)\rbrack \\ \frac{z_1}{z_2}=3\cdot0+3\cdot i \\ \frac{z_1}{z_2}=3i \end{gathered}[/tex]
Therefore, the answer is option D 3i
