polar coordinates - james madison university

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