resolve the problems of rectangular  to polar form

Let's solve the problems of converting complex numbers from rectangular form \(a + bi\) to polar form \(r \text{Cis} \theta\).

### The general steps are:
1. **Calculate the magnitude \( r \)**:
   \[
   r = \sqrt{a^2 + b^2}
   \]
2. **Determine the angle \( \theta \)** using:
   \[
   \theta = \tan^{-1}\left(\frac{b}{a}\right)
   \]
   Be cautious about the quadrant in which the complex number lies to determine the correct angle.

### Problem 1: \(4 + 4i\)
- **Magnitude \(r\)**:
  \[
  r = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
  \]
- **Angle \(\theta\)**:
  \[
  \theta = \tan^{-1}\left(\frac{4}{4}\right) = \tan^{-1}(1) = 45^\circ
  \]
- **Polar form**:
  \[
  4 + 4i = 4\sqrt{2} \text{Cis } 45^\circ
  \]

### Problem 2: \(1 + \sqrt{3}i\)
- **Magnitude \(r\)**:
  \[
  r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2
  \]
- **Angle \(\theta\)**:
  \[
  \theta = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \tan^{-1}(\sqrt{3}) = 60^\circ
  \]
- **Polar form**:
  \[
  1 + \sqrt{3}i = 2 \text{Cis } 60^\circ
  \]

### Problem 3: \(4 - 5i\)
- **Magnitude \(r\)**:
  \[
  r = \sqrt{4^2 + (-5)^2} = \sqrt{16 + 25} = \sqrt{41}
  \]
- **Angle \(\theta\)**:
  \[
  \theta = \tan^{-1}\left(\frac{-5}{4}\right) \approx \tan^{-1}(-1.25) \approx -51.34^\circ
  \]
  Since the number is in the fourth quadrant, \(\theta = -51.34^\circ\).
- **Polar form**:
  \[
  4 - 5i = \sqrt{41} \text{Cis } -51.34^\circ
  \]

### Problem 4: \(-\sqrt{27} + 3i\)
- **Magnitude \(r\)**:
  \[
  r = \sqrt{(-\sqrt{27})^2 + 3^2} = \sqrt{27 + 9} = \sqrt{36} = 6
  \]
- **Angle \(\theta\)**:
  \[
  \theta = \tan^{-1}\left(\frac{3}{-\sqrt{27}}\right) \approx \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) \approx 150^\circ
  \]
  Since the number is in the second quadrant, \(\theta = 150^\circ\).
- **Polar form**:
  \[
  -\sqrt{27} + 3i = 6 \text{Cis } 150^\circ
  \]

Would you like to review or discuss these solutions in more detail?

### Relative Questions:
1. How would you handle the conversion if the complex number was purely imaginary?
2. What is the effect of negative components on the angle \(\theta\) in polar form?
3. Why is it important to correctly determine the quadrant when finding the angle \(\theta\)?
4. How can you use the polar form to multiply two complex numbers?
5. What are the advantages of using the polar form of a complex number in certain operations?

**Tip:** When dealing with angles, always consider whether the context requires degrees or radians, and convert accordingly.

Learn how to convert complex numbers from rectangular form to polar form with detailed step-by-step solutions. Explore examples including 4 + 4i, 1 + sqrt(3)i, 4 - 5i, and -sqrt(27) + 3i, covering key concepts like magnitude calculation and angle determination.

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