Transcript Section P.5

The Complex Plane; De Moivre’s
Theorem
Polar Form
z  x  yi  r (cos  i sin  )
r0
0    2
1. Plot each complex number in the
complex plane and write it in polar
form. Express the argument in
degrees
(Similar to p.334 #11-22)
1  i
2. Plot each complex number in the
complex plane and write it in polar
form. Express the argument in
degrees
(Similar to p.334 #11-22)
5 3  5i
3. Plot each complex number in the
complex plane and write it in polar
form. Express the argument in
degrees
(Similar to p.334 #11-22)
7 i
4. Write each complex number in
rectangular form
(Similar to p.334 #23-32)
5(cos  i sin  )
5. Write each complex number in
rectangular form
(Similar to p.334 #23-32)



4 cos  i sin 
6
6

6. Write each complex number in
rectangular form
(Similar to p.334 #23-32)

7 cos22  i sin 22



Multiplication and Division
z1  r1 (cos1  i sin 1 )
z 2  r2 (cos 2  i sin  2 )
then
z1 z 2  r1r2 cos1   2   i sin 1   2 
z1 r1
 cos1   2   i sin 1   2 
z 2 r2
7. Find zw and z/w. Leave your
answers in polar form
(Similar to p.334 #33-40)
z  cos 200  i sin 200


w  cos120  i sin 120


8. Find zw and z/w. Leave your
answers in polar form
(Similar to p.334 #33-40)



z  8 cos  i sin 
4
4

3
3 

w  4 cos  i sin

8
8 

9. Find zw and z/w. Leave your
answers in polar form
(Similar to p.334 #33-40)
z  3  3i
w   3 i
De Moivre's T heorem
z  r (cos  i sin  )
then
z  r cos(n )  i sin(n )
(n  1)
n
n
10. Write each expression in the
standard form a + bi
(Similar to p.334 #41-52)
2cos105  i sin105 


4
11. Write each expression in the
standard form a + bi
(Similar to p.334 #41-52)
 
3
3 
2
cos

i
sin



20
20 
 
10
Let w = r(cos θo + i sin θo be a complex number,
and let n > 2 be an integer. There are n distinct
complex nth roots given by:
   0 2k 
  0 2k 
zk  r cos 
  i sin 

n 
n 
n
 n
where k  0,1,2,...,n  1
n
12. Find all the complex roots.
Leave your answers in polar form
with the argument in degrees
(Similar to p.335 #53-60)
T hecomplexfourthrootsof 4  4i
13. Solve the following equation.
Leave your answers in polar form
with the argument in degrees
(Similar to p.335 #53-60)
x  27
5