Download 6.5 Trigonometric Form of a Complex number

6.5 Trigonometric Form of a
Complex number
De Moivre’s theorem
Graph the Complex number 2+3i
On a Complex Plane
imaginary
real
Absolute Value of the Complex
number 4+3i
On a Complex Plane
a  bi  a  b
2
imaginary
2
4  3i  4  3  5
2
2
real
Trigonometric Form of a Complex
Number
Given: a + bi
z  r cos  i sin  
r cos , i  r sin  
r is called the Modulus
r

r  a b
θ is the Argument
2
b
tan 
a
2
Writing a Complex number in
Trigonometric form
Point: r cos , i  r sin  
z  4  5i
Writing a Complex number in
Trigonometric form
Point: r cos , i  r sin  
z  4  5i
r   4  5i 
 4  5  41
2
2
Writing a Complex number in
Trigonometric form
Point: r cos , i  r sin  
z  4  5i
r   4  5i 
 4  5  41
2
2
128 .7
5
 5 
tan  
 tan 
 
4
 4
1
  51.3 so   180  51.3  128.7
 51.3
Writing a Complex number in
Trigonometric form
Point: r cos , i  r sin   z   41Cos128.7  41Sin128.7
z  41Cos128.7  iSin128.7
z  4  5i
r   4  5i 
 4  5  41
2
2
z  41 cis128.7
tan  
5
 5 
 tan 
 
4
 4
1
  51.3 so   180  51.3  128.7
Product and Quotient of two
Complex Numbers
z  r Cos  iSin

z  r Cos  iSin 
1
1
1
2
2
2
1
2
z  z  r  r Cos     iSin  
1
2
1
2
1
2
1
z r
 Cos     iSin  
z r
1
1
1
2
2
z 0
2
2
1
2

2

Find the product
z  20cis17
1
z  15cis28
2
z  z  20 15cis45
1
2
 300cos 45  iSin45
Find the product
z  20cis17
1
z  15cis28
2
z  z  20 15cis45
1
2
 300cos 45  iSin45
 2
2 
 300

i   150 2  150i 2
2 
 2
Find the quotient
z  20cis17
1
z  15cis28
2
z  z  20 15cis(17  28)
1
2
 300cos(11)  iSin(11)
 3000.9816  (0.1908i )   294.48  57.24i
Power of Complex Numbers
De Moivre’s Theorem
z  r Cos  iSin 
z  r Cos2  iSin2 
2
2
z  r Cosn   iSinn 
n
n
Power of Complex Numbers
Solve
z  4cis135
z  16cis2 135
2
Power of Complex Numbers
Solve
z  4cis135
z  16cis2 135
2
z  16Cos270  iSin270
z  160   1i   16i
2
2
N th Roots of a Complex Number
Where k = 0, 1, 2, …., n - 1
n

   2k 
   2k  
r  Cos
  iSin

 n 
 n 

Find the Cube roots of 1000
K = 0, 1, 2..(3-1) done
3
2k
2k 

1000 Cos
 iSin

3
3 

20 
20  

10 Cos
 iSin
  10
3
3 

2(1)
21 

10 Cos
 iSin
  5  5i 3
3
3 

22 
22  

10 Cos
 iSin
  5  5i 3
3
3 

Homework
Page 457 – 459
# 1, 9, 17, 25, 33,
41, 49, 57, 65,
77, 85, 93, 101,
109
Homework
Page 457 – 459
# 5, 13, 21, 29,
37, 45, 53, 61,
73, 81, 89, 97,
105, 113
“I got the Power!!!”