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Trigonometry

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Moderate

Complex, Product, Trigonometric, Standard, Conversion

TRIG-@DJU9I

Given the two complex numbers ${ z }_{ 1 }=\cfrac { 1 }{ 2 } \left[ \cos { \left( \cfrac { \pi }{ 3 } \right) } +i\sin { \left( \cfrac { \pi }{ 3 } \right) } \right]$ and ${ z }_{ 2 }=6\left[ \cos { \left( \cfrac { 5\pi }{ 6 } \right) } +i\sin { \left( \cfrac { 5\pi }{ 6 } \right) } \right]$,

Find ${ { z }_{ 1 }\times z }_{ 2 }$. Write your answer in standard form.

A

$- \cfrac { 3\sqrt { 3 } }{ 2 } -\cfrac { 3 }{ 2 } i$

B

$- \cfrac { \sqrt { 3 } }{ 2 } -\cfrac { 1 }{ 2 } i$

C

$\cfrac { 3\sqrt { 3 } }{ 2 } +\cfrac { 3 }{ 2 } i$

D

$-\cfrac { 3 }{ 2 } +\cfrac { 3\sqrt { 3 } }{ 2 } i$