Download Complex Numbers, Trig Form of a Complex Number, DeMoivre`s

 Pre Calc 6.5 Day 1
Trigonometric Form
of a Complex Number
NOTES
COMPLETED
6.5 Trigonometric Form of a Complex Number
A complex number looks like a+
bi a.k.a.
Plot (2,3) in the coordinate plane
Plot (­1,­2) 3
2
1
­2 ­1­1
­2
1 2 3
Absolute Value
Real Number
1. 5 =
2. ­5 =
3. Find ­2 + 5i =
Plot 2+ 3i in the coordinate plane
Plot ­1­2i
Imaginary numbers always appear in pairs a ± bi
Find the conjugates:
Horizontal axis REAL
axis
1. 2+ 3i _______
Vertical 2. 5 ­ 6i _______
axis IMAGINARY axis
3. ­2 + 3i _______
4. 7 _______
5. ­3i ______ Complex Number: Distance between (0,0) and point (a,b). The absolute value of the complex number a + bi is
Trig Form of a Complex Number
r
θ
y
x
r = x 2 + y2
y
tan θ = ___
x
cos θ = adj sin θ = opp hyp
hyp
(a,b)
cos θ =
sin θ =
x =
y =
(x,y)
.
Radius of Circle
Trig Form of a Complex Number z = a + bi or z = x + yi is
z = a + bi z = x + y i
z = r cosθ + r sinθ i
z = r(cos θ + i sin θ)
You do not r
is called the ________ of z.
Modulus
have to is called an _________ of z.
Argument
know for θ
test
Write the complex number in TRIG FORM:
1. z = ­2 ­ 2 √3 i
or
Answers written 3 ways
using radians
using degrees
shortcut way using radians
2) 6 ­ 6i
4
3. z = ­ 3√ 3 +3i
4. ­7+ 0i
x= y=
r=
cos =
sin = x= y=
r=
cos =
sin =
0
0
5. z = 4 (cos120 + i sin120 )
1
6. z = (cos35 + i sin35 )
0
2
0
Note:
if the degree/radian given is off the Trig Circle, you can use decimals!! woohoo!!
Type the entire problem into your calculator and don't forget to round to 2 places.
Homework 6.5
pg 479:
11­23 odd
[
31­43 odd[
WPF/skip all graphs/skip #19
write all 3 formats of answer
WPF/skip all graphs
find answers like pg 3 notes
Pre Calc 6.5 Day 2
Trigonometric Form
of a Complex Number
NOTES
COMPLETED
6.5 Multiplication and Division of Complex Numbers
[
x and +
and ­
z1z2=
your work should keep 0 within the interval [0,2 ) ll
z1
z 2 =
4
Remember:
numbers need to be in the interval [0,2 ) ll
=
1
distance from zero
Use DeMoivre's Theorem to simplify
[We are not going to use DeMoivre's Thm. Just use your grapher]
­8i
(1 + i)6 = ____ 2035 ­ 828
­32
3
16
4096
5) (­1 + 3 i)12 = _______
­4
6) (­1 + i)4 = ____ Note: ­4 ­ 8E­13
= ­4 ­ 0
= ­4
Today's Homework 6.5
pg 479: 47­55 odd (AO).
Keep original units. Don't solve down to standard form.
Stop at the line before you get rid of the cos and sin.
Example of how to do problems 47­55 odd
pg 479: 59­66 all (part c only)
Simplify. Show all work and write answers in
standard form