Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal
places.
-[8( $\left. \left. \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) \right] \left[ 3 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) \right]$
A) $- 5.5000 + 9.5263 \mathrm { i }$
B) $- 0.5000 + 0.8660 \mathrm { i }$
C) $- 12.000 + 20.7846 \mathrm { i }$
D) $- 1.0000 + 1.7321 \mathrm { i }$

Question

Quizplus · Accepted Answer

This problem involves multiplying two complex numbers in polar form. The given complex numbers are $8(\cos 30^\circ + i\sin 30^\circ)$ and $3(\cos 90^\circ + i\sin 90^\circ)$. First, convert the angles to radians or use their sine and cosine values directly:- $\cos 30^\circ = \sqrt{3}/2$ and $\sin 30^\circ = 1/2$- $\cos 90^\circ = 0$ and $\sin 90^\circ = 1$The multiplication of the two complex numbers in rectangular form is:$$8(\sqrt{3}/2 + i/2) \times 3(0 + i) = 24(\sqrt{3}/2i + 1/2i^2)$$Since $i^2 = -1$, the expression simplifies to:$$24(\sqrt{3}/2i - 1/2) = 12\sqrt{3}i - 12$$Converting $\sqrt{3}$ to its decimal form ($\approx 1.732$):$$12 \times 1.732i - 12 = 20.784i - 12$$Thus, the answer in rectangular form is $-12 + 20.784i$, which matches choice C: $- 12.000 + 20.7846 \mathrm { i }$.

Perform the Indicated Operation cos⁡30∘+isin⁡30∘)][3(cos⁡90∘+isin⁡90∘)]\left. \left. \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) \right] \left[ 3 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) \right]cos30∘+isin30∘)][3(cos90∘+isin90∘)]

Perform the Indicated Operation $\left. \left. \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) \right] \left[ 3 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) \right]$