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Trigonometry

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Moderate

Complex Number, Trigonometric Form, Quotient

TRIG-GC8GWF

Recalling that $k\times cis\left( \theta \right)$ is an abbreviation for $k\left\{ \cos { \left( \theta \right) +i } \sin { \left( \theta \right) } \right\}$,

...given ${ z }_{ 1 }=4\times cis\left( \cfrac { \pi }{ 8 } \right)$ and ${ z }_{ 2 }=3\times cis\left( \cfrac { 9\pi }{ 16 } \right)$

What is $\cfrac { { z }_{ 1 } }{ { z }_{ 2 } }$? Leave your answer in abbreviated form and restrict the answer to being between $0\le \theta \le 2\pi$.

A

$\cfrac { 4 }{ 3 } cis\left( \cfrac { 25\pi }{ 16 } \right)$

B

$\cfrac { 3 }{ 4 } cis\left( \cfrac { 25\pi }{ 16 } \right)$

C

$\cfrac { 4 }{ 3 } cis\left( -\cfrac { 7\pi }{ 16 } \right)$

D

$\cfrac { 3 }{ 4 } cis\left( -\cfrac { 7\pi }{ 16 } \right)$