Perform the indicated operation using trigonometric form.Leave answer in trigonometric form. $( - 6 - 6 i ) ( - 5 - 5 i )$
A) $30 \left( \cos \frac { \pi } { 8 } + i \sin \frac { \pi } { 8 } \right)$
B) $- 60 \left( \cos \frac { \pi } { 8 } + i \sin \frac { \pi } { 8 } \right)$
C) $30 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
D) $60 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
E) $- 30 \left( \cos \frac { \pi } { 8 } + i \sin \frac { \pi } { 8 } \right)$

Question

Perform the indicated operation using trigonometric form.Leave answer in trigonometric form. $( - 6 - 6 i ) ( - 5 - 5 i )$
A) $30 \left( \cos \frac { \pi } { 8 } + i \sin \frac { \pi } { 8 } \right)$  
B) $- 60 \left( \cos \frac { \pi } { 8 } + i \sin \frac { \pi } { 8 } \right)$  
C) $30 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$  
D) $60 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$  
E) $- 30 \left( \cos \frac { \pi } { 8 } + i \sin \frac { \pi } { 8 } \right)$

Quizplus · Accepted Answer

To multiply complex numbers in trigonometric form, multiply their magnitudes and add their angles. The magnitude of $(-6-6i)$ and $(-5-5i)$ are both $\sqrt{6^2 + 6^2} = 6\sqrt{2}$ and $\sqrt{5^2 + 5^2} = 5\sqrt{2}$ respectively, and their angles are both $225^\circ$ or $\frac{5\pi}{4}$ radians since they lie in the third quadrant. Multiplying their magnitudes gives $6\sqrt{2} \times 5\sqrt{2} = 60$, and adding their angles gives $225^\circ + 225^\circ = 450^\circ$, which simplifies to $90^\circ$ or $\frac{\pi}{2}$ radians in the standard range of angles. Therefore, the product in trigonometric form is $60(\cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2}))$.

Perform the Indicated Operation Using Trigonometric Form (−6−6i)(−5−5i)( - 6 - 6 i ) ( - 5 - 5 i )(−6−6i)(−5−5i)

Perform the Indicated Operation Using Trigonometric Form $( - 6 - 6 i ) ( - 5 - 5 i )$