The Complex Plane; De Moivre’s Theorem

20

0

)sin(cos

r

iryixz

Polar Form

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)

i 1

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)

i535

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)

i7

4. Write each complex number in rectangular form

(Similar to p.334 #23-32)

)sin(cos5 i

5. Write each complex number in rectangular form

(Similar to p.334 #23-32)

6sin

6cos4

i

6. Write each complex number in rectangular form

(Similar to p.334 #23-32)

22sin22cos7 i

21212

1

2

1

21212121

2222

1111

sincos

sincos

)sin(cos

)sin(cos

ir

r

z

z

irrzz

then

irz

irz

Multiplication and Division

7. Find zw and z/w. Leave your answers in polar form

(Similar to p.334 #33-40)

120sin120cos

200sin200cos

iw

iz

8. Find zw and z/w. Leave your answers in polar form

(Similar to p.334 #33-40)

8

3sin

8

3cos4

4sin

4cos8

iw

iz

9. Find zw and z/w. Leave your answers in polar form

(Similar to p.334 #33-40)

iw

iz

3

33

)1(

)sin()cos(

)sin(cos

Theorem sMoivre' De

n

ninrz

then

irz

nn

10. Write each expression in the standard form a + bi

(Similar to p.334 #41-52)

4105sin105cos2 i

11. Write each expression in the standard form a + bi

(Similar to p.334 #41-52)

10

20

3sin

20

3cos2

i

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:

1,...,2,1,0

2sin

2cos 00

nkwhere

n

k

ni

n

k

nrz n

k

12. Find all the complex roots. Leave your answers in polar form

with the argument in degrees(Similar to p.335 #53-60)

i44 of rootsfourth complex The

13. Solve the following equation. Leave your answers in polar form

with the argument in degrees(Similar to p.335 #53-60)

275 x