Precalculus: Part 2
Module 5 Content Guide
Polar Coordinates cont.
Learning Outcomes
By the end of this module, you will be able to do the following:
1. Recognize and graph polar equations.
2. Write complex numbers in polar form.
3. Find products and quotients of complex numbers.
4. Find powers and roots of complex numbers.
Math Master Challenge
Your challenge this module is to demonstrate respect for the people who have helped you in
your efforts to learn. Write a thank you note to someone who has supported you. Your note
could be for a parent, a teacher, a friend, or a mentor in something completely unrelated to
math!
My Notes
Precalculus: Part 2
Module 5 Content Guide
Topic 5.1 Graphing Polar Equations: Lines and Circles
Fill in the blanks:
When we plot points in the rectangular plane, we use ■ and ■
coordinates to describe the location of a point. On a polar grid, we
use ■ and ■ to describe the location of the same point.
Finding the symmetry of a polar equation can be helpful for graphing,
just as it is with . Polar equations can
be with respect to the polar axis , the
𝜋𝜋
line 𝜃𝜃 = (𝑦𝑦-axis), or the pole .
2
We can find a few additional points that yield helpful information
about a given polar equation. First, we can find the
distance from the pole, denoted | ■ |. The maximum value of
𝑟𝑟 = 3 cos 𝜃𝜃 is ■ because the maximum value of cos 𝜃𝜃 is 1, which
occurs when .
Some of the most common are , ,
, , and .
Precalculus: Part 2
Module 5 Content Guide
Topic 5.2 Polar Form of Complex Numbers
Fill in the blanks:
When we find the absolute value of an integer, we find its distance
from ■ . The absolute value of a complex number is also called
its or , denoted | ■ | where 𝑧𝑧 = 𝑥𝑥 + 𝑦𝑦𝑦𝑦.
To convert complex numbers to polar form we use the same formulas
that we use when converting rectangular coordinates to polar
coordinates:
𝑟𝑟 = 𝑥𝑥 2 + 𝑦𝑦 2
Keep in mind that the (or modulus) is the absolute
value of a complex number and is its distance from the origin. This
means that the magnitude is equal to ■ when written in polar form.
The point 𝑧𝑧 = 𝑥𝑥 + 𝑦𝑦𝑦𝑦 can be written in polar form using the following
formula:
Where 𝑟𝑟 is the modulus, and 𝜃𝜃 is the argument.
Precalculus: Part 2
Module 5 Content Guide
Topic 5.3 Products and Quotients of Complex Numbers
Fill in the blanks:
To find the distance between two points in the complex plane, we
simply need to find the of the difference between
the points. In fact, when we find the of a complex
number, we calculate its distance from the .
Given two points in the complex plane, and , the
distance between the two points is:
(𝑎𝑎 − 𝑐𝑐)2 +(𝑏𝑏 − 𝑑𝑑)2
Or equivalently,
To find the midpoint between two points in the complex plane, we
just need to find the of both points.
Given two points in the complex plane, and , the
midpoint between the two points is:
Precalculus: Part 2
Module 5 Content Guide
Topic 5.4 Powers and Roots of Complex Numbers
Fill in the blanks:
De Moivre’s Theorem can be summarized as follows:
𝑧𝑧 𝑛𝑛 = 𝑟𝑟 𝑛𝑛 +
is the rectangular form of a complex number and
is the polar form.
While the process of finding roots of complex numbers is similar to
finding of complex numbers, it is slightly more
challenging. Aside from finding the root of 𝑟𝑟, we must also
2𝜋𝜋𝜋𝜋 𝜃𝜃
find the argument for each root by adding to for
𝑛𝑛 𝑛𝑛
. This is called the 𝑁𝑁th Root Theorem.
Given a complex number, 𝑧𝑧, where 𝑧𝑧 = 𝑟𝑟(cos 𝜃𝜃 + 𝑦𝑦 sin 𝜃𝜃), then
1 1 𝜃𝜃 2𝜋𝜋𝜋𝜋 𝜃𝜃 2𝜋𝜋𝜋𝜋
𝑧𝑧 𝑛𝑛 = 𝑟𝑟 𝑛𝑛 cos + + 𝑦𝑦 sin +
𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑛𝑛
Where ■ is a positive integer, and .
Precalculus: Part 2
Module 5 Content Guide
Math Masters Challenge
Write a heartfelt thank you note to someone who has made a significant impact on your life,
expressing your gratitude for their support and guidance, even if it's in a field completely
unrelated to your current module.
Reflection Question: Did you write a note to someone who has helped you in your learning? How
did this person help you? Who did you write to?
