Question

Convert z = (- 5 √3 )/ 2 + (5/2)i to polar coordinates

Answer 1

is there an answer choice

Answer 2

No. I have to show work and idk how to do these

Answer 3

let z=\[\left( \frac{ -5\sqrt{3} }{ 2}+\iota \frac{ 5 }{ 2 } \right)=\left( r \cos \theta +\iota~ r \sin \theta \right)\]
\[r~\cos \theta=\frac{ -5\sqrt{3} }{ 2 }....(1)\]
\[r~\sin \theta=\frac{ 5 }{ 2 } ...(2)\]
square and add
\[r^2 \left( \cos ^2 \theta +\sin ^2 \theta \right)=\frac{ 75 }{ 4 }+\frac{ 25 }{ 4 }=\frac{ 75+25 }{ 4 }=\frac{ 100 }{ 4 }=25\]
r=5
\[5 ~\cos \theta=\frac{ -5\sqrt{3} }{ 2 },\cos \theta=-\frac{ \sqrt{3} }{ 2 }=-\cos \frac{ \pi }{ 6 }=\cos \left( \pi-\frac{ \pi }{ 6 } \right)=\cos \frac{ 5~\pi } { 6 }\]
\[\theta=\frac{ 5~\pi }{ 6 }\]
\[z=\left( 5,\frac{ 5~\pi }{ 6 } \right)\]