Question
Given z subscript 1 baseline equals 2 left-parenthesis cosine Start Fraction pi over 6 End Fraction plus i sine Start Fraction pi over 6 End Fraction right-parenthesis and z subscript 2 baseline equals 3 left-parenthesis cosine Start Fraction pi over 4 End Fraction plus i sine Start Fraction pi over 4 End Fraction right-parenthesis, find z subscript 1 baseline z subscript 2 baseline where 0 less-than-or-equal-to theta less-than 2 pi.
(1 point)
Responses
z subscript 1 baseline z subscript 2 baseline equals 6 left-parenthesis cosine Start Fraction 11 pi over 12 End Fraction plus i sine Start Fraction 11 pi over 12 End Fraction right-parenthesis
Image with alt text: z subscript 1 baseline z subscript 2 baseline equals 6 left-parenthesis cosine Start Fraction 11 pi over 12 End Fraction plus i sine Start Fraction 11 pi over 12 End Fraction right-parenthesis
z subscript 1 baseline z subscript 2 baseline equals 5 left-parenthesis cosine Start Fraction pi over 24 End Fraction plus i sine Start Fraction pi over 24 End Fraction right-parenthesis
Image with alt text: z subscript 1 baseline z subscript 2 baseline equals 5 left-parenthesis cosine Start Fraction pi over 24 End Fraction plus i sine Start Fraction pi over 24 End Fraction right-parenthesis
z subscript 1 baseline z subscript 2 baseline equals 5 left-parenthesis cosine Start Fraction 5 pi over 12 End Fraction plus i sine Start Fraction 5 pi over 12 End Fraction right-parenthesis
Image with alt text: z subscript 1 baseline z subscript 2 baseline equals 5 left-parenthesis cosine Start Fraction 5 pi over 12 End Fraction plus i sine Start Fraction 5 pi over 12 End Fraction right-parenthesis
z subscript 1 baseline z subscript 2 baseline equals 6 left-parenthesis cosine Start Fraction 5 pi over 12 End Fraction plus i sine Start Fraction 5 pi over 12 End Fraction right-parenthesis
(1 point)
Responses
z subscript 1 baseline z subscript 2 baseline equals 6 left-parenthesis cosine Start Fraction 11 pi over 12 End Fraction plus i sine Start Fraction 11 pi over 12 End Fraction right-parenthesis
Image with alt text: z subscript 1 baseline z subscript 2 baseline equals 6 left-parenthesis cosine Start Fraction 11 pi over 12 End Fraction plus i sine Start Fraction 11 pi over 12 End Fraction right-parenthesis
z subscript 1 baseline z subscript 2 baseline equals 5 left-parenthesis cosine Start Fraction pi over 24 End Fraction plus i sine Start Fraction pi over 24 End Fraction right-parenthesis
Image with alt text: z subscript 1 baseline z subscript 2 baseline equals 5 left-parenthesis cosine Start Fraction pi over 24 End Fraction plus i sine Start Fraction pi over 24 End Fraction right-parenthesis
z subscript 1 baseline z subscript 2 baseline equals 5 left-parenthesis cosine Start Fraction 5 pi over 12 End Fraction plus i sine Start Fraction 5 pi over 12 End Fraction right-parenthesis
Image with alt text: z subscript 1 baseline z subscript 2 baseline equals 5 left-parenthesis cosine Start Fraction 5 pi over 12 End Fraction plus i sine Start Fraction 5 pi over 12 End Fraction right-parenthesis
z subscript 1 baseline z subscript 2 baseline equals 6 left-parenthesis cosine Start Fraction 5 pi over 12 End Fraction plus i sine Start Fraction 5 pi over 12 End Fraction right-parenthesis
Answers
GPT-4o mini
To find \( z_1 z_2 \), we use the properties of complex numbers in polar form.
Given:
\[
z_1 = 2 \left( \cos\frac{\pi}{6} + i \sin\frac{\pi}{6} \right)
\]
\[
z_2 = 3 \left( \cos\frac{\pi}{4} + i \sin\frac{\pi}{4} \right)
\]
In polar form, multiplying two complex numbers involves multiplying their magnitudes and adding their angles:
1. **Magnitude of \( z_1 z_2 \)**:
\[
|z_1 z_2| = |z_1| \cdot |z_2| = 2 \cdot 3 = 6
\]
2. **Angle of \( z_1 z_2 \)**:
\[
\theta_1 + \theta_2 = \frac{\pi}{6} + \frac{\pi}{4}
\]
To add these angles, we need a common denominator. The least common multiple of 6 and 4 is 12:
\[
\frac{\pi}{6} = \frac{2\pi}{12}, \quad \frac{\pi}{4} = \frac{3\pi}{12}
\]
Therefore:
\[
\theta_1 + \theta_2 = \frac{2\pi}{12} + \frac{3\pi}{12} = \frac{5\pi}{12}
\]
Now we can write \( z_1 z_2 \) in polar form:
\[
z_1 z_2 = 6 \left( \cos \frac{5\pi}{12} + i \sin \frac{5\pi}{12} \right)
\]
Thus, the final answer is:
\[
z_1 z_2 = 6 \left( \cos \frac{5\pi}{12} + i \sin \frac{5\pi}{12} \right)
\]
This corresponds to the option:
\[
\text{z subscript 1 baseline z subscript 2 baseline equals } 6 \left( \cos \frac{5\pi}{12} + i \sin \frac{5\pi}{12} \right)
\]
Given:
\[
z_1 = 2 \left( \cos\frac{\pi}{6} + i \sin\frac{\pi}{6} \right)
\]
\[
z_2 = 3 \left( \cos\frac{\pi}{4} + i \sin\frac{\pi}{4} \right)
\]
In polar form, multiplying two complex numbers involves multiplying their magnitudes and adding their angles:
1. **Magnitude of \( z_1 z_2 \)**:
\[
|z_1 z_2| = |z_1| \cdot |z_2| = 2 \cdot 3 = 6
\]
2. **Angle of \( z_1 z_2 \)**:
\[
\theta_1 + \theta_2 = \frac{\pi}{6} + \frac{\pi}{4}
\]
To add these angles, we need a common denominator. The least common multiple of 6 and 4 is 12:
\[
\frac{\pi}{6} = \frac{2\pi}{12}, \quad \frac{\pi}{4} = \frac{3\pi}{12}
\]
Therefore:
\[
\theta_1 + \theta_2 = \frac{2\pi}{12} + \frac{3\pi}{12} = \frac{5\pi}{12}
\]
Now we can write \( z_1 z_2 \) in polar form:
\[
z_1 z_2 = 6 \left( \cos \frac{5\pi}{12} + i \sin \frac{5\pi}{12} \right)
\]
Thus, the final answer is:
\[
z_1 z_2 = 6 \left( \cos \frac{5\pi}{12} + i \sin \frac{5\pi}{12} \right)
\]
This corresponds to the option:
\[
\text{z subscript 1 baseline z subscript 2 baseline equals } 6 \left( \cos \frac{5\pi}{12} + i \sin \frac{5\pi}{12} \right)
\]