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Transcript of Chapter 9 Polar Coordinates; Vectors shubbard/HPCanswers/PR_9e_ISM... Chapter 9 Polar Coordinates;...

  • 865

    Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall

    Chapter 9

    Polar Coordinates; Vectors Section 9.1

    1.

    The point lies in quadrant IV.

    2. 2

    6 9

    2   =   

    3. b

    a

    4. 4

    π−

    5. pole, polar axis

    6. True

    7. False

    8. cosr θ ; sinr θ

    9. A

    10. B

    11. C

    12. C

    13. B

    14. D

    15. A

    16. D

    17.

    18.

    19.

    20.

    21.

    22.

    23.

  • Chapter 9: Polar Coordinates; Vectors

    866

    Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall

    24.

    25.

    26.

    27.

    28.

    29.

    30.

    31.

    a. 4

    0, 2 0 5, 3

    r θ π > − π ≤ < −   

    b. 5

    0, 0 2 5, 3

    r θ π < ≤ < π −   

    c. 8

    0, 2 4 5, 3

    r θ π > π ≤ < π    

    32.

    a. 5

    0, 2 0 4, 4

    r θ π > − π ≤ < −   

    b. 7

    0, 0 2 4, 4

    r θ π < ≤ < π −   

    c. 11

    0, 2 4 4, 4

    r θ π > π ≤ < π    

    33.

    a. ( )0, 2 0 2, 2r θ> − π ≤ < − π b. ( )0, 0 2 2,r θ< ≤ < π − π c. ( )0, 2 4 2, 2r θ> π ≤ < π π

    34.

    a. ( )0, 2 0 3,r θ> − π ≤ < − π b. ( )0, 0 2 3, 0r θ< ≤ < π − c. ( )0, 2 4 3, 3r θ> π ≤ < π π

  • Section 9.1: Polar Coordinates

    867

    Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall

    35.

    a. 3

    0, 2 0 1, 2

    r θ π > − π ≤ < −   

    b. 3

    0, 0 2 1, 2

    r θ π < ≤ < π −   

    c. 5

    0, 2 4 1, 2

    r θ π > π ≤ < π    

    36.

    a. ( )0, 2 0 2,r θ> − π ≤ < − π

    b. ( )0, 0 2 2, 0r θ< ≤ < π −

    c. ( )0, 2 4 2, 3r θ> π ≤ < π π

    37.

    a. 5

    0, 2 0 3, 4

    r θ π > − π ≤ < −   

    b. 7

    0, 0 2 3, 4

    r θ π < ≤ < π −   

    c. 11

    ( 0, 2 4 3, 4

    r θ π > π ≤ < π    

    38.

    a. 5

    0, 2 0 2, 3

    r θ π > − π ≤ < −   

    b. 4

    0, 0 2 2, 3

    r θ π < ≤ < π −   

    c. 7

    0, 2 4 2, 3

    r θ π > π ≤ < π    

    39. cos 3cos 3 0 0 2

    x r θ π= = = ⋅ =

    sin 3sin 3 1 3 2

    y r θ π= = = ⋅ =

    Rectangular coordinates of the point 3, 2

    π     

    are

    ( )0, 3 .

    40. 3

    cos 4cos 4 0 0 2

    x r θ π= = = ⋅ =

    3 sin 4sin 4 ( 1) 4

    2 y r θ π= = = ⋅ − = −

    Rectangular coordinates of the point 3

    4, 2

    π     

    are

    ( )0, 4− .

    41. cos 2cos 0 2 1 2x r θ= = − = − ⋅ = − sin – 2sin 0 – 2 0 0y r θ= = = ⋅ =

    Rectangular coordinates of the point ( )– 2, 0 are ( )2, 0− .

    42. cos 3cos 3( 1) 3x r θ= = − π = − − = sin 3sin 3 0 0y r θ= = − π = − ⋅ =

    Rectangular coordinates of the point ( )3,− π are ( )3, 0 .

    43. 3

    cos 6cos150º 6 3 3 2

    x r θ  

    = = = − = −    

    1 sin 6sin150º 6 3

    2 y r θ= = = ⋅ =

    Rectangular coordinates of the point ( )6, 150º are ( )3 3, 3− .

    44. 1 5

    cos 5cos300º 5 2 2

    x r θ= = = ⋅ =

    3 5 3 sin 5sin 300º 5

    2 2 y r θ

      = = = − = −  

     

  • Chapter 9: Polar Coordinates; Vectors

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    Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall

    Rectangular coordinates of the point ( )5, 300º

    are 5 5 3

    , 2 2

      −  

      .

    45. 3 2

    cos 2cos 2 2 4 2

    x r θ  π= = − = − − =    

    3 2 sin 2sin 2 2

    4 2 y r θ π= = − = − ⋅ = −

    Rectangular coordinates of the point 3

    2, 4

    π −   

    are ( )2, 2− .

    46. 2 1

    cos 2cos 2 1 3 2

    x r θ π  = = − = − − =   

    2 3 sin 2sin 2 3

    3 2 y r θ π= = − = − ⋅ = −

    Rectangular coordinates of the point 2

    2, 3

    π −   

    are ( )1, 3− .

    47. 1 1

    cos 1cos 1 3 2 2

    x r θ π = = − − = − ⋅ = −   

    3 3 sin 1sin 1

    3 2 2 y r θ

     π = = − − = − − =       

    Rectangular coordinates of the point 1, 3

    π − −   

    are 1 3

    , 2 2

      −    

    .

    48. 3 2 3 2

    cos 3cos 3 4 2 2

    x r θ  π = = − − = − − =       

    3 2 3 2 sin 3sin 3

    4 2 2 y r θ

     π = = − − = − − =       

    Rectangular coordinates of the point 3

    3, 4

    π − −   

    are 3 2 3 2

    , 2 2

          

    .

    49. ( )cos 2cos( 180º ) 2 1 2x r θ= = − − = − − = sin – 2sin( 180º ) – 2 0 0y r θ= = − = ⋅ =

    Rectangular coordinates of the point

    ( )– 2, 180º− are ( )2, 0 .

    50. cos 3cos( 90º ) 3 0 0x r θ= = − − = − ⋅ = sin – 3sin( 90º ) 3( 1) 3y r θ= = − = − − =

    Rectangular coordinates of the point ( )– 3, 90º− are ( )0, 3 .

    51. cos 7.5cos110º 7.5( 0.3420) 2.57x r θ= = ≈ − ≈ − sin 7.5sin110º 7.5(0.9397) 7.05y r θ= = ≈ ≈

    Rectangular coordinates of the point ( )7.5, 110º are about ( )2.57, 7.05− .

    52. cos 3.1cos182º 3.1( 0.9994) 3.10x r θ= = − ≈ − − ≈ sin 3.1sin182º 3.1( 0.0349) 0.11y r θ= = − ≈ − − ≈

    Rectangular coordinates of the point ( )3.1, 182º− are about ( )3.10, 0.11 .

    53. ( )cos 6.3cos 3.8 6.3( 0.7910) 4.98x r θ= = ≈ − ≈ − ( )sin 6.3sin 3.8 6.3( 0.6119) 3.85y r θ= = ≈ − ≈ −

    Rectangular coordinates of the point ( )6.3, 3.8 are about ( )4.98, 3.85− − .

    54. ( )cos 8.1cos 5.2 8.1(0.4685) 3.79x r θ= = ≈ ≈ ( )sin 8.1sin 5.2 8.1( 0.8835) 7.16y r θ= = ≈ − ≈ −

    Rectangular coordinates of the point ( )8.1, 5.2 are about ( )3.79, 7.16− .

    55. 2 2 2 2

    1 1 1

    3 0 9 3

    0 tan tan tan 0 0

    3

    r x y

    y

    x θ − − −

    = + = + = =    = = = =       

    Polar coordinates of the point (3, 0) are (3, 0) .

    56. 2 2 2 20 2 4 2r x y= + = + = =

    1 1 2tan tan 0

    y

    x θ − −   = =   

       

    Since 2

    0 is undefined,

    2 θ π=

    Polar coordinates of the point (0, 2) are 2, 2

    π     

    .

  • Section 9.1: Polar Coordinates

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    Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall

    57. 2 2 2 2

    1 1 1

    ( 1) 0 1 1

    0 tan tan tan 0 0

    1

    r x y

    y

    x θ − − −

    = + = − + = =    = = = =   −   

    The point lies on the negative x-axis, so θ = π . Polar coordinates of the point ( 1, 0)− are ( )1, π .

    58. 2 2 2 2

    1 1

    0 ( 2) 4 2

    2 tan tan

    0

    r x y

    y

    x θ − −

    = + = + − = = −   = =   

       

    Since 2

    0

    − is undefined,

    2 θ π= .

    The point lies on the negative y-axis, so 2

    θ π= − .

    Polar coordinates of the point (0, 2)− are 2, 2

    π −   

    .

    59. The point (1, 1)− lies in quadrant IV. 2 2 2 2

    1 1 1

    1 ( 1) 2

    1 tan tan tan ( 1)

    1 4

    r x y

    y

    x θ − − −

    = + = + − = − π   = = = − = −   

       

    Polar coordinates of the point (1, 1)− are

    2, 4

    π −   

    .

    60. The point ( 3, 3)− lies in quadrant II. 2 2 2 2

    1 1 1

    ( 3) 3 3 2

    3 tan tan tan ( 1)

    3 4

    r x y

    y

    x θ − − −

    = + = − + = π   = = = − = −   −   

    Polar coordinates of the point ( )3, 3− are 3

    3 2, 4

    π     

    .

    61. The point ( )3, 1 lies in quadrant I. ( )22 2 2

    1 1

    3 1 4 2

    1 tan tan

    63

    r x y

    y