Polar CoordinatesA coordinate system uses a pair of numbers to represent a point on the plane.

We are familiar with the Cartesian or rectangular coordinate system, (x, y).

It is not always the most convenient system to use depending on the nature of our problem.

The polar coordinate system, (r , ✓), is convenient if we want to consider radial distance from a fixed

point (origin, or pole) and bearing (direction).

Figure: The polar coordinates (r , ✓) of a point P

Definition (Polar Coordinates)

Choose a point O on the plane and call it the pole (or origin). Call the horizontal ray drawn from the pole to

the right the polar axis, which coincides with the positive x-axis of the rectangular coordinate system.

For any point P on the plane,

let ✓ be the signed (radian) measure of an angle with the polar ray as initial ray and ±�!OP as terminal

ray, and

let r be the signed distance from O to P determined by the orientation of the terminal ray.

In the ordered pair (r , ✓), r and ✓ are called the polar coordinates of P.

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Each point P corresponds to infinitely many values of ✓, counter-clockwise measures being positive.

If we choose�!OP as the terminal ray, r � 0. If we choose ��!

OP as the terminal ray, r 0.

Thus, in the polar coordinate system, the ordered pairs (2, 2⇡), (�2,⇡) and (2, 0) all represent the same

point.

Figure: Three di↵erent polar coordinates representations of the same point P

(0, ✓) represents the pole for any value of ✓.

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Example

If a point P has polar coordinates

✓1,

⇡

3

◆, find all possible ordered pairs of polar coordinates for P.

Solution: The point P is located at a distance of 1 unit from O along the ray which makes a positive

(counter-clockwise) angle of⇡

3with the polar axis.

Using positive distance r = 1, ✓ can be any one of⇡

3+ 2n⇡ (n = 0,±1,±2, ...)

Using negative distance r = �1, ✓ can be any one of⇡

3+ ⇡ + 2n⇡ (n = 0,±1,±2, ...)

The ordered pairs of polar coordinates of P are

✓1,

⇡

3+ 2n⇡

◆and

✓�1,

4⇡

3+ 2n⇡

◆

(n = 0,±1,±2, ...)

Figure: In general, (r , ✓) is also represented by polar coordinates (r , ✓ + 2n⇡) and (�r , ✓ + ⇡ + 2n⇡)Math 267 (University of Calgary) Fall 2015, Winter 2016 3 / 19

Connection between Polar and Cartesian Coordinates

Theorem (Polar and Cartesian Coordinates)

Suppose point P has Cartesian coordinates (x, y) and polar coordinates (r , ✓). Then

From polar to Cartesian:

x = r cos ✓, y = r sin ✓

Figure: Conversions between Cartesian and Polar Coordinates

From Cartesian to polar:

r2= x

2+ y

2, tan ✓ =y

x

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Curves in Polar Coordinates

Example

1 The point with polar coordinates

✓2,

5⇡

6

◆has Cartesian coordinates

x = 2 cos5⇡

6= 2

�

p3

2

!= �

p3

and

y = 2 sin5⇡

6= 2

✓1

2

◆= 1.

Thus, (x, y) =⇣�p3, 1⌘.

2 The point with Cartesian coordinates (1,�1) is in the fourth quadrant (Quadrant IV).

It has polar coordinates satisfying

r2= 1

2+ (�1)

2= 2

and

tan ✓ =�1

1= �1.

One possible solution is (r , ✓) =

✓p2,

7⇡

4

◆.

Another solution is (r , ✓) =

✓�p2,

3⇡

4

◆.

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Curves in Polar CoordinatesWe would like to sketch the curve on the plane defined by a polar equation such as

r = 3

✓ =⇡

4

r = 2 sin ✓

r = cos 3✓

r = ✓

The graph of a polar equation consists of all points that have at least one pair of polar coordinates (r , ✓)satisfying the equation.

We do not require all pairs of polar coordinates of the point to satisfy the equation. One pair su�ces.

Figure: When does a point belong to a polar curve?

The pair (1, 1) satisfies r = ✓, but the pair (1, 1 + 2⇡) does not satisfy r = ✓. We would consider the

common point they represent as belonging to the curve.Math 267 (University of Calgary) Fall 2015, Winter 2016 6 / 19

In the polar equation r = 3, r refers to the distance from the pole to the point.

r = 3 is the circle centered at the pole with radius 3.

Figure: Polar curve r = 3 in the xy -plane; Line r = 3 in the r✓-plane

Note that r = 3 would be a vertical line on the r✓-plane (with horizontal r -axis and vertical ✓-axis).

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In the polar equation ✓ =⇡

4, ✓ refers to one of the polar angles of the point.

✓ =⇡

4is the straight line through the pole making an angle of

⇡

4radians with the polar axis.

Figure: Polar curve ✓ =⇡

4in the xy -plane; Line ✓ =

⇡

4in the r✓-plane

Note that ✓ =⇡

4would be a horizontal line on the r✓-plane.

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ExampleSketch and describe the curve defined by the polar equation r = 2 sin ✓.

Solution: Method 1 Make a table of values. Plot and connect the dots.

✓ r = 2 sin ✓0 0

⇡/6 1

⇡/4p2

⇡/3p3

⇡/2 0

2⇡/3p3

3⇡/4p2

5⇡/6 1

⇡ 0

Figure: Polar curve r = 2 sin ✓ (in the xy -plane)

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Method 2 Convert the polar equation into a Cartesian equation.

Multiply by r : r2= 2r sin ✓.

Using r2= x

2+ y

2and y = 2 sin ✓:

x2+ y

2= 2y () x

2+ y

2 � 2y = 0 () x2+ y

2 � 2y + 1 = 1 () x2+ (y � 1)

2= 1.

Circle with centre (0, 1) and radius 1.

Figure: r = 2 sin ✓ is the circle centered at (0, 1) with radius 1 in the xy -plane

Remark Let Q be the point (0, 2). The line PQ is perpendicular to the line OP. Studying the right-angled

triangle OPQ, we see that the length r of OP is 2 sin ✓.

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ExampleMore generally, here are two common polar curves. We will encounter them again in future examples.

1 r = 2a sin ✓ () x2+ (y � a)

2= a

2, circle with centre (0, a) and radius |a|.

2 r = 2a cos ✓ () (x � a)2+ y

2= a

2, circle with centre (a, 0) and radius |a|.

Figure: r = 2a sin ✓ and r = 2a cos ✓, where a > 0

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ExampleSketch the curve r = cos 3✓.

Solution: The curve r = cos ✓ is a circle with centre

✓1

2, 0

◆and radius

1

2.

Replace ✓ by 3✓ in the equation: The curve loops around three times as fast.

The curve r = cos ✓ traces the circle once when ✓ increases from �⇡

2to

⇡

2.

The curve r = cos 3✓ traces a loop once when ✓ increases from �⇡

6to

⇡

6.

Figure: A plot of r = cos 3✓ using Maple 18

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Recall: The curve r = cos ✓ traces the same circle a second time when ✓ increases from⇡

2to

3⇡

2.

In this interval (✓ in Quadrants II and III), r becomes negative and the curve is traced in Quadrants IV

and I. So, it retraces the same circle.

Figure: Examine how the circle on the right is retraced when ✓ increases from⇡

2to

3⇡

2

The curve r = cos 3✓ traces the loop a second time when ✓ increases from⇡

6to

⇡

2.

Like the case with the circle, r is negative.

But unlike the case with the circle, this new loop does not coincide with the first loop. The second loop

appears in Quadrant III.

The completed graph has exactly three loops.

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Further examples/exercises

Example

Sketch the following curves by (a) making a table of values and plotting, or by (b) using WolframAlpha.

1 r = 3 sin 4✓

2 r = 1 + cos ✓

3 r = 1 + 2 cos ✓

4 r = sin9⇡

5

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AreaRecall: The formula for the area of a sector:

A =1

2r2✓

Theorem (Area in Polar Coordinates)

Suppose that D is a region on the xy -plane bounded between polar curves as follows:

0 r f (✓), ↵ ✓ �,

where f is non-negative and continuous.

Figure: 0 r f (✓),↵ ✓ �

Then the area of D is given by

A =1

2

Z �

↵(f (✓))2 d✓

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Proof (Outline):

Slice D radially into pieces resembling sectors.

Figure: 0 r f (✓),↵ ✓ �

Each small piece has small angle d✓ and approximate radius r = f (✓).

The area of a small piece is approximately1

2(f (✓))2 d✓.

The total area is the integral.

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Example

Find the area of bounded inside one of the loops of the polar curve r = cos(3✓).

Solution:

Figure: A plot of r = cos 3✓ using Maple 18

Consider the loop bounded as ✓ increases from �⇡

6to

⇡

6.

A =1

2

Z ⇡/6

�⇡/6(cos (3✓))2 d✓ =

1

2

Z ⇡/6

�⇡/6

1 + cos (6✓)

2d✓

=1

4

✓✓ +

sin(6✓)

2

◆����⇡/6

⇡/6

=⇡

12.

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Theorem (Area in Polar Coordinates)

Suppose that D is a region on the xy -plane described by

f (✓) r g(✓), ↵ ✓ �,

where f is non-negative and continuous.

Figure: 0 f (✓) r g(✓),↵ ✓ �

Then the area of D is given by

A =1

2

Z �

↵

⇣(g(✓))2 � (f (✓))2

⌘d✓

Math 267 (University of Calgary) Fall 2015, Winter 2016 18 / 19

Further examples/exercises

Example

1 Find the area of the bounded region enclosed by the given polar curves (considering only r � 0).

1 r = ✓, 0 ✓ 2⇡

2 r =pcos ✓, �

⇡

2 ✓

⇡

23 One of the loops of r = sin 2✓4 r = 2� cos ✓

2 Consider the circle r = 6 sin ✓ and the cardioid r = 2 + 2 sin ✓.

1 Sketch them.

2 Find their intersection.

3 Identify the bounded region inside the circle and outside the cardioid. Find its area.

4 Identify the bounded region outside the circle and inside the cardioid. Find its area.

5 Find the area of the region inside both the circle and the cardioid.

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