Trigonometry

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Moderate

Complex, DeMoivre’s Theorem, Standard Form

TRIG-VA97LB

Given the complex number $ z=3\left[ \cos { \left( \cfrac { \pi }{ 12 } \right) } +i\sin { \left( \cfrac { \pi }{ 12 } \right) } \right]$,

What is $ { z }^{ 4 }$ in standard form?

A

${ z }^{ 4 }=\cfrac { 1 }{ 2 } +\cfrac { \sqrt { 3 } }{ 2 } i$

B

${ z }^{ 4 }=\cfrac { 81 }{ 2 } +\cfrac { 81\sqrt { 3 } }{ 2 } i$

C

${ z }^{ 4 }=\cfrac { 3 }{ 2 } +\cfrac { 3\sqrt { 3 } }{ 2 } i$

D

${ z }^{ 4 }=\cfrac { 81\sqrt { 3 } }{ 2 } +\cfrac { 81 }{ 2 } i$