Let's solve the given problems involving operations on complex numbers in polar form:

### Problem 15
\[ \frac{8 \text{Cis } 315^\circ}{4 \text{Cis } 180^\circ} \]

#### Solution:
- **Magnitude:** 
  \[
  \frac{8}{4} = 2
  \]
- **Angle:** 
  \[
  315^\circ - 180^\circ = 135^\circ
  \]
- **Result:**
  \[
  2 \text{Cis } 135^\circ
  \]

### Problem 16
\[ \frac{8\sqrt{3} \text{Cis } 80^\circ}{\sqrt{3} \text{Cis } 240^\circ} \]

#### Solution:
- **Magnitude:** 
  \[
  \frac{8\sqrt{3}}{\sqrt{3}} = 8
  \]
- **Angle:** 
  \[
  80^\circ - 240^\circ = -160^\circ
  \]
  We can express the angle as \(200^\circ\) by adding \(360^\circ\) to it.
- **Result:**
  \[
  8 \text{Cis } 200^\circ
  \]

### Problem 17
\[ \frac{(2 \text{Cis } 110^\circ)^2(4 \text{Cis } 25^\circ)}{8 \text{Cis } 220^\circ} \]

#### Solution:
- **Numerator:**
  \[
  (2 \text{Cis } 110^\circ)^2 = 4 \text{Cis } 220^\circ
  \]
  \[
  4 \text{Cis } 220^\circ \times 4 \text{Cis } 25^\circ = 16 \text{Cis } (220^\circ + 25^\circ) = 16 \text{Cis } 245^\circ
  \]
- **Denominator:**
  \[
  8 \text{Cis } 220^\circ
  \]
- **Division:**
  \[
  \frac{16 \text{Cis } 245^\circ}{8 \text{Cis } 220^\circ} = 2 \text{Cis } (245^\circ - 220^\circ) = 2 \text{Cis } 25^\circ
  \]

### Problem 18
\[ \frac{\sqrt{6} \text{Cis } 310^\circ}{(\sqrt{3} \text{Cis } 110^\circ)(2\sqrt{2} \text{Cis } 60^\circ)} \]

#### Solution:
- **Denominator:**
  \[
  (\sqrt{3} \text{Cis } 110^\circ)(2\sqrt{2} \text{Cis } 60^\circ) = 2\sqrt{6} \text{Cis } (110^\circ + 60^\circ) = 2\sqrt{6} \text{Cis } 170^\circ
  \]
- **Division:**
  \[
  \frac{\sqrt{6} \text{Cis } 310^\circ}{2\sqrt{6} \text{Cis } 170^\circ} = \frac{1}{2} \text{Cis } (310^\circ - 170^\circ) = \frac{1}{2} \text{Cis } 140^\circ
  \]

**Results:**
1. **Problem 15:** \( 2 \text{Cis } 135^\circ \)
2. **Problem 16:** \( 8 \text{Cis } 200^\circ \)
3. **Problem 17:** \( 2 \text{Cis } 25^\circ \)
4. **Problem 18:** \( \frac{1}{2} \text{Cis } 140^\circ \)

Would you like further details on any of these steps?

---

Here are 5 related questions to consider:
1. How do you determine the angle when dividing two complex numbers in polar form?
2. What is the significance of the magnitude in complex number operations?
3. How do you simplify a complex number expression when both magnitude and angle are involved?
4. What are the advantages of using polar form for multiplication and division of complex numbers?
5. How can you convert a negative angle in polar form to its positive equivalent?

**Tip:** When performing operations with angles, always ensure the final angle is within the standard range of \(0^\circ\) to \(360^\circ\) for consistency.

Explore solutions to complex number problems in polar form with step-by-step explanations. This page covers operations on complex numbers, including multiplication and division in polar form. Ideal for students and enthusiasts of advanced mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation