1. 8.3 Polar Form of
Complex Numbers
Matthew 10:32-33 "So everyone who acknowledges me
before men, I also will acknowledge before my Father
who is in heaven, but whoever denies me before men, I
also will deny before my Father who is in heaven."

2. When graphing complex numbers, we use the
Complex Plane

3. When graphing complex numbers, we use the
Complex Plane
imaginary axis
real axis

4. When graphing complex numbers, we use the
Complex Plane
imaginary axis
P ( a,b )
real axis
P is a complex number
a + bi

5. Let’s plot some points:
i
R

6. Let’s plot some points:
i
P 3 + 5i
R

7. Let’s plot some points:
i
P 3 + 5i
P ( 3, 5 )
R

8. Let’s plot some points:
i
P 3 + 5i
P ( 3, 5 )
Q 2−i
R

9. Let’s plot some points:
i
P 3 + 5i
P ( 3, 5 )
Q 2−i
R
Q ( 2, − 1)

10. Let’s plot some points:
i
P 3 + 5i
P ( 3, 5 )
Q 2−i
R 4i R
Q ( 2, − 1)

11. Let’s plot some points:
i
P 3 + 5i
P ( 3, 5 )
Q 2−i R ( 0, 4 )
R 4i R
Q ( 2, − 1)

12. Let’s plot some points:
i
P 3 + 5i
P ( 3, 5 )
Q 2−i R ( 0, 4 )
R 4i R
Q ( 2, − 1)
S − 2

13. Let’s plot some points:
i
P 3 + 5i
P ( 3, 5 )
Q 2−i R ( 0, 4 )
R 4i R
Q ( 2, − 1)
S − 2
(
S − 2, 0 )

14. Let’s plot some points:
i
P 3 + 5i
P ( 3, 5 )
Q 2−i R ( 0, 4 )
R 4i R
Q ( 2, − 1)
S − 2
(
S − 2, 0 )
A complex number in rectangular form is a + bi
and has coordinates ( a, b ) in the complex plane.

15. We can also use the complex plane but express
the complex number in polar form rather than
rectangular form.

16. We can also use the complex plane but express
the complex number in polar form rather than
rectangular form.
i
P ( a, b )
r
b
θ
a
R

17. We can also use the complex plane but express
the complex number in polar form rather than
rectangular form.
i a = r cosθ
P ( a, b ) b = r sin θ
r 2 2
b r = a +b
θ
a
R

18. We can also use the complex plane but express
the complex number in polar form rather than
rectangular form.
i a = r cosθ
P ( a, b ) b = r sin θ
r 2 2
b r = a +b
θ
a
R
so ... a + bi = r cosθ + ( r sin θ ) i

19. We can also use the complex plane but express
the complex number in polar form rather than
rectangular form.
i a = r cosθ
P ( a, b ) b = r sin θ
r 2 2
b r = a +b
θ
a
R
so ... a + bi = r cosθ + ( r sin θ ) i
= r ( cosθ + i sin θ )

20. The polar form of a complex number
r ( cosθ + i sin θ )
is abbreviated: r cis θ

21. The polar form of a complex number
r ( cosθ + i sin θ )
is abbreviated: r cis θ
a + bi rectangular form of a complex number
r cis θ polar form of a complex number

22. Express in rectangular form:
⎛ π π ⎞
1. 2 ⎜ cos + i sin ⎟
⎝ 3 3 ⎠

23. Express in rectangular form:
⎛ π π ⎞
1. 2 ⎜ cos + i sin ⎟
⎝ 3 3 ⎠
⎛ 1 3 ⎞
2 ⎜ + i ⎟
⎝ 2 2 ⎠

24. Express in rectangular form:
⎛ π π ⎞
1. 2 ⎜ cos + i sin ⎟
⎝ 3 3 ⎠
⎛ 1 3 ⎞
2 ⎜ + i ⎟
⎝ 2 2 ⎠
2 6
+ i
2 2

25. Express in rectangular form:
2. 5 cis π

26. Express in rectangular form:
2. 5 cis π
5 ( cos π + i sin π )

27. Express in rectangular form:
2. 5 cis π
5 ( cos π + i sin π )
5 ( −1+ 0i )

28. Express in rectangular form:
2. 5 cis π
5 ( cos π + i sin π )
5 ( −1+ 0i )
−5

29. Express in polar form:
3. 5 − 5i

30. Express in polar form:
3. 5 − 5i we need r and θ

31. Express in polar form:
3. 5 − 5i we need r and θ
2 2
r = 5 + ( −5 )

32. Express in polar form:
3. 5 − 5i we need r and θ
2 2
r = 5 + ( −5 )
= 50

33. Express in polar form:
3. 5 − 5i we need r and θ
2 2
r = 5 + ( −5 )
= 50
=5 2

34. Express in polar form:
3. 5 − 5i we need r and θ
2 2 ( 5, − 5 ) is in QIV
r = 5 + ( −5 )
= 50
=5 2

35. Express in polar form:
3. 5 − 5i we need r and θ
2 2 ( 5, − 5 ) is in QIV
r = 5 + ( −5 )
b −5
tan θ = = = −1
= 50 a 5
=5 2

36. Express in polar form:
3. 5 − 5i we need r and θ
2 2 ( 5, − 5 ) is in QIV
r = 5 + ( −5 )
b −5
tan θ = = = −1
= 50 a 5
7π
=5 2 θ=
4

37. Express in polar form:
3. 5 − 5i we need r and θ
2 2 ( 5, − 5 ) is in QIV
r = 5 + ( −5 )
b −5
tan θ = = = −1
= 50 a 5
7π
=5 2 θ=
4
⎛ 7π 7π ⎞
5 2 ⎜ cos + i sin ⎟
⎝ 4 4 ⎠

38. Express in polar form: (in your groups)
4. 3 + i 3

39. Express in polar form: (in your groups)
4. 3 + i 3
2 2 θ is in QI
r= 3 + 3
3
tan θ =
= 12 3
π
=2 3 θ=
6
π
2 3 cis
6

40. HW #4
Someone’s opinion of you does not have to
become your reality.
Les Brown