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

Honors Pre-Calculus

11-4 Roots of Complex Numbers

Objective: Find roots of complex numbers

Graph complexequations

Solve the following over the set of complex numbers:

13 zWe 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:

n

k

ni

n

k

nrz n

k

2sin

2cos

1 , ,2 ,1 ,0 where nk

101 22 r

n

k

ni

n

k

nrz n

k

2sin

2cos

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)

01

0tan 1

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.

2,1,0for ,3

2

3

0sin

3

2

3

0cos13

k

ki

kzk

10sin0cos1 iHere's the root we already knew.

3

12

3

0sin

3

12

3

0cos13

1

iz

ii2

3

2

1

3

2sin

3

2cos1

3

22

3

0sin

3

22

3

0cos13

2

iz

ii2

3

2

1

3

4sin

3

4cos1

If you cube any of these numbers

you get 1. (Try it and see!)

3

02

3

0sin

3

02

3

0cos13

0

iz

2,1,0for ,3

2

3

0sin

3

2

3

0cos13

k

ki

kzk

ii2

3

2

1,

2

3

2

1,1 We found the cube roots of 1 were:

Let's plot these on the complex plane

about 0.9

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.

each line is 1/2 unit

This representation known as Argand

diagram

Steps to Find Roots of Complex Numbers

1) Change complex number to polar form

z = r cis θ

2) Find the nth roots:

3) Change back to complex numbers

Find the complex fifth roots of

The five complex roots are:

for k = 0, 1, 2, 3,4 .

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.

yx 42

cosrx

sinry sin4cos 2 rr

sin4cos 22 rr

substitute in for x and y

Notice Polar Equations are different from our typical rectangular equations since the independent and dependent have switched locations.

What are the polar conversions we found for x and y?

Converting a Cartesian Equation to a Polar Equation

2

4sin

cosr

4 tan secr

Find a rectangular equation for r = 4 cos θ

Convert a Cartesian Equation to a Polar Equation

3x + 2y = 4Let x = r cos and y = r sin to get

.sin2cos3

4or4sin2cos3

rrr

Cartesian Equation Polar Equation

2r

ry

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

2y r y

rr y r

For the polar equation

(a) convert to a rectangular equation,

(b) use a graphing calculator to graph the polar equation for 0 2, and

(c) use a graphing calculator to graph the rectangular equation.

(a) Multiply both sides by the denominator.

,sin14

r

4 sin 4

1 sinr r r

sin 4 4r r r y 2 24 (4 )r y r y

2 2 2(4 )x y y

Convert to a rectangular equation:

Multiply both sides by the denominator.

,sin14

r

4 sin 4

1 sinr r r

sin 4 4r r r y

2 24 (4 )r y r y

2 2 2(4 )x y y

2 2 2

2

2

16 88 168( 2)

x y y yx yx y

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.

(b) The figure shows (c) Solving x2 = –8(y – 2)a graph with polar for y, we obtaincoordinates.

.2 281 xy

Theorem Tests for Symmetry

Symmetry with Respect to the Polar Axis (x-axis):

Theorem Tests for Symmetry

Theorem Tests for Symmetry

Symmetry with Respect to the Pole (Origin):

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.

Identify points on the graph:

Polar axis:

Symmetric with respect to the polar axis.

Check Symmetry of:

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

The pole:

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

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.