Perform the indicated operation using trigonometric form. Leave answer in trigonometric form.
$$( 3 + 3 i ) ( 5 + 5 i )$$
A) $- 15 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$
B) $- 30 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
C) $15 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
D) $30 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
E) $15 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$

Question

Quizplus · Accepted Answer

To multiply complex numbers in trigonometric form, you multiply their magnitudes and add their angles. The magnitude of $3 + 3i$ is $\sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}$, and its angle is $\frac{\pi}{4}$ because it lies on the line $y = x$ in the first quadrant. The magnitude of $5 + 5i$ is $\sqrt{5^2 + 5^2} = \sqrt{50} = 5\sqrt{2}$, and its angle is also $\frac{\pi}{4}$. Multiplying the magnitudes gives $3\sqrt{2} \times 5\sqrt{2} = 30$, and adding the angles gives $\frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2}$. Therefore, the product in trigonometric form is $30 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right)$.

Perform the Indicated Operation Using Trigonometric Form (3+3i)(5+5i)( 3 + 3 i ) ( 5 + 5 i )(3+3i)(5+5i)

Perform the Indicated Operation Using Trigonometric Form $( 3 + 3 i ) ( 5 + 5 i )$