Perform the operation and leave the result in trigonometric form. $\frac { 2 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) } { 3 \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right) }$
A) $\frac { 2 } { 3 } \left( \cos 270 ^ { \circ } + i \sin 270 ^ { \circ } \right)$
B) $\frac { 2 } { 3 } \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$
C) $\frac { 2 } { 3 } \left( \cos 270 ^ { \circ } - i \sin 270 ^ { \circ } \right)$
D) $\frac { 2 } { 3 } \left( \cos 30 ^ { \circ } - i \sin 30 ^ { \circ } \right)$
E) $\frac { 2 } { 3 } \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right)$

Question

Perform the operation and leave the result in trigonometric form.   $\frac { 2 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) } { 3 \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right) }$  
A) $\frac { 2 } { 3 } \left( \cos 270 ^ { \circ } + i \sin 270 ^ { \circ } \right)$
B)  $\frac { 2 } { 3 } \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$
C) $\frac { 2 } { 3 } \left( \cos 270 ^ { \circ } - i \sin 270 ^ { \circ } \right)$
D)  $\frac { 2 } { 3 } \left( \cos 30 ^ { \circ } - i \sin 30 ^ { \circ } \right)$
E) $\frac { 2 } { 3 } \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right)$

Quizplus · Accepted Answer

The division of complex numbers in trigonometric form is performed by dividing the magnitudes and subtracting the angles. Here, the magnitudes are $2/3$ and the angle difference is $30^\circ - 120^\circ = -90^\circ$, which corresponds to $270^\circ$ in the unit circle.