Expressing Complex Numbers in Polar Coordinates

Oct 13, 2025 by wbrum.org 109 views

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. A complex number is a number that can be expressed in the form $a + bi$ , where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$ . In this article, we will explore the concept of complex numbers in polar coordinates and learn how to express them in the form $(r \cos \theta) + i(r \sin \theta)$ .

Polar Coordinates

Polar coordinates are a way of representing points in a two-dimensional plane using a distance from a reference point (the origin) and an angle from a reference direction (the positive x-axis). The polar coordinates of a point are given by the radius $r$ and the angle $\theta$ . The relationship between polar coordinates and Cartesian coordinates is given by the following equations:

$x = r \cos \theta$ $y = r \sin \theta$

Expressing Complex Numbers in Polar Coordinates

To express a complex number in polar coordinates, we need to find the radius $r$ and the angle $\theta$ . The radius $r$ is given by the magnitude of the complex number, which is calculated using the following formula:

$r = \sqrt{a^2 + b^2}$

The angle $\theta$ is given by the inverse tangent of the ratio of the imaginary part to the real part:

$\theta = \tan^{-1}\left(\frac{b}{a}\right)$

Example

Let's consider the complex number $10+10i$ . We need to find the radius $r$ and the angle $\theta$ to express it in polar coordinates.

Step 1: Calculate the radius

Using the formula for the radius, we get:

$r = \sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2}$

Step 2: Calculate the angle

Using the formula for the angle, we get:

$\theta = \tan^{-1}\left(\frac{10}{10}\right) = \tan^{-1}(1) = \frac{\pi}{4}$

Step 3: Express the complex number in polar coordinates

Now that we have the radius $r$ and the angle $\theta$ , we can express the complex number in polar coordinates:

$(r \cos \theta) + i(r \sin \theta) = (10\sqrt{2} \cos \left(\frac{\pi}{4}\right)) + i(10\sqrt{2} \sin \left(\frac{\pi}{4}\right))$

Simplifying the Expression

We can simplify the expression by using the values of $\cos \left(\frac{\pi}{4}\right)$ and $\sin \left(\frac{\pi}{4}\right)$ , which are both equal to $\frac{1}{\sqrt{2}}$ .

$(10\sqrt{2} \cos \left(\frac{\pi}{4}\right)) + i(10\sqrt{2} \sin \left(\frac{\pi}{4}\right)) = (10\sqrt{2} \cdot \frac{1}{\sqrt{2}}) + i(10\sqrt{2} \cdot \frac{1}{\sqrt{2}})$

$= 10 + 10i$

Conclusion

In this article, we learned how to express complex numbers in polar coordinates using the form $(r \cos \theta) + i(r \sin \theta)$ . We applied this concept to the complex number $10+10i$ and found that it can be expressed in polar coordinates as $(10\sqrt{2} \cos \left(\frac{\pi}{4}\right)) + i(10\sqrt{2} \sin \left(\frac{\pi}{4}\right))$ . We also simplified the expression using the values of $\cos \left(\frac{\pi}{4}\right)$ and $\sin \left(\frac{\pi}{4}\right)$ .

Answer

The correct answer is:

$\boxed{\left(\frac{1}{10} \cos \left(\frac{\pi}{4}\right)\right) + i\left(\frac{1}{10} \sin \left(\frac{\pi}{4}\right)\right)}$

Introduction

In our previous article, we explored the concept of complex numbers in polar coordinates and learned how to express them in the form $(r \cos \theta) + i(r \sin \theta)$ . In this article, we will answer some frequently asked questions about polar coordinates of complex numbers.

Q: What is the relationship between polar coordinates and Cartesian coordinates?

A: The relationship between polar coordinates and Cartesian coordinates is given by the following equations:

$x = r \cos \theta$ $y = r \sin \theta$

Q: How do I find the radius $r$ of a complex number in polar coordinates?

A: To find the radius $r$ , you need to calculate the magnitude of the complex number using the following formula:

$r = \sqrt{a^2 + b^2}$

Q: How do I find the angle $\theta$ of a complex number in polar coordinates?

A: To find the angle $\theta$ , you need to calculate the inverse tangent of the ratio of the imaginary part to the real part:

$\theta = \tan^{-1}\left(\frac{b}{a}\right)$

Q: Can I use a calculator to find the radius and angle of a complex number in polar coordinates?

A: Yes, you can use a calculator to find the radius and angle of a complex number in polar coordinates. Most calculators have a built-in function to calculate the magnitude and angle of a complex number.

Q: How do I express a complex number in polar coordinates using the form $(r \cos \theta) + i(r \sin \theta)$ ?

A: To express a complex number in polar coordinates using the form $(r \cos \theta) + i(r \sin \theta)$ , you need to follow these steps:

Calculate the radius $r$ using the formula $r = \sqrt{a^2 + b^2}$ .
Calculate the angle $\theta$ using the formula $\theta = \tan^{-1}\left(\frac{b}{a}\right)$ .
Substitute the values of $r$ and $\theta$ into the formula $(r \cos \theta) + i(r \sin \theta)$ .

Q: Can I simplify the expression $(r \cos \theta) + i(r \sin \theta)$ ?

A: Yes, you can simplify the expression $(r \cos \theta) + i(r \sin \theta)$ by using the values of $\cos \theta$ and $\sin \theta$ . For example, if $\theta = \frac{\pi}{4}$ , then $\cos \theta = \frac{1}{\sqrt{2}}$ and $\sin \theta = \frac{1}{\sqrt{2}}$ .

Q: What are some common mistakes to avoid when working with polar coordinates of complex numbers?

A: Some common mistakes to avoid when working with polar coordinates of complex numbers include:

Not calculating the radius and angle correctly
Not using the correct formula for the radius and angle
Not simplifying the expression $(r \cos \theta) + i(r \sin \theta)$
Not checking the units of the radius and angle

Conclusion

In this article, we answered some frequently asked questions about polar coordinates of complex numbers. We hope that this article has been helpful in clarifying any confusion you may have had about polar coordinates of complex numbers. If you have any further questions, please don't hesitate to ask.

Additional Resources

Practice Problems

Express the complex number $3+4i$ in polar coordinates using the form $(r \cos \theta) + i(r \sin \theta)$ .
Find the radius and angle of the complex number $5+12i$ in polar coordinates.
Simplify the expression $(r \cos \theta) + i(r \sin \theta)$ for the complex number $2+3i$ .

Answer Key

$(\sqrt{3^2 + 4^2} \cos \tan^{-1}\left(\frac{4}{3}\right)) + i(\sqrt{3^2 + 4^2} \sin \tan^{-1}\left(\frac{4}{3}\right))$
$r = \sqrt{5^2 + 12^2} = 13$ , $\theta = \tan^{-1}\left(\frac{12}{5}\right) = 1.249$
$(2 \cos \tan^{-1}\left(\frac{3}{2}\right)) + i(2 \sin \tan^{-1}\left(\frac{3}{2}\right))$