Section 6.5 Notes

1st Day

Today we will learn how to change from rectangular (standard) complex form to trigonometric (polar) complex form and vice versus.

A complex numberz = a + bi

can be represented as a point (a, b) in a coordinate plane called the complex plane.The horizontal axis is called the real axis and the vertical axis is called the imaginary axis.

real axis

imaginary axis

Example 1

Graph.a. 2 + 3ib. -1 – 2i

I

R

(2, 3)2 + 3i

(-1, -2)

-1 - 2i

In Algebra II you learned how to add, subtract, multiply, and divide complex numbers.In Pre-Calculus you will learn how to work with powers and roots of complex numbers.To do this you must write the complex numbers in trigonometric form (or polar form).

On the next slide you will see how we change a rectangular (standard) complex number into a trigonometric (polar) complex number.

θ

(a, b)

ab

r

I

R

cos ar

a cosr

sin br

b sinr

2 2a b

a + bi

r

The trigonometric form of the complex number z = a + bi is

z = r(cos θ + i sin θ)where a = r cos θ, b = r sin θ,

In most cases 0 ≤ θ < 2π or 0° ≤ θ < 360°.

2 2 , and tan . br a ba

There is a shortcut for writing a trigonometric complex number. The shortcut is

z = rcis θ = r(cos θ + i sin θ)

Example 2

Write the complex number z = 6 – 6i in trigonometric (polar) form in radians.1. The point is in what quadrant?

4th quadrant

226 6 r 6 2

6tan 16

2. Find r.

3. Find θ.

74

Remember θ is in the 4th quad.

7 76 2 cos sin4 4

z i 76 2cis

4

Example 3Represent the complex number graphically and then find the rectangular (standard) form of the number. No rounding.

z = 6(cos 135° + isin 135°)

135°

6

Find a and b.

6cos135 a 262

6sin135 b 262

3 2 3 2 z i

3 2

3 2

Now we will learn how to multiply and divide complex numbers in trigonometric (polar) form.

Product of Complex Numbers

1 1 1 1 2 2 2 2Let cos sin and cos sin

be complex numbers.

z r i z r i

1 2 1 2 1 2 1 2

1 2 1 2

cos sin where

0 2 or 0 360

z z r r i

Example 4

Find the product z1z2 of the complex numbers. Write your answer in standard form.

1 26 cos sin and 4 cos sin3 3 6 6

z i z i

1 2 z z 6 4 cos sin3 6 3 6

i

24 cos sin2 2

i

24 0 1 i

24 i

Quotient of Complex Numbers

1 1 1 1 2 2 2 2Let cos sin and cos sin

be complex numbers.

z r i z r i

1 11 2 1 2

2 2

1 2 1 2

cos sin

where 0 2 or 0 360

z r iz r

Example 5

1

2

Find the quotient of the complex numbers.

Write your answer in standard form.

zz

1 26 cos40 sin 40 and 2 cos10 sin10 z i z i

1

2

6 cos40 sin 402 cos10 sin10

iz

z i

6 cos 40 10 sin 40 102

i

3 cos 30 sin 30 i

3 132 2

i

3 3 32 2

i

END OF 1ST DAY

2nd DayToday we will learn to:1. Raise complex numbers to a power.2. Find the roots of complex numbers.

Multiply:(4 + 2i)10

DeMoivre’s Theorem

If z = r(cos θ + isin θ) is a complex number and n is a positive integer, then

cos sin nnz r i

cos sin n nz r n i n

Example 6

12Use DeMoivre's Theorem to find 1 3 .

Write your answer in standard form.

i

In what quadrant is this complex number?2nd Quadrant

1. Change to polar form.

2. Find θ.

221 3 r 2

3tan 31

23

12

12 2 21 3 2 cos sin3 3

i i

12 12 2 12 22 cos sin

3 3

i

4096 cos8 sin8 i

4096 1 0 i

4096

The nth Root of a Complex Number

The complex number u = a + bi is an nth root of the complex number z if

z = un = (a + bi)n

Finding the nth Root of a Complex Number

For a positive integer n, the complex number z = r(cos θ + i sin θ)

has exactly n distinct nth roots given by

2 2cos sin

where 0,1,2,..., 1

n k kr in n

k nor

360 360cos sin

where 0,1,2,..., 1

n k kr in n

k n

Example 7

Find all the fourth roots of 1.This means: x4 = 1.1. Change 1 into a polar complex number.

1 = cos(0π) + isin(0π)r = 1, n = 4 and k = 0, 1, 2, 3

k = 0

0 2 0 0 2 01 cos sin

4 4

x i

1

1 cos0 sin 0 i

1 1 0 i

k = 1 0 2 1 0 2 1

1 cos sin4 4

x i

i

1 cos sin2 2

i

1 0 i

k = 2

0 2 2 0 2 21 cos sin

4 4

x i

1

1 cos sin i

1 1 0 i

k = 3

0 6 0 61 cos sin4 4

x i

i

3 31 cos sin2 2

i

1 0 i

Notice that the roots in example 2 all have a magnitude of 1 and are equally spaced around the unit circle. Also notice that the complex roots occur in conjugate pairs.The n distinct nth roots of 1 are called the nth roots of unity.

Example 8

Find the three cube roots of z = -6 + 6i to the nearest thousandth.This means x3 = -6 + 6i.

1. Change to polar complex form in degrees.

2. Now find the three cube roots.

72cis 135 z

k = 0

6 135 072cis3

x

6 72cis 45

1.442 1.442 i

k = 1

6 135 36072cis3

x

6 72cis 165

1.970 0.528 i

k = 2

6 135 72072cis3

x

6 72cis 285

0.528 1.970 i