Example: barber

Polar Coordinates (r,θ

Polar Coordinates (r, ) Polar Coordinates (r, )in the plane are describedbyr=distance from the originand [0,2 )is the counter-clockwise make the convention( r, ) = (r, + ). Polar Coordinates (r, ) Polar Coordinates (r, )in the plane are describedbyr=distance from the originand [0,2 )is the counter-clockwise make the convention( r, ) = (r, + ). Polar Coordinates (r, ) Polar Coordinates (r, )in the plane are describedbyr=distance from the originand [0,2 )is the counter-clockwise make the convention( r, ) = (r, + ). Polar Coordinates (r, ) Polar Coordinates (r, )in the plane are describedbyr=distance from the originand [0,2 )is the counter-clockwise make the convention( r, ) = (r, + ). Polar Coordinates (r, ) Polar Coordinates (r, )in the plane are describedbyr=distance from the originand [0,2 )is the counter-clockwise make the convention( r, ) = (r, + ).]]]]]

Polar Coordinates (r,θ) Polar Coordinates (r,θ) in the plane are described by r = distance from the origin and θ ∈ [0,2π) is the counter-clockwise angle.

Tags:

  Coordinates, Panels, Polar, Polar coordinates

Information

Domain:

Source:

Link to this page:

Please notify us if you found a problem with this document:

Other abuse

Transcription of Polar Coordinates (r,θ

1 Polar Coordinates (r, ) Polar Coordinates (r, )in the plane are describedbyr=distance from the originand [0,2 )is the counter-clockwise make the convention( r, ) = (r, + ). Polar Coordinates (r, ) Polar Coordinates (r, )in the plane are describedbyr=distance from the originand [0,2 )is the counter-clockwise make the convention( r, ) = (r, + ). Polar Coordinates (r, ) Polar Coordinates (r, )in the plane are describedbyr=distance from the originand [0,2 )is the counter-clockwise make the convention( r, ) = (r, + ). Polar Coordinates (r, ) Polar Coordinates (r, )in the plane are describedbyr=distance from the originand [0,2 )is the counter-clockwise make the convention( r, ) = (r, + ). Polar Coordinates (r, ) Polar Coordinates (r, )in the plane are describedbyr=distance from the originand [0,2 )is the counter-clockwise make the convention( r, ) = (r, + ).]]]]]

2 Plotting pointsExamplePlot the points whosepolarcoordinates are given.(a)(1,5 4)(b) (2,3 ) (c)(2, 2 3)(d)( 3,3 4)SolutionThe points are plotted in Figure part (d) the point( 3,3 4)is located three units from the pole in the fourthquadrant because the angle 3 4is in the second quadrant andr= 3 is pointsExamplePlot the points whosepolarcoordinates are given.(a)(1,5 4)(b) (2,3 ) (c)(2, 2 3)(d)( 3,3 4)SolutionThe points are plotted in Figure part (d) the point( 3,3 4)is located three units from the pole in the fourthquadrant because the angle 3 4is in the second quadrant andr= 3 is pointsExamplePlot the points whosepolarcoordinates are given.(a)(1,5 4)(b) (2,3 ) (c)(2, 2 3)(d)( 3,3 4)SolutionThe points are plotted in Figure part (d) the point( 3,3 4)is located three units from the pole in the fourthquadrant because the angle 3 4is in the second quadrant andr= 3 is pointsExamplePlot the points whosepolarcoordinates are given.

3 (a)(1,5 4)(b) (2,3 ) (c)(2, 2 3)(d)( 3,3 4)SolutionThe points are plotted in Figure part (d) the point( 3,3 4)is located three units from the pole in the fourthquadrant because the angle 3 4is in the second quadrant andr= 3 is pointsExamplePlot the points whosepolarcoordinates are given.(a)(1,5 4)(b) (2,3 ) (c)(2, 2 3)(d)( 3,3 4)SolutionThe points are plotted in Figure part (d) the point( 3,3 4)is located three units from the pole in the fourthquadrant because the angle 3 4is in the second quadrant andr= 3 is conversion - Polar /Cartesianx=rcos y=rsin r2=x2+y2tan =yxCoordinate conversion - Polar /Cartesianx=rcos y=rsin r2=x2+y2tan =yxCoordinate conversion - Polar /Cartesianx=rcos y=rsin r2=x2+y2tan =yxExampleConvert the point(2, 3) 2 and = 3,x=rcos = 2 cos 3= 2 12= 1y=rsin = 2 sin 3= 2 32= 3 Therefore, the point is (1, 3) the point(2, 3) 2 and = 3,x=rcos = 2 cos 3= 2 12= 1y=rsin = 2 sin 3= 2 32= 3 Therefore, the point is (1, 3) the point(2, 3)

4 2 and = 3,x=rcos = 2 cos 3= 2 12= 1y=rsin = 2 sin 3= 2 32= 3 Therefore, the point is (1, 3) the point(2, 3) 2 and = 3,x=rcos = 2 cos 3= 2 12= 1y=rsin = 2 sin 3= 2 32= 3 Therefore, the point is (1, 3) the point(2, 3) 2 and = 3,x=rcos = 2 cos 3= 2 12= 1y=rsin = 2 sin 3= 2 32= 3 Therefore, the point is (1, 3) the point(2, 3) 2 and = 3,x=rcos = 2 cos 3= 2 12= 1y=rsin = 2 sin 3= 2 32= 3 Therefore, the point is (1, 3) the point(2, 3) 2 and = 3,x=rcos = 2 cos 3= 2 12= 1y=rsin = 2 sin 3= 2 32= 3 Therefore, the point is (1, 3) the point(2, 3) 2 and = 3,x=rcos = 2 cos 3= 2 12= 1y=rsin = 2 sin 3= 2 32= 3 Therefore, the point is (1, 3) the point(2, 3) 2 and = 3,x=rcos = 2 cos 3= 2 12= 1y=rsin = 2 sin 3= 2 32= 3 Therefore, the point is (1, 3) the point(2, 3) 2 and = 3,x=rcos = 2 cos 3= 2 12= 1y=rsin = 2 sin 3= 2 32= 3 Therefore, the point is (1, 3) the point(2, 3) 2 and = 3,x=rcos = 2 cos 3= 2 12= 1y=rsin = 2 sin 3= 2 32= 3 Therefore, the point is (1, 3) the point(2, 3) 2 and = 3,x=rcos = 2 cos 3= 2 12= 1y=rsin = 2 sin 3= 2 32= 3 Therefore, the point is (1, 3) the point withCartesiancoordinates (1, 1) in terms we chooserto be positive, thenr= x2+y2= 12+ ( 1)2= 2tan =yx= 1 Since the point (1, 1) lies in the fourth quadrant,we choose = 4or = 7 , one possibleanswer is( 2, 4); another is( 2,7 4).

5 ExampleRepresent the point withCartesiancoordinates (1, 1) in terms we chooserto be positive, thenr= x2+y2= 12+ ( 1)2= 2tan =yx= 1 Since the point (1, 1) lies in the fourth quadrant,we choose = 4or = 7 , one possibleanswer is( 2, 4); another is( 2,7 4).ExampleRepresent the point withCartesiancoordinates (1, 1) in terms we chooserto be positive, thenr= x2+y2= 12+ ( 1)2= 2tan =yx= 1 Since the point (1, 1) lies in the fourth quadrant,we choose = 4or = 7 , one possibleanswer is( 2, 4); another is( 2,7 4).ExampleRepresent the point withCartesiancoordinates (1, 1) in terms we chooserto be positive, thenr= x2+y2= 12+ ( 1)2= 2tan =yx= 1 Since the point (1, 1) lies in the fourth quadrant,we choose = 4or = 7 , one possibleanswer is( 2, 4); another is( 2,7 4).

6 ExampleRepresent the point withCartesiancoordinates (1, 1) in terms we chooserto be positive, thenr= x2+y2= 12+ ( 1)2= 2tan =yx= 1 Since the point (1, 1) lies in the fourth quadrant,we choose = 4or = 7 , one possibleanswer is( 2, 4); another is( 2,7 4).ExampleRepresent the point withCartesiancoordinates (1, 1) in terms we chooserto be positive, thenr= x2+y2= 12+ ( 1)2= 2tan =yx= 1 Since the point (1, 1) lies in the fourth quadrant,we choose = 4or = 7 , one possibleanswer is( 2, 4); another is( 2,7 4).ExampleRepresent the point withCartesiancoordinates (1, 1) in terms we chooserto be positive, thenr= x2+y2= 12+ ( 1)2= 2tan =yx= 1 Since the point (1, 1) lies in the fourth quadrant,we choose = 4or = 7 , one possibleanswer is( 2, 4); another is( 2,7 4).

7 ExampleRepresent the point withCartesiancoordinates (1, 1) in terms we chooserto be positive, thenr= x2+y2= 12+ ( 1)2= 2tan =yx= 1 Since the point (1, 1) lies in the fourth quadrant,we choose = 4or = 7 , one possibleanswer is( 2, 4); another is( 2,7 4).ExampleRepresent the point withCartesiancoordinates (1, 1) in terms we chooserto be positive, thenr= x2+y2= 12+ ( 1)2= 2tan =yx= 1 Since the point (1, 1) lies in the fourth quadrant,we choose = 4or = 7 , one possibleanswer is( 2, 4); another is( 2,7 4).ExampleRepresent the point withCartesiancoordinates (1, 1) in terms we chooserto be positive, thenr= x2+y2= 12+ ( 1)2= 2tan =yx= 1 Since the point (1, 1) lies in the fourth quadrant,we choose = 4or = 7 , one possibleanswer is( 2, 4); another is( 2,7 4).

8 Graph of a Polar equationDefinitionThegraph of a Polar equationr=f( ), or more generallyF(r, ) = 0, consists ofall pointsPthat have at least one polarrepresentation (r, ) whose Coordinates satisfy curve is represented by thepolarequation r=2?SolutionThe curve consists of all points(r, )withr= the distance from thepoint to the pole, the curver=2represents thecircle with centerOand radius general, theequationr=arepresents a circle with centerOandradius|a|.ExampleWhat curve is represented by thepolarequation r=2?SolutionThe curve consists of all points(r, )withr= the distance from thepoint to the pole, the curver=2represents thecircle with centerOand radius general, theequationr=arepresents a circle with centerOandradius|a|.ExampleWhat curve is represented by thepolarequation r=2?

9 SolutionThe curve consists of all points(r, )withr= the distance from thepoint to the pole, the curver=2represents thecircle with centerOand radius general, theequationr=arepresents a circle with centerOandradius|a|.ExampleWhat curve is represented by thepolarequation r=2?SolutionThe curve consists of all points(r, )withr= the distance from thepoint to the pole, the curver=2represents thecircle with centerOand radius general, theequationr=arepresents a circle with centerOandradius|a|.ExampleWhat curve is represented by thepolarequation r=2?SolutionThe curve consists of all points(r, )withr= the distance from thepoint to the pole, the curver=2represents thecircle with centerOand radius general, theequationr=arepresents a circle with centerOandradius|a|.

10 Sketch the curve withpolar equation r=2cos .SolutionPlotting points we find what seems to bea circle:Sketch the curve withpolar equation r=2cos .SolutionPlotting points we find what seems to bea circle:Sketch the curve withpolar equation r=2cos .SolutionPlotting points we find what seems to bea circle:ExampleFind theCartesiancoordinates forr= 2 cos .SolutionSincex=rcos , the equationr= 2 cos becomesr=2xror2x=r2=x2+y2orx2 2x+y2= 0or(x 1)2+y2= is the equation of a circle of radius 1 centeredat (1,0).ExampleFind theCartesiancoordinates forr= 2 cos .SolutionSincex=rcos , the equationr= 2 cos becomesr=2xror2x=r2=x2+y2orx2 2x+y2= 0or(x 1)2+y2= is the equation of a circle of radius 1 centeredat (1,0).ExampleFind theCartesiancoordinates forr= 2 cos.


Related search queries