Perform the indicated operation using trigonometric form.Leave the answer in trigonometric forms. $$( 2 + 2 i ) ( - 1 - i )$$
A) $- 2 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
B) $4 \left( \cos \frac { 3 \pi } { 2 } - i \sin \frac { 3 \pi } { 2 } \right)$
C) $4 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
D) $- 2 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$
E) $2 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$

Question

Perform the indicated operation using trigonometric form.Leave the answer in trigonometric forms. $$( 2 + 2 i ) ( - 1 - i )$$ ​
A) $- 2 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
B)​ $4 \left( \cos \frac { 3 \pi } { 2 } - i \sin \frac { 3 \pi } { 2 } \right)$
C)​ $4 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
D)​ $- 2 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$
E)​ $2 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$

Quizplus · Accepted Answer

To multiply complex numbers in trigonometric form, we first convert the given complex numbers to their trigonometric forms. The complex number $2 + 2i$ has a magnitude of $\sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}$ and an angle of $\frac{\pi}{4}$ because it forms a 45-degree angle in the first quadrant. Thus, its trigonometric form is $2\sqrt{2}(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4})$.The complex number $-1 - i$ has a magnitude of $\sqrt{(-1)^2 + (-1)^2} = \sqrt{2}$ and an angle of $\frac{5\pi}{4}$ because it lies in the third quadrant, making a 45-degree angle with the negative x-axis. Its trigonometric form is $\sqrt{2}(\cos\frac{5\pi}{4} + i\sin\frac{5\pi}{4})$.Multiplying these gives a new magnitude of $2\sqrt{2} \times \sqrt{2} = 4$ and an angle of $\frac{\pi}{4} + \frac{5\pi}{4} = \frac{6\pi}{4} = \frac{3\pi}{2}$. Therefore, the product in trigonometric form is $4(\cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2})$.

Perform the Indicated Operation Using Trigonometric Form (2+2i)(−1−i)( 2 + 2 i ) ( - 1 - i )(2+2i)(−1−i)

Perform the Indicated Operation Using Trigonometric Form $( 2 + 2 i ) ( - 1 - i )$