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This activity asks to convert the following complex numbers into polar form. To express a complex number in polar form, we need to find both its **modulus** (magnitude) and **argument** (angle).

For a complex number \( z = a + bi \), the polar form is:
\[ z = r (\cos \theta + i \sin \theta) \]
where:
- \( r = \sqrt{a^2 + b^2} \) (the modulus)
- \( \theta = \arg(z) = \tan^{-1} \left( \frac{b}{a} \right) \) (the argument, in radians or degrees).

Let me start by converting the numbers step by step. I will solve for each one:

1. **\( -5 - 8i \)**:
   - \( a = -5 \), \( b = -8 \)
   - Modulus: \( r = \sqrt{(-5)^2 + (-8)^2} = \sqrt{25 + 64} = \sqrt{89} \)
   - Argument: \( \theta = \tan^{-1} \left( \frac{-8}{-5} \right) = \tan^{-1}(1.6) \) (in the third quadrant)

I'll now compute the solution for all the problems in the table.Here are the polar forms of the given complex numbers. Each solution includes the modulus \( r \) and the argument \( \theta \) in degrees:

1. **\( -5 - 8i \)**:  
   - Modulus: \( 9.43 \)  
   - Argument: \( -122.01^\circ \)

2. **\( -6 + 4i \)**:  
   - Modulus: \( 7.21 \)  
   - Argument: \( 146.31^\circ \)

3. **\( -7 - 9i \)**:  
   - Modulus: \( 11.40 \)  
   - Argument: \( -127.87^\circ \)

4. **\( -6 - 3i \)**:  
   - Modulus: \( 6.71 \)  
   - Argument: \( -153.43^\circ \)

5. **\( -8 + 10i \)**:  
   - Modulus: \( 12.81 \)  
   - Argument: \( 128.66^\circ \)

6. **\( -10 - 6i \)**:  
   - Modulus: \( 11.66 \)  
   - Argument: \( -149.04^\circ \)

7. **\( -7 + 6i \)**:  
   - Modulus: \( 9.22 \)  
   - Argument: \( 139.40^\circ \)

8. **\( -4 - 5i \)**:  
   - Modulus: \( 6.40 \)  
   - Argument: \( -128.66^\circ \)

9. **\( -12 + 7i \)**:  
   - Modulus: \( 13.89 \)  
   - Argument: \( 149.74^\circ \)

10. **\( -3 - 8i \)**:  
    - Modulus: \( 8.54 \)  
    - Argument: \( -110.56^\circ \)

Let me know if you'd like to dive deeper into any of these results, or have any questions! Here are some related questions:

1. How can I convert polar form back to rectangular form?
2. What is the difference between the modulus and argument in polar form?
3. How does \( \tan^{-1} \) work in determining the argument of a complex number?
4. How are quadrants important in determining the angle for the argument?
5. What are the advantages of using polar form over rectangular form?

**Tip:** Always ensure your angles are expressed in the correct quadrant when converting to polar form, as \( \tan^{-1} \) only gives the principal value.

Determine the polar form of the following complex numbers: -5-8i, -6+4i, -7-9i, -6-3i, -8+10i, -10-6i, -7+6i, -4-5i, -12+7i, -3-8i.

Explore the process of converting complex numbers like -5-8i, -6+4i, and others into their polar forms using modulus and argument formulas. This page provides step-by-step solutions and explanations suitable for college-level students studying advanced mathematics.

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