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8.1 Polar Coordinates                     Day TwoPsalm 33:22 "May your unfailing love rest upon us, OLORD, even as we put ...
We are now going to overlay a RectangularCoordinate system over a Polar Coordinate system inorder to identify the relation...
y    x
y    P ( x, y )                 x
y        P ( x, y )    θ                     x   polar axis
y                 P ( x, y )                 P ( r,θ ) , r > 0             2         +y        x2    r=    θ              ...
y                     P ( x, y )                     P ( r,θ ) , r > 0             2          +y        x2    r=    θ     ...
y                     P ( x, y )                     P ( r,θ ) , r > 0             2          +y        x2                ...
y                                    P ( x, y )                                    P ( r,θ ) , r > 0                      ...
Find the rectangular coordinates:     ⎛    π ⎞1. P ⎜ −3, ⎟     ⎝    6 ⎠
Find the rectangular coordinates:     ⎛    π ⎞1. P ⎜ −3, ⎟     ⎝    6 ⎠   x = r cosθ
Find the rectangular coordinates:     ⎛    π ⎞1. P ⎜ −3, ⎟     ⎝    6 ⎠   x = r cosθ           y = r sin θ
Find the rectangular coordinates:     ⎛    π ⎞1. P ⎜ −3, ⎟     ⎝    6 ⎠   x = r cosθ           y = r sin θ          ...
Find the rectangular coordinates:     ⎛    π ⎞1. P ⎜ −3, ⎟     ⎝    6 ⎠   x = r cosθ           y = r sin θ          ...
Find the rectangular coordinates:     ⎛    π ⎞1. P ⎜ −3, ⎟     ⎝    6 ⎠   x = r cosθ           y = r sin θ          ...
Find the rectangular coordinates:     ⎛    π ⎞1. P ⎜ −3, ⎟     ⎝    6 ⎠   x = r cosθ           y = r sin θ          ...
Find the rectangular coordinates:     ⎛    π ⎞1. P ⎜ −3, ⎟     ⎝    6 ⎠   x = r cosθ           y = r sin θ          ...
Find the rectangular coordinates:     ⎛    π ⎞1. P ⎜ −3, ⎟     ⎝    6 ⎠   x = r cosθ           y = r sin θ          ...
Find the rectangular coordinates:     ⎛    π ⎞1. P ⎜ −3, ⎟     ⎝    6 ⎠   x = r cosθ                     y = r sin θ...
Find the rectangular coordinates:      ⎛    2π ⎞2. Q ⎜ 10,    ⎟      ⎝     9 ⎠
Find the rectangular coordinates:      ⎛    2π ⎞2. Q ⎜ 10,    ⎟      ⎝     9 ⎠   x = r cosθ
Find the rectangular coordinates:      ⎛    2π ⎞2. Q ⎜ 10,    ⎟      ⎝     9 ⎠   x = r cosθ           y = r sin θ
Find the rectangular coordinates:      ⎛    2π ⎞2. Q ⎜ 10,    ⎟      ⎝     9 ⎠   x = r cosθ           y = r sin θ   ...
Find the rectangular coordinates:      ⎛    2π ⎞2. Q ⎜ 10,    ⎟      ⎝     9 ⎠   x = r cosθ           y = r sin θ   ...
Find the rectangular coordinates:      ⎛    2π ⎞2. Q ⎜ 10,    ⎟      ⎝     9 ⎠   x = r cosθ           y = r sin θ   ...
Find the rectangular coordinates:      ⎛    2π ⎞2. Q ⎜ 10,    ⎟      ⎝     9 ⎠   x = r cosθ           y = r sin θ   ...
Find the rectangular coordinates:      ⎛    2π ⎞2. Q ⎜ 10,    ⎟      ⎝     9 ⎠   x = r cosθ                   y = r ...
Polar coordinates ( r, θ ) can be obtained from therectangular coordinates ( x, y ) by:                         ⎧        ...
Find the polar coordinates:1. R (10, − 10 )
Find the polar coordinates:1. R (10, − 10 )           2           2    r = 10 + ( −10 )
Find the polar coordinates:1. R (10, − 10 )           2    r = 10 + ( −10 )                       2               ⎛ 10 ⎞...
Find the polar coordinates:1. R (10, − 10 )           2    r = 10 + ( −10 )                       2               ⎛ 10 ⎞...
Find the polar coordinates:1. R (10, − 10 )           2    r = 10 + ( −10 )                       2               ⎛ 10 ⎞...
Find the polar coordinates:1. R (10, − 10 )           2    r = 10 + ( −10 )                       2               ⎛ 10 ⎞...
Find the polar coordinates:1. R (10, − 10 )           2    r = 10 + ( −10 )                       2               ⎛ 10 ⎞...
Find the polar coordinates:1. R (10, − 10 )           2    r = 10 + ( −10 )                       2                    ⎛ ...
Find the polar coordinates:     (2. S −4, 4 3   )
Find the polar coordinates:     (2. S −4, 4 3          )                                  2   r=    ( −4 )   2            ...
Find the polar coordinates:     (2. S −4, 4 3          )                                  2               ⎛ 4 3 ⎞   r=  ...
Find the polar coordinates:     (2. S −4, 4 3          )                                  2               ⎛ 4 3 ⎞   r=  ...
Find the polar coordinates:     (2. S −4, 4 3          )                                  2               ⎛ 4 3 ⎞   r=  ...
Find the polar coordinates:     (2. S −4, 4 3          )                                  2               ⎛ 4 3 ⎞   r=  ...
Find the polar coordinates:     (2. S −4, 4 3          )                                  2               ⎛ 4 3 ⎞   r=  ...
Find the polar coordinates:     (2. S −4, 4 3          )                                       2               ⎛ 4 3 ⎞  ...
Convert y = −1 to a polar equation.
Convert y = −1 to a polar equation.               y = −1
Convert y = −1 to a polar equation.               y = −1           r sin θ = −1
Convert y = −1 to a polar equation.               y = −1           r sin θ = −1                      1                r=− ...
Convert y = −1 to a polar equation.               y = −1           r sin θ = −1                      1                r=− ...
Convert r = 5 cosθ to a rectangular equation.
Convert r = 5 cosθ to a rectangular equation.               r = 5 cosθ
Convert r = 5 cosθ to a rectangular equation.               r = 5 cosθ               2              r = 5r cosθ
Convert r = 5 cosθ to a rectangular equation.               r = 5 cosθ                2               r = 5r cosθ         ...
Convert r = 5 cosθ to a rectangular equation.                r = 5 cosθ                 2                r = 5r cosθ      ...
Convert r = 5 cosθ to a rectangular equation.                r = 5 cosθ                 2                r = 5r cosθ      ...
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0802 ch 8 day 2

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  • Transcript of "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

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