Bentuk Polar Bilangan Kompleks | Matematika Tingkat Lanjut SMA Kelas XI Kurikulum Merdeka

Introduction

This tutorial will guide you through understanding and converting complex numbers into their polar form. The polar form is a useful representation of complex numbers that can simplify calculations in mathematics, particularly in advanced high school curricula. We will break down the definitions and steps required to express complex numbers in polar form.

Step 1: Understand Complex Numbers

Before diving into polar forms, it’s essential to have a grasp of what complex numbers are. A complex number is generally expressed as:

• z = x + iy

Where:

• x is the real part
• y is the imaginary part
• i is the imaginary unit, satisfying i² = -1

Key Concept

Complex numbers can also be represented graphically on a plane, where the x-axis represents the real part and the y-axis represents the imaginary part.

Step 2: Learn the Polar Form of Complex Numbers

The polar form of a complex number is expressed as:

• z = r(cos θ + i sin θ)

Where:

• r is the modulus (distance from the origin)
• θ is the argument (angle from the positive x-axis)

Important Formulas

• The modulus (r) is calculated using:

  • r = √(x² + y²)

• The angle (θ) can be determined using:

  • sin θ = y / r
  • cos θ = x / r

Step 3: Calculate the Modulus

To convert a complex number into its polar form, start by calculating the modulus using the formula from Step 2.

1. Identify the real part (x) and the imaginary part (y) of the complex number.
2. Plug these values into the modulus formula:

   r = √(x² + y²)
   

Practical Tip

Ensure your calculations are accurate, as errors in calculating r will affect the final polar form.

Step 4: Determine the Argument

Next, find the angle (θ) using the sine and cosine relationships.

1. Once you have r, compute θ using:
   • θ = arctan(y/x) (considering the correct quadrant based on the signs of x and y).

Common Pitfalls

• Make sure to adjust θ based on the quadrant:
  • Quadrant I: θ is as calculated.
  • Quadrant II: θ = π - calculated value.
  • Quadrant III: θ = π + calculated value.
  • Quadrant IV: θ = 2π - calculated value.

Step 5: Write the Polar Form

Finally, substitute the values of r and θ into the polar form equation.

1. The polar form is now:

   z = r(cos θ + i sin θ)
   

Example

For the complex number z = 3 + 4i:

• Calculate modulus:
  • r = √(3² + 4²) = √(9 + 16) = 5
• Determine argument:
  • θ = arctan(4/3)
• Write polar form:

  z = 5(cos θ + i sin θ)
  

Conclusion

Converting complex numbers to their polar form involves understanding their components, calculating the modulus, finding the argument, and then writing the final expression. This form is particularly useful for multiplication and division of complex numbers. Practice with several examples to gain confidence in your skills. For further learning, consider exploring related topics such as Cartesian form and their applications in advanced mathematics.