MTH 253Calculus (Other Topics)

Chapter 11 – Analytic Geometry in Calculus

Section 11.1 – Polar Coordinates

Copyright © 2006 by Ron Wallace, all rights reserved.

Rectangular (aka: Cartesian) Coordinates

positive x-axisnegative x-axis

positive y-axis

negative y-axis

x

y(x, y)

origin

For any point there is a unique ordered pair (x, y) that

specifies the location of that point.

Polar Coordinates

polar axis

(r, )

r

pole

Is (r, ) unique for

every point?

NO!

All of the following refer to the same point:

(5, 120º)(5, 480º)(-5, 300º)(-5, -60º)etc ...

The angle may be expressed in degrees or radians.

Polar Graph PaperLocating and Graphing Points

0

30

6090

180

120

150

210

240270

300

330

(5, 150) (6, 75)

(3, 300)(3, -60)(-3, 120)

(-4, 30)

(7, 0)

(-7, 180)

Converting CoordinatesPolar Rectangular

2 2 2r x y

tany

x

sinry

x

y(r, ) (x, y)

r

Recommendation: Find (r, ) wherer > 0 and0 ≤ < 2 or 0 ≤ < 360.

cosx r

Relationships between r, , x, & y

R P

P R

Examples: Converting CoordinatesPolar Rectangular

sinry cosrx

(3,210 )

(3cos 210 ,3sin 210 )

2

13 ,

2

33

2

3 ,

2

33

6 ,2

2cos , 2sin6 6

2

12 ,

2

32

1- ,3

Examples: Converting CoordinatesPolar Rectangular

222 yxr x

ytan

Quadrant I

)7 ,3( 5873 22 r

8.663

7tan 1

)8.66 ,58( )7 ,3(

7tan

3

Examples: Converting CoordinatesPolar Rectangular

222 yxr x

ytan

Quadrant II

)7 ,3( 587)3( 22 r

8.663

7tan 1

)2.113 ,58()1808.66 ,58( )7 ,3(

3

7tan

)8.66 ,58( )7 ,3( OR

Examples: Converting CoordinatesPolar Rectangular

222 yxr x

ytan

Quadrant III

)7 ,3( 58)7()3( 22 r

8.663

7tan 1

)8.246 ,58()1808.66 ,58( )7 ,3(

3

7tan

)8.66 ,58( )7 ,3( OR

Examples: Converting CoordinatesPolar Rectangular

222 yxr x

ytan

Quadrant IV

)7 ,3( 58)7(3 22 r

8.663

7tan 1

)93.22 ,58()3608.66 ,58( )7 ,3(

3

7tan

)8.66 ,58( )7 ,3( OR

Converting EquationsPolar Rectangular

Use the same identities:

222 yxr

x

ytan sinry

cosrx

Converting EquationsPolar Rectangular

Replace all occurrences of xx with r cos .

Replace all occurrences of yy with r sin .

Simplify Solve for rr (if possible).

Converting EquationsPolar Rectangular

Express the equation in terms of sine and cosine only.

If possible, manipulate the equation so that all occurrences of cos and sin are multiplied by r.

Replace all occurrences of …

Simplify (solve for y if possible)

r cos with x

r sin with y

r2 with x2 + y2

Or, if all else fails, use:

2 2cos

x

x y

22sin

yx

y

22 yxr

Graphing Polar Equations

Reminder: How do you graph rectangular equations? Method 1:

Create a table of values. Plot ordered pairs. Connect the dots in order as x increases.

Method 2: Recognize and graph various common

forms. Examples: linear equations, quadratic

equations, conics, …

The same basic approach can be applied to polar equations.

Graphing Polar EquationsMethod 1: Plotting and Connecting Points

1. Create a table of values.2. Plot ordered pairs.3. Connect the dots in order as

increases.

NOTE: Since most of these equations involve periodic functions (esp. sine and cosine), at some point the graph will start repeating itself (but not always).

Graphing Polar EquationsMethod 1: Plotting and Connecting Points

wrt x-axis• Replacing with - doesn’t change the function

Symmetry Tests

(r,)

(r,-)

Graphing Polar EquationsMethod 1: Plotting and Connecting Points

wrt y-axis• Replacing with - doesn’t change the function

Symmetry Tests

(r,)(r,-)

Graphing Polar EquationsMethod 1: Plotting and Connecting Points

wrt the origin• Replacing r with –r doesn’t change the function.• Replacing with doesn’t change the function.

Symmetry Tests

(r,)

(-r,)(r, )

Graphing Polar EquationsMethod 2: Recognizing Common Forms

Circles Centered at the origin: r = a

radius: a period = 360

Tangent to the x-axis at the origin: r = a sin center: (a/2, 90) radius: a/2 period = 180 a > 0 above a < 0 below

Tangent to the y-axis at the origin: r = a cos center: (a/2, 90) radius: a/2 period = 180 a > 0 right a < 0 left

r = 4

r = 4 sin

r = 4 cos

Graphing Polar EquationsMethod 2: Recognizing Common Forms

Flowers (centered at the origin) r = a cos n or r = a sin n

radius: |a| n is even 2n petals

petal every 180/n period = 360

n is odd n petals petal every 360/n period = 180

cos 1st petal @ 0 sin 1st petal @ 90/n

r = 4 sin 2

r = 4 cos 3

Graphing Polar EquationsMethod 2: Recognizing Common Forms

Spirals Spiral of Archimedes: r = k

|k| large loose |k| small tight

r = r = ¼

Other spirals … see page 726.

Graphing Polar EquationsMethod 2: Recognizing Common Forms

Heart (actually: cardioid if a = b … otherwise: limaçon)

r = a ± b cos or r = a ± b sin

r = 3 + 3 cos r = 2 - 5 cos r = 3 + 2 sin r = 3 - 3 sin

Graphing Polar EquationsMethod 2: Recognizing Common Forms

Lines Through the Origin: y = mx = tan-1m

Horizontal: y = k r sin = k r = k csc

Vertical: x = h r cos = h r = h sec

Others:

ax + by = c

y = mx + b

cos sin

cr

a b

sin cos

br

m

Graphing Polar EquationsMethod 2: Recognizing Common Forms

Parabolas (w/ vertex on an axis)

NOTE: With these forms, the vertex will never be at the origin.

cos1

ar

sin1

ar

cos1

3

r

cos1

7

r

sin1

5

r

sin1

1

r

Graphing Polar EquationsMethod 2: Recognizing Common Forms

Parabolas (w/ vertex at the origin)

2

sin

cosr

a

2

cos

sinr

a

2y ax 2x ay

Graphing Polar EquationsMethod 2: Recognizing Common Forms

Leminscate2 cos 2r a 2 sin 2r a

a = 16

Replacing the 2 w/ n will give 2n petals if n is odd and n petals if n is even.(these are not considered to be leminscates)