1. Honors Pre-Calculus 11-4 Roots of Complex Numbers
Objective: Find roots of complex numbers
Graph complex equations

2. Solve the following over the set of complex numbers:
We know that if we cube root both sides we could get 1 but we know that there are 3 roots. So we want the complex cube roots of 1.
Using DeMoivre's Theorem with the power being a rational exponent (and therefore meaning a root), we can develop a method for finding complex roots. This leads to the following formula:

3. Let's try this on our problem. We want the cube roots of 1.
We want cube root so our n = 3. Can you convert 1 to polar form? (hint: 1 = 1 + 0i)
We want cube root so use 3 numbers here
Once we build the formula, we use it first with k = 0 and get one root, then with k = 1 to get the second root and finally with k = 2 for last root.

4. If you cube any of these numbers you get 1. (Try it and see!)
Here's the root we already knew.
If you cube any of these numbers you get 1. (Try it and see!)

5. This representation known as Argand diagram
We found the cube roots of 1 were:
Let's plot these on the complex plane
about 0.9
each line is 1/2 unit
Notice each of the complex roots has the same magnitude. Also the three points are evenly spaced on a circle. This will always be true of complex roots.
This representation known as Argand diagram

6. Steps to Find Roots of Complex Numbers
Change complex number to polar form
z = r cis θ
Find the nth roots:
Change back to complex numbers

7. Find the complex fifth roots of
The five complex roots are:
for k = 0, 1, 2, 3,4 .

9. Graphs of Polar Equations
Equations such as
r = 3 sin , r = 2 + cos , or r = ,
are examples of polar equations where r and  are the variables.
The simplest equation for many types of curves turns out to be a polar equation.
Evaluate r in terms of  until a pattern appears.

10. substitute in for x and y
Converting a Cartesian Equation to a Polar Equation
What are the polar conversions we found for x and y?
substitute in for x and y
Notice Polar Equations are different from our typical rectangular equations since the independent and dependent have switched locations.

11. Find a rectangular equation for r = 4 cos θ

12. Convert a Cartesian Equation to a Polar Equation
3x + 2y = 4
Let x = r cos  and y = r sin  to get
Cartesian Equation
Polar Equation

13. Now you try: Convert r = 2 csc to rectangular form.
Since csc = r/y, substitute for csc.
Multiply both sides by y/r.
Simplify, we have (a horizontal line) is the rectangular form.
y = 2

14. For the polar equation
convert to a rectangular equation,
use a graphing calculator to graph the polar equation for 0    2, and
use a graphing calculator to graph the rectangular equation.
(a) Multiply both sides by the denominator.

15. Convert to a rectangular equation:
Multiply both sides by the denominator.

16. The figure shows (c) Solving x2 = –8(y – 2)
Square both sides.
rectangular equation
It is a parabola vertex at (0, 2) opening down and p = –2, focusing at (0, 0), and with diretrix at y = 4.
The figure shows (c) Solving x2 = –8(y – 2)
a graph with polar for y, we obtain
coordinates.

17. Theorem Tests for Symmetry
Symmetry with Respect to the Polar Axis (x-axis):

18. Theorem Tests for Symmetry

19. Theorem Tests for Symmetry
Symmetry with Respect to the Pole (Origin):

20. The tests for symmetry just presented are sufficient conditions for symmetry, but not necessary.
In class, an instructor might say a student will pass provided he/she has perfect attendance. Thus, perfect attendance is sufficient for passing, but not necessary.

21. Identify points on the graph:

22. Check Symmetry of:
Polar axis:
Symmetric with respect to the polar axis.

23. The test fails so the graph may or may not be symmetric with respect to the above line.

24. The pole:
The test fails, so the graph may or may not be symmetric with respect to the pole.

25. Graphing a polar Equation Using a Graphing Utility
Solve the equation for r in terms of θ.
Select a viewing window in POLar mode. The polar mode requires setting max and min and step values for θ. Use a square window.
Enter the expression from Step1.
Graph.