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Polar Coordinates Department of Mathematics and Statistics October 9, 2012 Calculus III (James Madison University) Math 237 October 9, 2012 1/5
Polar Coordinates
Department of Mathematics and Statistics
October 9, 2012
Calculus III (James Madison University) Math 237 October 9, 2012 1 / 5
Polar Coordinates
The polar coordinate system is an alternate method for locating pointsin the plane.
In this system we start with a point called the pole and a horizontal linethrough the pole called the polar axis.
The coordinate pair, (r , θ), is used to locate points.
r is the “signed” distance from the origin.
θ is the counterclockwise rotation from the polar axis.
Calculus III (James Madison University) Math 237 October 9, 2012 2 / 5
Polar Coordinates
The polar coordinate system is an alternate method for locating pointsin the plane.
In this system we start with a point called the pole and a horizontal linethrough the pole called the polar axis.
The coordinate pair, (r , θ), is used to locate points.
r is the “signed” distance from the origin.
θ is the counterclockwise rotation from the polar axis.
Calculus III (James Madison University) Math 237 October 9, 2012 2 / 5
Polar Coordinates
The polar coordinate system is an alternate method for locating pointsin the plane.
In this system we start with a point called the pole and a horizontal linethrough the pole called the polar axis.
The coordinate pair, (r , θ), is used to locate points.
r is the “signed” distance from the origin.
θ is the counterclockwise rotation from the polar axis.
Calculus III (James Madison University) Math 237 October 9, 2012 2 / 5
Polar Coordinates
The polar coordinate system is an alternate method for locating pointsin the plane.
In this system we start with a point called the pole and a horizontal linethrough the pole called the polar axis.
The coordinate pair, (r , θ), is used to locate points.
r is the “signed” distance from the origin.
θ is the counterclockwise rotation from the polar axis.
Calculus III (James Madison University) Math 237 October 9, 2012 2 / 5
Polar Coordinates
The polar coordinate system is an alternate method for locating pointsin the plane.
In this system we start with a point called the pole and a horizontal linethrough the pole called the polar axis.
The coordinate pair, (r , θ), is used to locate points.
r is the “signed” distance from the origin.
θ is the counterclockwise rotation from the polar axis.
Calculus III (James Madison University) Math 237 October 9, 2012 2 / 5
Polar Coordinates
The polar coordinate system is an alternate method for locating pointsin the plane.
In this system we start with a point called the pole and a horizontal linethrough the pole called the polar axis.
The coordinate pair, (r , θ), is used to locate points.
r is the “signed” distance from the origin.
θ is the counterclockwise rotation from the polar axis.
Calculus III (James Madison University) Math 237 October 9, 2012 2 / 5
Equivalent Polar Coordinates
Unlike the rectangular coordinate system, every point in the polar planecan be represented using infinitely many different pairs of polarcoordinates.
Theorem
The polar coordinates (r , θ + 2πk) represent the same point for everyinteger k.
The polar coordinates (−r , θ + π) represent the same point as (r , θ)for any value of θ.
The polar coordinates (0, θ) represent the pole for any value of θ.
Calculus III (James Madison University) Math 237 October 9, 2012 3 / 5
Equivalent Polar Coordinates
Unlike the rectangular coordinate system, every point in the polar planecan be represented using infinitely many different pairs of polarcoordinates.
Theorem
The polar coordinates (r , θ + 2πk) represent the same point for everyinteger k.
The polar coordinates (−r , θ + π) represent the same point as (r , θ)for any value of θ.
The polar coordinates (0, θ) represent the pole for any value of θ.
Calculus III (James Madison University) Math 237 October 9, 2012 3 / 5
Equivalent Polar Coordinates
Unlike the rectangular coordinate system, every point in the polar planecan be represented using infinitely many different pairs of polarcoordinates.
Theorem
The polar coordinates (r , θ + 2πk) represent the same point for everyinteger k.
The polar coordinates (−r , θ + π) represent the same point as (r , θ)for any value of θ.
The polar coordinates (0, θ) represent the pole for any value of θ.
Calculus III (James Madison University) Math 237 October 9, 2012 3 / 5
Equivalent Polar Coordinates
Unlike the rectangular coordinate system, every point in the polar planecan be represented using infinitely many different pairs of polarcoordinates.
Theorem
The polar coordinates (r , θ + 2πk) represent the same point for everyinteger k.
The polar coordinates (−r , θ + π) represent the same point as (r , θ)for any value of θ.
The polar coordinates (0, θ) represent the pole for any value of θ.
Calculus III (James Madison University) Math 237 October 9, 2012 3 / 5
Equivalent Polar Coordinates
Unlike the rectangular coordinate system, every point in the polar planecan be represented using infinitely many different pairs of polarcoordinates.
Theorem
The polar coordinates (r , θ + 2πk) represent the same point for everyinteger k.
The polar coordinates (−r , θ + π) represent the same point as (r , θ)for any value of θ.
The polar coordinates (0, θ) represent the pole for any value of θ.
Calculus III (James Madison University) Math 237 October 9, 2012 3 / 5
Equivalent Polar Coordinates
Unlike the rectangular coordinate system, every point in the polar planecan be represented using infinitely many different pairs of polarcoordinates.
Theorem
The polar coordinates (r , θ + 2πk) represent the same point for everyinteger k.
The polar coordinates (−r , θ + π) represent the same point as (r , θ)for any value of θ.
The polar coordinates (0, θ) represent the pole for any value of θ.
Calculus III (James Madison University) Math 237 October 9, 2012 3 / 5
Converting between Polar and Rectangular Coordinates
Theorem
If a point in the plane is represented by (r , θ) in polar coordinates,then the rectangular coordinates of the point are given by (x , y) where
x = r cos θ and y = r sin θ.
If a point in the plane is represented by (x , y) in rectangularcoordinates, then the polar coordinates (r , θ) of the point satisfy thefollowing formulas:
r2 = x2 + y2 and tan θ = yx .
Calculus III (James Madison University) Math 237 October 9, 2012 4 / 5
Converting between Polar and Rectangular Coordinates
Theorem
If a point in the plane is represented by (r , θ) in polar coordinates,then the rectangular coordinates of the point are given by (x , y) where
x = r cos θ and y = r sin θ.
If a point in the plane is represented by (x , y) in rectangularcoordinates, then the polar coordinates (r , θ) of the point satisfy thefollowing formulas:
r2 = x2 + y2 and tan θ = yx .
Calculus III (James Madison University) Math 237 October 9, 2012 4 / 5
Converting between Polar and Rectangular Coordinates
Theorem
If a point in the plane is represented by (r , θ) in polar coordinates,then the rectangular coordinates of the point are given by (x , y) where
x = r cos θ and y = r sin θ.
If a point in the plane is represented by (x , y) in rectangularcoordinates, then the polar coordinates (r , θ) of the point satisfy thefollowing formulas:
r2 = x2 + y2 and tan θ = yx .
Calculus III (James Madison University) Math 237 October 9, 2012 4 / 5
Converting between Polar and Rectangular Coordinates
Theorem
If a point in the plane is represented by (r , θ) in polar coordinates,then the rectangular coordinates of the point are given by (x , y) where
x = r cos θ and y = r sin θ.
If a point in the plane is represented by (x , y) in rectangularcoordinates, then the polar coordinates (r , θ) of the point satisfy thefollowing formulas:
r2 = x2 + y2 and tan θ = yx .
Calculus III (James Madison University) Math 237 October 9, 2012 4 / 5
Some Circles and Lines using Polar Coordinates
The graph of the equation θ = c is a straight line through the pole.
The graph of the equation r = a is a circle with radius |a| centered at thepole.
The graph of the equation r = 2a cos θ is a circle with radius |a| centeredat (a, 0).
The graph of the equation r = 2a sin θ is a circle with radius |a| centeredat
(a, π
2
).
Calculus III (James Madison University) Math 237 October 9, 2012 5 / 5
Some Circles and Lines using Polar Coordinates
The graph of the equation θ = c is
a straight line through the pole.
The graph of the equation r = a is a circle with radius |a| centered at thepole.
The graph of the equation r = 2a cos θ is a circle with radius |a| centeredat (a, 0).
The graph of the equation r = 2a sin θ is a circle with radius |a| centeredat
(a, π
2
).
Calculus III (James Madison University) Math 237 October 9, 2012 5 / 5
Some Circles and Lines using Polar Coordinates
The graph of the equation θ = c is a straight line through the pole.
The graph of the equation r = a is a circle with radius |a| centered at thepole.
The graph of the equation r = 2a cos θ is a circle with radius |a| centeredat (a, 0).
The graph of the equation r = 2a sin θ is a circle with radius |a| centeredat
(a, π
2
).
Calculus III (James Madison University) Math 237 October 9, 2012 5 / 5
Some Circles and Lines using Polar Coordinates
The graph of the equation θ = c is a straight line through the pole.
The graph of the equation r = a is
a circle with radius |a| centered at thepole.
The graph of the equation r = 2a cos θ is a circle with radius |a| centeredat (a, 0).
The graph of the equation r = 2a sin θ is a circle with radius |a| centeredat
(a, π
2
).
Calculus III (James Madison University) Math 237 October 9, 2012 5 / 5
Some Circles and Lines using Polar Coordinates
The graph of the equation θ = c is a straight line through the pole.
The graph of the equation r = a is a circle with radius |a| centered at thepole.
The graph of the equation r = 2a cos θ is a circle with radius |a| centeredat (a, 0).
The graph of the equation r = 2a sin θ is a circle with radius |a| centeredat
(a, π
2
).
Calculus III (James Madison University) Math 237 October 9, 2012 5 / 5
Some Circles and Lines using Polar Coordinates
The graph of the equation θ = c is a straight line through the pole.
The graph of the equation r = a is a circle with radius |a| centered at thepole.
The graph of the equation r = 2a cos θ is
a circle with radius |a| centeredat (a, 0).
The graph of the equation r = 2a sin θ is a circle with radius |a| centeredat
(a, π
2
).
Calculus III (James Madison University) Math 237 October 9, 2012 5 / 5
Some Circles and Lines using Polar Coordinates
The graph of the equation θ = c is a straight line through the pole.
The graph of the equation r = a is a circle with radius |a| centered at thepole.
The graph of the equation r = 2a cos θ is a circle with radius |a| centeredat (a, 0).
The graph of the equation r = 2a sin θ is a circle with radius |a| centeredat
(a, π
2
).
Calculus III (James Madison University) Math 237 October 9, 2012 5 / 5
Some Circles and Lines using Polar Coordinates
The graph of the equation θ = c is a straight line through the pole.
The graph of the equation r = a is a circle with radius |a| centered at thepole.
The graph of the equation r = 2a cos θ is a circle with radius |a| centeredat (a, 0).
The graph of the equation r = 2a sin θ is
a circle with radius |a| centeredat
(a, π
2
).
Calculus III (James Madison University) Math 237 October 9, 2012 5 / 5
Some Circles and Lines using Polar Coordinates
The graph of the equation θ = c is a straight line through the pole.
The graph of the equation r = a is a circle with radius |a| centered at thepole.
The graph of the equation r = 2a cos θ is a circle with radius |a| centeredat (a, 0).
The graph of the equation r = 2a sin θ is a circle with radius |a| centeredat
(a, π
2
).
Calculus III (James Madison University) Math 237 October 9, 2012 5 / 5
