Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form.
-$$\frac { 20 \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right) } { 4 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right) }$$
A) $- 5 \mathrm { i }$
B) $5 \sqrt { 2 } - 5 i \sqrt { 2 }$
C) $5 \mathrm { i }$
D) $- 5 \mathrm { i } \sqrt { 2 }$

Question

Quizplus · Accepted Answer

To find the quotient, divide the magnitudes and subtract the angles in trigonometric form. The magnitude of the numerator is 20, and the magnitude of the denominator is 4, so their quotient is 5. The angle in the numerator is 45 degrees, and the angle in the denominator is 135 degrees, so the difference is -90 degrees. Thus, the quotient in trigonometric form is $5(\cos(-90^\circ) + i\sin(-90^\circ))$, which simplifies to $5(0 - i) = -5i$, corresponding to choice A.

Find the Quotient and Write in Rectangular Form 20(cos⁡45∘+isin⁡45∘)4(cos⁡135∘+isin⁡135∘)\frac { 20 \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right) } { 4 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right) }4(cos135∘+isin135∘)20(cos45∘+isin45∘)​

Find the Quotient and Write in Rectangular Form $\frac { 20 \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right) } { 4 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right) }$