Question

For z1 = 9cis 5pi/6 and z2=3cis pi/3, find z1/z2 in rectangular form

Answer 1

We have the following:

are the complex number

`\begin{gathered} z_1=9cis(5\pi)/(6)_{} \\ z_2=3\text{cis}(\pi)/(3) \\ (z_1)/(z_2) \end{gathered}`

So magnitudes are r₁ = 9, and r₂ = 3 and arguments are ∅₁ = 5π/6, and ∅₂ = π/3

`(z_1)/(z_1)=(r_1)/(r_2)\cdot\text{cis(}\emptyset_1\cdot\emptyset_(2))`

replacing:

`\begin{gathered} (z_1)/(z_2)=(9)/(3)\cdot\text{cis}((5\pi)/(6)-(\pi)/(3)) \\ (z_1)/(z_2)=3\cdot\text{cis}((5\pi)/(6)-(2\pi)/(6)) \\ (z_1)/(z_2)=3\cdot\text{cis}((3\pi)/(6)) \\ (z_1)/(z_2)=3\cdot\text{cis}((\pi)/(2))\rightarrow\text{cis}((\pi)/(2))=\cos \mleft((\pi)/(2)\mright)+3i\sin \mleft((\pi)/(2)\mright) \\ (z_1)/(z_2)=3\cdot\lbrack\cos ((\pi)/(2))+i\sin ((\pi)/(2))\rbrack \\ (z_1)/(z_2)=3\cdot\lbrack0+i\cdot1)\rbrack \\ (z_1)/(z_2)=3\cdot0+3\cdot i \\ (z_1)/(z_2)=3i \end{gathered}`

Therefore, the answer is option D 3i