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0802 ch 8 day 2
 

0802 ch 8 day 2

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0802 ch 8 day 2 0802 ch 8 day 2 Presentation Transcript

  • 8.1 Polar Coordinates Day TwoPsalm 33:22 "May your unfailing love rest upon us, OLORD, even as we put our hope in you."
  • We are now going to overlay a RectangularCoordinate system over a Polar Coordinate system inorder to identify the relationships between polar andrectangular coordinates.
  • 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 x polar axis
  • y P ( x, y ) P ( r,θ ) , r > 0 2 +y x2 r= θ x polar axis x = r cosθ
  • y P ( x, y ) P ( r,θ ) , r > 0 2 +y x2 y = r sin θ r= θ x polar axis x = r cosθ
  • 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
  • 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 π = −3cos 6
  • Find the rectangular coordinates: ⎛ π ⎞1. P ⎜ −3, ⎟ ⎝ 6 ⎠ x = r cosθ y = r sin θ π = −3cos 6 ⎛ 3 ⎞ = −3 ⎜ ⎝ 2 ⎟ ⎠
  • 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
  • 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
  • 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
  • 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
  • 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 ⎟ ⎝ ⎠
  • 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 2π = 10 cos 9
  • Find the rectangular coordinates: ⎛ 2π ⎞2. Q ⎜ 10, ⎟ ⎝ 9 ⎠ x = r cosθ y = r sin θ 2π = 10 cos 9 ≈ 7.66
  • 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
  • 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
  • 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)
  • 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 ⎟ ⎠ ⎪ ⎩
  • 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 ⎞ θ = Arc tan ⎜ − ⎟ ⎝ 10 ⎠
  • Find the polar coordinates:1. R (10, − 10 ) 2 r = 10 + ( −10 ) 2 ⎛ 10 ⎞ θ = Arc tan ⎜ − ⎟ ⎝ 10 ⎠ = 200
  • Find the polar coordinates:1. R (10, − 10 ) 2 r = 10 + ( −10 ) 2 ⎛ 10 ⎞ θ = Arc tan ⎜ − ⎟ ⎝ 10 ⎠ = 200 = 10 2
  • Find the polar coordinates:1. R (10, − 10 ) 2 r = 10 + ( −10 ) 2 ⎛ 10 ⎞ Q IV θ = Arc tan ⎜ − ⎟ ⎝ 10 ⎠ = 200 = 10 2
  • 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
  • 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 ⎠
  • Find the polar coordinates: (2. S −4, 4 3 )
  • Find the polar coordinates: (2. S −4, 4 3 ) 2 r= ( −4 ) 2 ( + 4 3 )
  • Find the polar coordinates: (2. S −4, 4 3 ) 2 ⎛ 4 3 ⎞ r= ( −4 ) 2 ( + 4 3 ) θ = Arc tan ⎜ − ⎝ 4 ⎟ ⎠
  • Find the polar coordinates: (2. S −4, 4 3 ) 2 ⎛ 4 3 ⎞ r= ( −4 ) 2 ( + 4 3 ) θ = Arc tan ⎜ − ⎝ 4 ⎟ ⎠ = 64
  • Find the polar coordinates: (2. S −4, 4 3 ) 2 ⎛ 4 3 ⎞ r= ( −4 ) 2 ( + 4 3 ) θ = Arc tan ⎜ − ⎝ 4 ⎟ ⎠ = 64 =8
  • 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
  • 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
  • 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 ⎠
  • 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=− sin θ
  • Convert y = −1 to a polar equation. y = −1 r sin θ = −1 1 r=− sin θ r = − cscθ
  • 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 2 2 x + y = 5x
  • 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
  • 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