### Problem Statement2. Find the exact value of: $[2 \text{cis} \frac{\pi}{4}] [3 \text{cis} \frac{2\pi}{3}]$#### Explanation:Given a problem involving the multiplication of two complex numbers in polar form. The notation $\text{cis}$ stands for $ \cos(\theta) + i\sin(\theta) $. Therefore, the problem can be interpreted as multiplying two complex numbers expressed in polar coordinates.#### Step-by-Step Solution:1. **Identify the given complex numbers in polar form:** \[ z_1 = 2 \text{cis} \frac{\pi}{4} \] \[ z_2 = 3 \text{cis} \frac{2\pi}{3} \]2. **Multiply the magnitudes:** \[ |z_1| \times |z_2| = 2 \times 3 = 6 \]3. **Add the angles:** \[ \theta_1 + \theta_2 = \frac{\pi}{4} + \frac{2\pi}{3} = \frac{3\pi}{12} + \frac{8\pi}{12} = \frac{11\pi}{12} \]4. **Combine them into the polar form:** \[ 6 \text{cis} \frac{11\pi}{12} \]#### Final Result:\[ [2 \text{cis} \frac{\pi}{4}] [3 \text{cis} \frac{2\pi}{3}] = 6 \text{cis} \frac{11\pi}{12} \]This represents the exact value of the given expression.

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Answered: 2. Find the exact value of: of:[2cis ]…

Math

Trigonometry

2. Find the exact value of: of:[2cis ] [3cis 5]

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Complex Numbers

Section4.4: Trigonometric Form Of A Complex Number

Problem 2ECP: Write the complex number z=3+4i in trigonometric form.

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Question

### Problem Statement
2. Find the exact value of: $[2 \text{cis} \frac{\pi}{4}] [3 \text{cis} \frac{2\pi}{3}]$

#### Explanation:
Given a problem involving the multiplication of two complex numbers in polar form. The notation $\text{cis}$ stands for $ \cos(\theta) + i\sin(\theta) $. Therefore, the problem can be interpreted as multiplying two complex numbers expressed in polar coordinates.

#### Step-by-Step Solution:
1. **Identify the given complex numbers in polar form:**
\[ z_1 = 2 \text{cis} \frac{\pi}{4} \]
\[ z_2 = 3 \text{cis} \frac{2\pi}{3} \]

2. **Multiply the magnitudes:**
\[ |z_1| \times |z_2| = 2 \times 3 = 6 \]

3. **Add the angles:**
\[ \theta_1 + \theta_2 = \frac{\pi}{4} + \frac{2\pi}{3} = \frac{3\pi}{12} + \frac{8\pi}{12} = \frac{11\pi}{12} \]

4. **Combine them into the polar form:**
\[ 6 \text{cis} \frac{11\pi}{12} \]

#### Final Result:
\[ [2 \text{cis} \frac{\pi}{4}] [3 \text{cis} \frac{2\pi}{3}] = 6 \text{cis} \frac{11\pi}{12} \]

This represents the exact value of the given expression.

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