What Is The Polar Form Of \[$-2 \sqrt{3} - 6i\$\]?A. \[$2 \sqrt{6}\left(\cos \left(\frac{2 \pi}{3}\right) + I \sin \left(\frac{2 \pi}{3}\right)\right)\$\]B. \[$4 \sqrt{3}\left(\cos \left(\frac{2 \pi}{3}\right) + I \sin \left(\frac{2

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Introduction to Polar Form

In mathematics, complex numbers are often represented in the form a + bi, where a and b are real numbers and i is the imaginary unit. However, there is another way to represent complex numbers, known as polar form. The polar form of a complex number is a way of expressing it in terms of its magnitude (or length) and angle. This form is particularly useful for performing operations such as multiplication and division of complex numbers.

What is the Polar Form of a Complex Number?

The polar form of a complex number is given by:

r(cosθ + isinθ)

where r is the magnitude of the complex number, and θ is the angle it makes with the positive real axis.

How to Convert a Complex Number to Polar Form

To convert a complex number to polar form, we need to find its magnitude and angle. The magnitude of a complex number is given by:

r = √(a^2 + b^2)

where a and b are the real and imaginary parts of the complex number, respectively.

The angle θ is given by:

θ = arctan(b/a)

Example: Converting a Complex Number to Polar Form

Let's consider the complex number -2√3 - 6i. To convert this number to polar form, we need to find its magnitude and angle.

Calculating the Magnitude

The magnitude of the complex number is given by:

r = √((-2√3)^2 + (-6)^2) = √(12 + 36) = √48 = 4√3

Calculating the Angle

The angle θ is given by:

θ = arctan(-6/-2√3) = arctan(3/√3) = arctan(√3) = π/3

However, since the complex number is in the third quadrant, the angle is actually:

θ = π + π/3 = 4π/3

Writing the Complex Number in Polar Form

Now that we have the magnitude and angle, we can write the complex number in polar form:

-2√3 - 6i = 4√3(cos(4π/3) + isin(4π/3))

Comparing with the Given Options

Let's compare our result with the given options:

A. 2√6(cos(2π/3) + isin(2π/3)) B. 4√3(cos(2π/3) + isin(2π/3))

Our result matches option B.

Conclusion

In this article, we discussed the polar form of complex numbers and how to convert a complex number to polar form. We used the complex number -2√3 - 6i as an example and calculated its magnitude and angle. We then wrote the complex number in polar form and compared our result with the given options. We found that our result matches option B.

Frequently Asked Questions

  • What is the polar form of a complex number? The polar form of a complex number is a way of expressing it in terms of its magnitude (or length) and angle.
  • How to convert a complex number to polar form? To convert a complex number to polar form, we need to find its magnitude and angle. The magnitude is given by r = √(a^2 + b^2), and the angle is given by θ = arctan(b/a).
  • What is the magnitude of the complex number -2√3 - 6i? The magnitude of the complex number is 4√3.
  • What is the angle of the complex number -2√3 - 6i? The angle of the complex number is 4Ï€/3.

References

  • "Complex Numbers" by Khan Academy
  • "Polar Form of Complex Numbers" by Math Open Reference
  • "Converting Complex Numbers to Polar Form" by Purplemath

Introduction

In our previous article, we discussed the polar form of complex numbers and how to convert a complex number to polar form. In this article, we will answer some frequently asked questions about the polar form of complex numbers.

Q&A

Q: What is the polar form of a complex number?

A: The polar form of a complex number is a way of expressing it in terms of its magnitude (or length) and angle.

Q: How to convert a complex number to polar form?

A: To convert a complex number to polar form, we need to find its magnitude and angle. The magnitude is given by r = √(a^2 + b^2), and the angle is given by θ = arctan(b/a).

Q: What is the magnitude of the complex number -2√3 - 6i?

A: The magnitude of the complex number is 4√3.

Q: What is the angle of the complex number -2√3 - 6i?

A: The angle of the complex number is 4Ï€/3.

Q: How to find the magnitude of a complex number?

A: The magnitude of a complex number is given by r = √(a^2 + b^2), where a and b are the real and imaginary parts of the complex number, respectively.

Q: How to find the angle of a complex number?

A: The angle of a complex number is given by θ = arctan(b/a), where a and b are the real and imaginary parts of the complex number, respectively.

Q: What is the relationship between the polar form and the rectangular form of a complex number?

A: The polar form and the rectangular form of a complex number are related by the following equations:

r(cosθ + isinθ) = a + bi

where r is the magnitude, θ is the angle, a is the real part, and b is the imaginary part.

Q: How to perform operations on complex numbers in polar form?

A: To perform operations on complex numbers in polar form, we can use the following rules:

  • Addition: r1(cosθ1 + isinθ1) + r2(cosθ2 + isinθ2) = (r1 + r2)(cos(θ1 + θ2) + isin(θ1 + θ2))
  • Subtraction: r1(cosθ1 + isinθ1) - r2(cosθ2 + isinθ2) = (r1 - r2)(cos(θ1 - θ2) + isin(θ1 - θ2))
  • Multiplication: r1(cosθ1 + isinθ1) × r2(cosθ2 + isinθ2) = r1r2(cos(θ1 + θ2) + isin(θ1 + θ2))

Q: What are some common applications of the polar form of complex numbers?

A: The polar form of complex numbers has many applications in mathematics and engineering, including:

  • Representing complex numbers in a more intuitive and visual way
  • Performing operations on complex numbers more easily
  • Solving equations involving complex numbers
  • Analyzing and designing electrical circuits

Conclusion

In this article, we answered some frequently asked questions about the polar form of complex numbers. We hope that this article has been helpful in clarifying any doubts you may have had about the polar form of complex numbers.

Frequently Asked Questions

  • What is the polar form of a complex number?
  • How to convert a complex number to polar form?
  • What is the magnitude of the complex number -2√3 - 6i?
  • What is the angle of the complex number -2√3 - 6i?
  • How to find the magnitude of a complex number?
  • How to find the angle of a complex number?
  • What is the relationship between the polar form and the rectangular form of a complex number?
  • How to perform operations on complex numbers in polar form?
  • What are some common applications of the polar form of complex numbers?

References

  • "Complex Numbers" by Khan Academy
  • "Polar Form of Complex Numbers" by Math Open Reference
  • "Converting Complex Numbers to Polar Form" by Purplemath