Perform the indicated operation using trigonometric form. Leave answer in trigonometric form.
$$( 8 + 8 i ) ( 2 + 2 i )$$
A) $- 16 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$
B) $- 32 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
C) $16 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
D) $32 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
E) $16 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$

Question

Quizplus · Accepted Answer

To multiply complex numbers in trigonometric form, you multiply their magnitudes and add their angles. The magnitude of $8 + 8i$ is $\sqrt{8^2 + 8^2} = \sqrt{128} = 8\sqrt{2}$, and its angle is $\frac{\pi}{4}$ because it lies on the line $y = x$ in the first quadrant. The magnitude of $2 + 2i$ is $\sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}$, and its angle is also $\frac{\pi}{4}$. Multiplying the magnitudes gives $8\sqrt{2} \times 2\sqrt{2} = 32$, and adding the angles gives $\frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2}$. Therefore, the product in trigonometric form is $32 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right)$.

Perform the Indicated Operation Using Trigonometric Form (8+8i)(2+2i)( 8 + 8 i ) ( 2 + 2 i )(8+8i)(2+2i)

Perform the Indicated Operation Using Trigonometric Form $( 8 + 8 i ) ( 2 + 2 i )$