1. 8.1 Polar Coordinates
Day Two
Psalm 33:22 "May your unfailing love rest upon us, O
LORD, even as we put our hope in you."

2. We are now going to overlay a Rectangular
Coordinate system over a Polar Coordinate system in
order to identify the relationships between polar and
rectangular coordinates.

3. y
x

4. y
P ( x, y )
x

5. y
P ( x, y )
θ
x polar axis

6. y
P ( x, y )
P ( r,θ ) , r > 0
2
+y
x2
r=
θ
x polar axis

7. y
P ( x, y )
P ( r,θ ) , r > 0
2
+y
x2
r=
θ
x polar axis
x = r cosθ

8. y
P ( x, y )
P ( r,θ ) , r > 0
2
+y
x2
y = r sin θ
r=
θ
x polar axis
x = r cosθ

9. y
P ( x, y )
P ( r,θ ) , r > 0
2
+y
x 2
y = r sin θ
r=
θ
x polar axis
x = r cosθ
2 2 2 y
x = r cosθ y = r sin θ x +y =r tan θ =
x

10. Find the rectangular coordinates:
⎛ π ⎞
1. P ⎜ −3, ⎟
⎝ 6 ⎠

11. Find the rectangular coordinates:
⎛ π ⎞
1. P ⎜ −3, ⎟
⎝ 6 ⎠
x = r cosθ

12. Find the rectangular coordinates:
⎛ π ⎞
1. P ⎜ −3, ⎟
⎝ 6 ⎠
x = r cosθ y = r sin θ

13. Find the rectangular coordinates:
⎛ π ⎞
1. P ⎜ −3, ⎟
⎝ 6 ⎠
x = r cosθ y = r sin θ
π
= −3cos
6

14. Find the rectangular coordinates:
⎛ π ⎞
1. P ⎜ −3, ⎟
⎝ 6 ⎠
x = r cosθ y = r sin θ
π
= −3cos
6
⎛ 3 ⎞
= −3 ⎜
⎝ 2 ⎟
⎠

15. Find the rectangular coordinates:
⎛ π ⎞
1. P ⎜ −3, ⎟
⎝ 6 ⎠
x = r cosθ y = r sin θ
π
= −3cos
6
⎛ 3 ⎞
= −3 ⎜
⎝ 2 ⎟
⎠
3 3
=−
2

16. Find the rectangular coordinates:
⎛ π ⎞
1. P ⎜ −3, ⎟
⎝ 6 ⎠
x = r cosθ y = r sin θ
π π
= −3cos = −3sin
6 6
⎛ 3 ⎞
= −3 ⎜
⎝ 2 ⎟
⎠
3 3
=−
2

17. Find the rectangular coordinates:
⎛ π ⎞
1. P ⎜ −3, ⎟
⎝ 6 ⎠
x = r cosθ y = r sin θ
π π
= −3cos = −3sin
6 6
⎛ 3 ⎞ ⎛ 1 ⎞
= −3 ⎜ = −3 ⎜ ⎟
⎝ 2 ⎟
⎠ ⎝ 2 ⎠
3 3
=−
2

18. Find the rectangular coordinates:
⎛ π ⎞
1. P ⎜ −3, ⎟
⎝ 6 ⎠
x = r cosθ y = r sin θ
π π
= −3cos = −3sin
6 6
⎛ 3 ⎞ ⎛ 1 ⎞
= −3 ⎜ = −3 ⎜ ⎟
⎝ 2 ⎟
⎠ ⎝ 2 ⎠
3 3 3
=− =−
2 2

19. Find the rectangular coordinates:
⎛ π ⎞
1. P ⎜ −3, ⎟
⎝ 6 ⎠
x = r cosθ y = r sin θ
π π
= −3cos = −3sin
6 6
⎛ 3 ⎞ ⎛ 1 ⎞
= −3 ⎜ = −3 ⎜ ⎟
⎝ 2 ⎟
⎠ ⎝ 2 ⎠
3 3 3
=− =−
2 2
⎛ 3 3 3 ⎞
⎜ − 2 , − 2 ⎟
⎝ ⎠

20. Find the rectangular coordinates:
⎛ 2π ⎞
2. Q ⎜ 10, ⎟
⎝ 9 ⎠

21. Find the rectangular coordinates:
⎛ 2π ⎞
2. Q ⎜ 10, ⎟
⎝ 9 ⎠
x = r cosθ

22. Find the rectangular coordinates:
⎛ 2π ⎞
2. Q ⎜ 10, ⎟
⎝ 9 ⎠
x = r cosθ y = r sin θ

23. Find the rectangular coordinates:
⎛ 2π ⎞
2. Q ⎜ 10, ⎟
⎝ 9 ⎠
x = r cosθ y = r sin θ
2π
= 10 cos
9

24. Find the rectangular coordinates:
⎛ 2π ⎞
2. Q ⎜ 10, ⎟
⎝ 9 ⎠
x = r cosθ y = r sin θ
2π
= 10 cos
9
≈ 7.66

25. Find the rectangular coordinates:
⎛ 2π ⎞
2. Q ⎜ 10, ⎟
⎝ 9 ⎠
x = r cosθ y = r sin θ
2π 2π
= 10 cos = 10sin
9 9
≈ 7.66

26. Find the rectangular coordinates:
⎛ 2π ⎞
2. Q ⎜ 10, ⎟
⎝ 9 ⎠
x = r cosθ y = r sin θ
2π 2π
= 10 cos = 10sin
9 9
≈ 7.66 ≈ 6.43

27. Find the rectangular coordinates:
⎛ 2π ⎞
2. Q ⎜ 10, ⎟
⎝ 9 ⎠
x = r cosθ y = r sin θ
2π 2π
= 10 cos = 10sin
9 9
≈ 7.66 ≈ 6.43
( 7.66, 6.43)

28. Polar coordinates ( r, θ ) can be obtained from the
rectangular coordinates ( x, y ) by:
⎧ y
⎪ Arc tan , x > 0
2 2 ⎪ x
r= x +y θ = ⎨
⎛ y
⎪ Arc tan ⎜ + π ⎞ , x < 0
⎝ x ⎟
⎠
⎪
⎩

29. Find the polar coordinates:
1. R (10, − 10 )

30. Find the polar coordinates:
1. R (10, − 10 )
2 2
r = 10 + ( −10 )

31. Find the polar coordinates:
1. R (10, − 10 )
2
r = 10 + ( −10 )
2 ⎛ 10 ⎞
θ = Arc tan ⎜ − ⎟
⎝ 10 ⎠

32. Find the polar coordinates:
1. R (10, − 10 )
2
r = 10 + ( −10 )
2 ⎛ 10 ⎞
θ = Arc tan ⎜ − ⎟
⎝ 10 ⎠
= 200

33. Find the polar coordinates:
1. R (10, − 10 )
2
r = 10 + ( −10 )
2 ⎛ 10 ⎞
θ = Arc tan ⎜ − ⎟
⎝ 10 ⎠
= 200
= 10 2

34. Find the polar coordinates:
1. R (10, − 10 )
2
r = 10 + ( −10 )
2 ⎛ 10 ⎞ Q IV
θ = Arc tan ⎜ − ⎟
⎝ 10 ⎠
= 200
= 10 2

35. Find the polar coordinates:
1. R (10, − 10 )
2
r = 10 + ( −10 )
2 ⎛ 10 ⎞ Q IV
θ = Arc tan ⎜ − ⎟
⎝ 10 ⎠
7π
= 200 =
4
= 10 2

36. Find the polar coordinates:
1. R (10, − 10 )
2
r = 10 + ( −10 )
2 ⎛ 10 ⎞ Q IV
θ = Arc tan ⎜ − ⎟
⎝ 10 ⎠
7π
= 200 =
4
= 10 2
⎛ 7π ⎞
⎜ 10 2,
⎝ ⎟
4 ⎠

37. Find the polar coordinates:
(
2. S −4, 4 3 )

38. Find the polar coordinates:
(
2. S −4, 4 3 )
2
r= ( −4 ) 2
(
+ 4 3 )

39. Find the polar coordinates:
(
2. S −4, 4 3 )
2 ⎛ 4 3 ⎞
r= ( −4 ) 2
(
+ 4 3 ) θ = Arc tan ⎜ −
⎝ 4 ⎟
⎠

40. Find the polar coordinates:
(
2. S −4, 4 3 )
2 ⎛ 4 3 ⎞
r= ( −4 ) 2
(
+ 4 3 ) θ = Arc tan ⎜ −
⎝ 4 ⎟
⎠
= 64

41. Find the polar coordinates:
(
2. S −4, 4 3 )
2 ⎛ 4 3 ⎞
r= ( −4 ) 2
(
+ 4 3 ) θ = Arc tan ⎜ −
⎝ 4 ⎟
⎠
= 64
=8

42. Find the polar coordinates:
(
2. S −4, 4 3 )
2 ⎛ 4 3 ⎞
r= ( −4 ) 2
(
+ 4 3 ) θ = Arc tan ⎜ −
⎝ 4 ⎟
⎠
Q II
= 64
=8

43. Find the polar coordinates:
(
2. S −4, 4 3 )
2 ⎛ 4 3 ⎞
r= ( −4 ) 2
(
+ 4 3 ) θ = Arc tan ⎜ −
⎝ 4 ⎟
⎠
Q II
2π
= 64 =
3
=8

44. Find the polar coordinates:
(
2. S −4, 4 3 )
2 ⎛ 4 3 ⎞
r= ( −4 ) 2
(
+ 4 3 ) θ = Arc tan ⎜ −
⎝ 4 ⎟
⎠
Q II
2π
= 64 =
3
=8
⎛ 2π ⎞
⎜ 8,
⎝ ⎟
3 ⎠

45. Convert y = −1 to a polar equation.

46. Convert y = −1 to a polar equation.
y = −1

47. Convert y = −1 to a polar equation.
y = −1
r sin θ = −1

48. Convert y = −1 to a polar equation.
y = −1
r sin θ = −1
1
r=−
sin θ

49. Convert y = −1 to a polar equation.
y = −1
r sin θ = −1
1
r=−
sin θ
r = − cscθ

50. Convert r = 5 cosθ to a rectangular equation.

51. Convert r = 5 cosθ to a rectangular equation.
r = 5 cosθ

52. Convert r = 5 cosθ to a rectangular equation.
r = 5 cosθ
2
r = 5r cosθ

53. Convert r = 5 cosθ to a rectangular equation.
r = 5 cosθ
2
r = 5r cosθ
2 2
x + y = 5x

54. Convert r = 5 cosθ to a rectangular equation.
r = 5 cosθ
2
r = 5r cosθ
2 2
x + y = 5x
2 2
x − 5x + y = 0

55. Convert r = 5 cosθ to a rectangular equation.
r = 5 cosθ
2
r = 5r cosθ
2 2
x + y = 5x
2 2
x − 5x + y = 0
HW #1
Take your life in your own hands, and what happens?
A terrible thing: no one to blame.
Erica Jong