1. 8.1 Polar Coordinates
Day Three
1 Corinthians 1:17 "For Christ did not send me to baptize
but to preach the gospel, and not with words of eloquent
wisdom, lest the cross of Christ be emptied of its power."

2. Find the rectangular coordinates for the point whose
⎛ 5π ⎞
polar coordinates are ⎜ −4, ⎟
⎝ 2 ⎠

3. Find the rectangular coordinates for the point whose
⎛ 5π ⎞
polar coordinates are ⎜ −4, ⎟
⎝ 2 ⎠
x = r cosθ

4. Find the rectangular coordinates for the point whose
⎛ 5π ⎞
polar coordinates are ⎜ −4, ⎟
⎝ 2 ⎠
x = r cosθ y = r sin θ

5. Find the rectangular coordinates for the point whose
⎛ 5π ⎞
polar coordinates are ⎜ −4, ⎟
⎝ 2 ⎠
x = r cosθ y = r sin θ
5π π
θ= =
2 2

6. Find the rectangular coordinates for the point whose
⎛ 5π ⎞
polar coordinates are ⎜ −4, ⎟
⎝ 2 ⎠
x = r cosθ y = r sin θ
5π π
θ= =
2 2
π
= −4 cos
2

7. Find the rectangular coordinates for the point whose
⎛ 5π ⎞
polar coordinates are ⎜ −4, ⎟
⎝ 2 ⎠
x = r cosθ y = r sin θ
5π π
θ= =
2 2
π
= −4 cos
2
= −4 ( 0 )

8. Find the rectangular coordinates for the point whose
⎛ 5π ⎞
polar coordinates are ⎜ −4, ⎟
⎝ 2 ⎠
x = r cosθ y = r sin θ
5π π
θ= =
2 2
π
= −4 cos
2
= −4 ( 0 )
=0

9. Find the rectangular coordinates for the point whose
⎛ 5π ⎞
polar coordinates are ⎜ −4, ⎟
⎝ 2 ⎠
x = r cosθ y = r sin θ
5π π
θ= =
2 2
π π
= −4 cos = −4 sin
2 2
= −4 ( 0 )
=0

10. Find the rectangular coordinates for the point whose
⎛ 5π ⎞
polar coordinates are ⎜ −4, ⎟
⎝ 2 ⎠
x = r cosθ y = r sin θ
5π π
θ= =
2 2
π π
= −4 cos = −4 sin
2 2
= −4 ( 0 ) = −4 (1)
=0

11. Find the rectangular coordinates for the point whose
⎛ 5π ⎞
polar coordinates are ⎜ −4, ⎟
⎝ 2 ⎠
x = r cosθ y = r sin θ
5π π
θ= =
2 2
π π
= −4 cos = −4 sin
2 2
= −4 ( 0 ) = −4 (1)
=0 = −4

12. Find the rectangular coordinates for the point whose
⎛ 5π ⎞
polar coordinates are ⎜ −4, ⎟
⎝ 2 ⎠
x = r cosθ y = r sin θ
5π π
θ= =
2 2
π π
= −4 cos = −4 sin
2 2
= −4 ( 0 ) = −4 (1)
=0 = −4
( 0, − 4 )

13. Convert the rectangular coordinates to polar
coordinates with r > 0 and 0 ≤ θ < 2π :
(3 )
3, − 3

14. Convert the rectangular coordinates to polar
coordinates with r > 0 and 0 ≤ θ < 2π :
(3 3, − 3 )
2 2 2
r =x +y

15. Convert the rectangular coordinates to polar
coordinates with r > 0 and 0 ≤ θ < 2π :
(3 3, − 3 )
2 2 2
r =x +y
2
2
(
r = 3 3 + ( −3) ) 2

16. Convert the rectangular coordinates to polar
coordinates with r > 0 and 0 ≤ θ < 2π :
(3 3, − 3 )
2 2 2
r =x +y
2
2
(
r = 3 3 + ( −3) ) 2
2
r = 27 + 9

17. Convert the rectangular coordinates to polar
coordinates with r > 0 and 0 ≤ θ < 2π :
(3 3, − 3 )
2 2 2
r =x +y
2
2
(
r = 3 3 + ( −3) ) 2
2
r = 27 + 9
2
r = 36

18. Convert the rectangular coordinates to polar
coordinates with r > 0 and 0 ≤ θ < 2π :
(3 3, − 3 )
2 2 2
r =x +y
2
2
(
r = 3 3 + ( −3) ) 2
2
r = 27 + 9
2
r = 36
r=±6

19. Convert the rectangular coordinates to polar
coordinates with r > 0 and 0 ≤ θ < 2π :
(3 3, − 3 )
2 2 2
r =x +y
2
2
(
r = 3 3 + ( −3) ) 2
2
r = 27 + 9
2
r = 36
r=±6
r=6

20. Convert the rectangular coordinates to polar
coordinates with r > 0 and 0 ≤ θ < 2π :
(3 3, − 3 )
2
r =x +y 2 2
3
tan θ = −
2 3 3
2
(
r = 3 3 + ( −3) ) 2
2
r = 27 + 9
2
r = 36
r=±6
r=6

21. Convert the rectangular coordinates to polar
coordinates with r > 0 and 0 ≤ θ < 2π :
(3 3, − 3 )
2
r =x +y 2 2
3
tan θ = −
2 3 3
2
(
r = 3 3 + ( −3) ) 2
1
2 tan θ = −
r = 27 + 9 3
2
r = 36
r=±6
r=6

22. Convert the rectangular coordinates to polar
coordinates with r > 0 and 0 ≤ θ < 2π :
(3 3, − 3 )
2
r =x +y 2 2
3
tan θ = −
2 3 3
2
(
r = 3 3 + ( −3) ) 2
1
2 tan θ = −
r = 27 + 9 3
2
r = 36 (3 )
3, − 3 is in QIV
r=±6
r=6

23. Convert the rectangular coordinates to polar
coordinates with r > 0 and 0 ≤ θ < 2π :
(3 3, − 3 )
2
r =x +y 2 2
3
tan θ = −
2 3 3
2
(
r = 3 3 + ( −3) ) 2
1
2 tan θ = −
r = 27 + 9 3
2
r = 36 (3 )
3, − 3 is in QIV
r=±6 11π
θ=
r=6 6

24. Convert the rectangular coordinates to polar
coordinates with r > 0 and 0 ≤ θ < 2π :
(3 3, − 3 )
2
r =x +y 2 2
3
tan θ = −
2 3 3
2
(
r = 3 3 + ( −3) ) 2
1
2 tan θ = −
r = 27 + 9 3
2
r = 36 (3 )
3, − 3 is in QIV
r=±6 11π
θ=
r=6 6
⎛ 11π ⎞
⎜ 6,
⎝ ⎟
6 ⎠

25. Convert x = 1 (the equation) to polar form:

26. Convert x = 1 (the equation) to polar form:
x =1

27. Convert x = 1 (the equation) to polar form:
x =1
r cosθ = 1

28. Convert x = 1 (the equation) to polar form:
x =1
r cosθ = 1
1
r=
cosθ

29. Convert x = 1 (the equation) to polar form:
x =1
r cosθ = 1
1
r=
cosθ
r = secθ

30. Convert y = 4x to polar form:
2

31. Convert y = 4x to polar form:
2
2
y = 4x

32. Convert y = 4x to polar form:
2
2
y = 4x
2
( r sinθ ) = 4 ( r cosθ )

33. Convert y = 4x to polar form:
2
2
y = 4x
2
( r sinθ ) = 4 ( r cosθ )
2 2
r sin θ = 4r cosθ

34. Convert y = 4x to polar form:
2
2
y = 4x
2
( r sinθ ) = 4 ( r cosθ )
2 2
r sin θ = 4r cosθ
4 cosθ
r= 2
sin θ

35. Convert y = 4x to polar form:
2
2
y = 4x
2
( r sinθ ) = 4 ( r cosθ )
2 2
r sin θ = 4r cosθ
4 cosθ
r= 2
sin θ
1 cosθ
r = 4⋅ ⋅
sin θ sin θ

36. Convert y = 4x to polar form:
2
2
y = 4x
2
( r sinθ ) = 4 ( r cosθ )
2 2
r sin θ = 4r cosθ
4 cosθ
r= 2
sin θ
1 cosθ
r = 4⋅ ⋅
sin θ sin θ
r = 4 cscθ cot θ

37. Convert the polar equation to rectangular form.
r = 6 cosθ

38. Convert the polar equation to rectangular form.
r = 6 cosθ
2
r = 6r cosθ

39. Convert the polar equation to rectangular form.
r = 6 cosθ
2
r = 6r cosθ
2 2
x + y = 6x

40. Convert the polar equation to rectangular form.
r = 6 cosθ
2
r = 6r cosθ
2 2
x + y = 6x
Bonus Math ... what geometric shape is this?

41. Convert the polar equation to rectangular form.
r = 6 cosθ
2
r = 6r cosθ
2 2
x + y = 6x
Bonus Math ... what geometric shape is this?
(x 2
− 6x + 9 ) + y = 9
2

42. Convert the polar equation to rectangular form.
r = 6 cosθ
2
r = 6r cosθ
2 2
x + y = 6x
Bonus Math ... what geometric shape is this?
(x 2
− 6x + 9 ) + y = 9
2
2 2 2
( x − 3) +y =3

43. Convert the polar equation to rectangular form.
r = 6 cosθ
2
r = 6r cosθ
2 2
x + y = 6x
Bonus Math ... what geometric shape is this?
(x 2
− 6x + 9 ) + y = 9
2
2 2 2
( x − 3) +y =3
Circle of radius 3 centered at ( 3, 0 )

45. Convert the polar equation to rectangular form.
r = 5 + 4 cosθ

46. Convert the polar equation to rectangular form.
r = 5 + 4 cosθ
2
r = 5r + 4r cosθ

47. Convert the polar equation to rectangular form.
r = 5 + 4 cosθ
2
r = 5r + 4r cosθ
2 2
x + y = 5r + 4x

48. Convert the polar equation to rectangular form.
r = 5 + 4 cosθ
2
r = 5r + 4r cosθ
2 2
x + y = 5r + 4x
2 2 2 2
x + y = 5 x + y + 4x

49. Convert the polar equation to rectangular form.
r = 5 + 4 cosθ
2
r = 5r + 4r cosθ
2 2
x + y = 5r + 4x
2 2 2 2
x + y = 5 x + y + 4x
Good enough. We could simplify more, but it wouldn’t
really achieve much ...

50. HW #2
We need to restore the full meaning of that old
word, duty. It is the other side of rights.
Pearl Buck