Question
Find the product of z subscript 1 baseline equals Start Fraction 2 over 3 End Fraction left-parenthesis cosine 60 degree-sign plus i sine 60 degree-sign right-parenthesis and z subscript 1 baseline equals 9 left-parenthesis cosine 20 degree-sign plus i sine 20 degree-sign right-parenthesis where 0 less-than-or-equal-to theta less-than 360 degree-sign.
(1 point)
Responses
z subscript 1 baseline z subscript 2 baseline equals 9.67 cis 80 degree-sign
Image with alt text: z subscript 1 baseline z subscript 2 baseline equals 9.67 cis 80 degree-sign
z subscript 1 baseline z subscript 2 baseline equals 6 cis 80 degree-sign
Image with alt text: z subscript 1 baseline z subscript 2 baseline equals 6 cis 80 degree-sign
z subscript 1 baseline z subscript 2 baseline equals 6 cis 40 degree-sign
Image with alt text: z subscript 1 baseline z subscript 2 baseline equals 6 cis 40 degree-sign
z subscript 1 baseline z subscript 2 baseline equals Start Fraction 2 over 27 End Fraction cis 40 degree-sign
(1 point)
Responses
z subscript 1 baseline z subscript 2 baseline equals 9.67 cis 80 degree-sign
Image with alt text: z subscript 1 baseline z subscript 2 baseline equals 9.67 cis 80 degree-sign
z subscript 1 baseline z subscript 2 baseline equals 6 cis 80 degree-sign
Image with alt text: z subscript 1 baseline z subscript 2 baseline equals 6 cis 80 degree-sign
z subscript 1 baseline z subscript 2 baseline equals 6 cis 40 degree-sign
Image with alt text: z subscript 1 baseline z subscript 2 baseline equals 6 cis 40 degree-sign
z subscript 1 baseline z subscript 2 baseline equals Start Fraction 2 over 27 End Fraction cis 40 degree-sign
Answers
Answered by
GPT-4o mini
To find the product of the two complex numbers given in polar form, we can use the formula for multiplying complex numbers in polar form:
\[
z_1 z_2 = r_1 r_2 \left(\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)\right)
\]
Where:
- \( r_1 \) and \( r_2 \) are the magnitudes of the complex numbers.
- \( \theta_1 \) and \( \theta_2 \) are the angles of the complex numbers.
Given the complex numbers:
1. \( z_1 = \frac{2}{3} \left( \cos(60^\circ) + i \sin(60^\circ) \right) \)
- Here, \( r_1 = \frac{2}{3} \) and \( \theta_1 = 60^\circ \).
2. \( z_2 = 9 \left( \cos(20^\circ) + i \sin(20^\circ) \right) \)
- Here, \( r_2 = 9 \) and \( \theta_2 = 20^\circ \).
Now let's calculate the product \( z_1 z_2 \):
1. **Multiply the magnitudes:**
\[
r_1 r_2 = \frac{2}{3} \times 9 = 6
\]
2. **Add the angles:**
\[
\theta_1 + \theta_2 = 60^\circ + 20^\circ = 80^\circ
\]
Thus, the product is:
\[
z_1 z_2 = 6 \left( \cos(80^\circ) + i \sin(80^\circ) \right) = 6 \, \text{cis} \, 80^\circ
\]
Therefore, the answer is:
\[
z_1 z_2 = 6 \, \text{cis} \, 80^\circ
\]
So, the correct response is:
**z subscript 1 baseline z subscript 2 baseline equals 6 cis 80 degree-sign.**
\[
z_1 z_2 = r_1 r_2 \left(\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)\right)
\]
Where:
- \( r_1 \) and \( r_2 \) are the magnitudes of the complex numbers.
- \( \theta_1 \) and \( \theta_2 \) are the angles of the complex numbers.
Given the complex numbers:
1. \( z_1 = \frac{2}{3} \left( \cos(60^\circ) + i \sin(60^\circ) \right) \)
- Here, \( r_1 = \frac{2}{3} \) and \( \theta_1 = 60^\circ \).
2. \( z_2 = 9 \left( \cos(20^\circ) + i \sin(20^\circ) \right) \)
- Here, \( r_2 = 9 \) and \( \theta_2 = 20^\circ \).
Now let's calculate the product \( z_1 z_2 \):
1. **Multiply the magnitudes:**
\[
r_1 r_2 = \frac{2}{3} \times 9 = 6
\]
2. **Add the angles:**
\[
\theta_1 + \theta_2 = 60^\circ + 20^\circ = 80^\circ
\]
Thus, the product is:
\[
z_1 z_2 = 6 \left( \cos(80^\circ) + i \sin(80^\circ) \right) = 6 \, \text{cis} \, 80^\circ
\]
Therefore, the answer is:
\[
z_1 z_2 = 6 \, \text{cis} \, 80^\circ
\]
So, the correct response is:
**z subscript 1 baseline z subscript 2 baseline equals 6 cis 80 degree-sign.**