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# 0802 ch 8 day 2

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• ### 0802 ch 8 day 2

1. 1. 8.1 Polar Coordinates Day TwoPsalm 33:22 "May your unfailing love rest upon us, OLORD, even as we put our hope in you."
2. 2. We are now going to overlay a RectangularCoordinate system over a Polar Coordinate system inorder to identify the relationships between polar andrectangular coordinates.
3. 3. y x
4. 4. y P ( x, y ) x
5. 5. y P ( x, y ) θ x polar axis
6. 6. y P ( x, y ) P ( r,θ ) , r > 0 2 +y x2 r= θ x polar axis
7. 7. y P ( x, y ) P ( r,θ ) , r > 0 2 +y x2 r= θ x polar axis x = r cosθ
8. 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. 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 yx = r cosθ y = r sin θ x +y =r tan θ = x
10. 10. Find the rectangular coordinates: ⎛ π ⎞1. P ⎜ −3, ⎟ ⎝ 6 ⎠
11. 11. Find the rectangular coordinates: ⎛ π ⎞1. P ⎜ −3, ⎟ ⎝ 6 ⎠ x = r cosθ
12. 12. Find the rectangular coordinates: ⎛ π ⎞1. P ⎜ −3, ⎟ ⎝ 6 ⎠ x = r cosθ y = r sin θ
13. 13. Find the rectangular coordinates: ⎛ π ⎞1. P ⎜ −3, ⎟ ⎝ 6 ⎠ x = r cosθ y = r sin θ π = −3cos 6
14. 14. Find the rectangular coordinates: ⎛ π ⎞1. P ⎜ −3, ⎟ ⎝ 6 ⎠ x = r cosθ y = r sin θ π = −3cos 6 ⎛ 3 ⎞ = −3 ⎜ ⎝ 2 ⎟ ⎠
15. 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. 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. 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. 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. 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. 20. Find the rectangular coordinates: ⎛ 2π ⎞2. Q ⎜ 10, ⎟ ⎝ 9 ⎠
21. 21. Find the rectangular coordinates: ⎛ 2π ⎞2. Q ⎜ 10, ⎟ ⎝ 9 ⎠ x = r cosθ
22. 22. Find the rectangular coordinates: ⎛ 2π ⎞2. Q ⎜ 10, ⎟ ⎝ 9 ⎠ x = r cosθ y = r sin θ
23. 23. Find the rectangular coordinates: ⎛ 2π ⎞2. Q ⎜ 10, ⎟ ⎝ 9 ⎠ x = r cosθ y = r sin θ 2π = 10 cos 9
24. 24. Find the rectangular coordinates: ⎛ 2π ⎞2. Q ⎜ 10, ⎟ ⎝ 9 ⎠ x = r cosθ y = r sin θ 2π = 10 cos 9 ≈ 7.66
25. 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. 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. 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. 28. Polar coordinates ( r, θ ) can be obtained from therectangular coordinates ( x, y ) by: ⎧ y ⎪ Arc tan , x > 0 2 2 ⎪ x r= x +y θ = ⎨ ⎛ y ⎪ Arc tan ⎜ + π ⎞ , x < 0 ⎝ x ⎟ ⎠ ⎪ ⎩
29. 29. Find the polar coordinates:1. R (10, − 10 )
30. 30. Find the polar coordinates:1. R (10, − 10 ) 2 2 r = 10 + ( −10 )
31. 31. Find the polar coordinates:1. R (10, − 10 ) 2 r = 10 + ( −10 ) 2 ⎛ 10 ⎞ θ = Arc tan ⎜ − ⎟ ⎝ 10 ⎠
32. 32. Find the polar coordinates:1. R (10, − 10 ) 2 r = 10 + ( −10 ) 2 ⎛ 10 ⎞ θ = Arc tan ⎜ − ⎟ ⎝ 10 ⎠ = 200
33. 33. Find the polar coordinates:1. R (10, − 10 ) 2 r = 10 + ( −10 ) 2 ⎛ 10 ⎞ θ = Arc tan ⎜ − ⎟ ⎝ 10 ⎠ = 200 = 10 2
34. 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. 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. 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. 37. Find the polar coordinates: (2. S −4, 4 3 )
38. 38. Find the polar coordinates: (2. S −4, 4 3 ) 2 r= ( −4 ) 2 ( + 4 3 )
39. 39. Find the polar coordinates: (2. S −4, 4 3 ) 2 ⎛ 4 3 ⎞ r= ( −4 ) 2 ( + 4 3 ) θ = Arc tan ⎜ − ⎝ 4 ⎟ ⎠
40. 40. Find the polar coordinates: (2. S −4, 4 3 ) 2 ⎛ 4 3 ⎞ r= ( −4 ) 2 ( + 4 3 ) θ = Arc tan ⎜ − ⎝ 4 ⎟ ⎠ = 64
41. 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. 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. 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. 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. 45. Convert y = −1 to a polar equation.
46. 46. Convert y = −1 to a polar equation. y = −1
47. 47. Convert y = −1 to a polar equation. y = −1 r sin θ = −1
48. 48. Convert y = −1 to a polar equation. y = −1 r sin θ = −1 1 r=− sin θ
49. 49. Convert y = −1 to a polar equation. y = −1 r sin θ = −1 1 r=− sin θ r = − cscθ
50. 50. Convert r = 5 cosθ to a rectangular equation.
51. 51. Convert r = 5 cosθ to a rectangular equation. r = 5 cosθ
52. 52. Convert r = 5 cosθ to a rectangular equation. r = 5 cosθ 2 r = 5r cosθ
53. 53. Convert r = 5 cosθ to a rectangular equation. r = 5 cosθ 2 r = 5r cosθ 2 2 x + y = 5x
54. 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. 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 #1Take your life in your own hands, and what happens?A terrible thing: no one to blame. Erica Jong