Advanced Engineering Mathematics
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ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Problem Statement:**
Find the product \( z_1z_2 \) and the quotient \( \frac{z_1}{z_2} \). Express your answers in polar form.
Given:
\[ z_1 = 10(\cos(110^\circ) + i \sin(110^\circ)) \]
\[ z_2 = 2(\cos(20^\circ) + i \sin(20^\circ)) \]
**Solution:**
\[ z_1z_2 = \_\_\_\_ \]
\[ \frac{z_1}{z_2} = \_\_\_\_ \]
**Explanation:**
In polar form, the complex number \( z = r(\cos \theta + i \sin \theta) \) can be treated using the modulus \( r \) and argument \( \theta \).
For the product \( z_1z_2 \), multiply the moduli and add the arguments:
\[ z_1z_2 = (10 \times 2)\left(\cos(110^\circ + 20^\circ) + i \sin(110^\circ + 20^\circ)\right) = 20(\cos(130^\circ) + i \sin(130^\circ)) \]
For the quotient \( \frac{z_1}{z_2} \), divide the moduli and subtract the arguments:
\[ \frac{z_1}{z_2} = \left(\frac{10}{2}\right)\left(\cos(110^\circ - 20^\circ) + i \sin(110^\circ - 20^\circ)\right) = 5(\cos(90^\circ) + i \sin(90^\circ)) \]
\( \cos(90^\circ) = 0 \) and \( \sin(90^\circ) = 1 \), so:
\[ \frac{z_1}{z_2} = 5i \]](https://content.bartleby.com/qna-images/question/61d8af2b-7fdf-4a8e-a1bf-75f5ea698050/2bc5a761-8630-4ebd-a67e-b941a35a7af4/h7dvaa9_processed.jpeg)
Transcribed Image Text:**Problem Statement:**
Find the product \( z_1z_2 \) and the quotient \( \frac{z_1}{z_2} \). Express your answers in polar form.
Given:
\[ z_1 = 10(\cos(110^\circ) + i \sin(110^\circ)) \]
\[ z_2 = 2(\cos(20^\circ) + i \sin(20^\circ)) \]
**Solution:**
\[ z_1z_2 = \_\_\_\_ \]
\[ \frac{z_1}{z_2} = \_\_\_\_ \]
**Explanation:**
In polar form, the complex number \( z = r(\cos \theta + i \sin \theta) \) can be treated using the modulus \( r \) and argument \( \theta \).
For the product \( z_1z_2 \), multiply the moduli and add the arguments:
\[ z_1z_2 = (10 \times 2)\left(\cos(110^\circ + 20^\circ) + i \sin(110^\circ + 20^\circ)\right) = 20(\cos(130^\circ) + i \sin(130^\circ)) \]
For the quotient \( \frac{z_1}{z_2} \), divide the moduli and subtract the arguments:
\[ \frac{z_1}{z_2} = \left(\frac{10}{2}\right)\left(\cos(110^\circ - 20^\circ) + i \sin(110^\circ - 20^\circ)\right) = 5(\cos(90^\circ) + i \sin(90^\circ)) \]
\( \cos(90^\circ) = 0 \) and \( \sin(90^\circ) = 1 \), so:
\[ \frac{z_1}{z_2} = 5i \]
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