10.3

Polar Functions

Quick Review

1. Find the component form of a vector with magnitude 4 and direction

angle 30 .

2. Find the area of a 30 sector of a circle of radius 6.

3. Find the area of a sector if a circle of radius 8 that has a

central

angle of /8 radians.

4. Find the rectangular equation of a circle of radius 5 centered at

the origin.

2 ,323

4

2522 yx

Quick Review

Given 3cos , 5sin , 0 2 .

5. Find / .

6. Find the slope of the curve at 2.

7. Find the points on the curve where the slope is zero.

8. Find the points on the curve where the slope is undefined.

x t y t t

dy dx

t

9. Find the length of the curve from 0 to .t t

5. Find dy / dx.

6. Find the slope of the curve at t = 2.

7. Find the points on the curve where the slope is zero.

8. Find the points on the curve where the slope is undefined.

9. Find the length of the curve from t = 0 to t = .

tcot3

5

2cot3

5

5 ,0 and 5 ,0

0 ,3 and 0 ,3

763.12

What you’ll learn about Polar Coordinates Polar Curves Slopes of Polar Curves Areas Enclosed by Polar Curves A Small Polar Gallery

Essential QuestionsHow can we use polar equations to define some interesting and important curves that would be difficultor impossible to define in the form y=f(x)?

Rectangular and Polar Coordinates1. Find rectangular coordinates for the following polar coordinates

a. (4, /2)

b. (8, 30o)

c. (8, 240o)

d. (6, 5/6)

4 ,0

4 ,7

7 ,4

3 ,5

Example Rectangular and Polar Coordinates

1. Find two different sets of polar coordinates for the point with the rectangular coordinate (3, 3).

4 /3 ,23

4 /7 ,23

3

323

Circles

Rose Curves

Rose Curves

Limaçon Curves

Limaçon Curves

Lemniscate Curves

Spiral of Archimedes

Polar-Rectangular Conversion Formulas

2 2 2cos

sin tan

x r r x y

yy r

x

Parametric Equations of Polar Curves The polar graph of ( ) is the curve defined parametrically by:

cos ( )cos

sin ( )sin

r f

x r f

y r f

Converting Polar to Rectangular3. Replace the polar equation by an equivalent rectangular equation.

Then identify the graph.

22 yx

2r

___22 xx 1

1 ,1

cos2r sin2rx2 y2

___22 yy _____0

sin2cos2 rMultiply both sides by r.

11 1

21x 21 y 2

A circle with center:

and radius: 2

Example Finding Slope of a Polar Curve4. Find the slope of the rose curve r = 2sin3 at the point where = /6.

Define parametrically.

ddxddy

dx

dy

3sin2x

2

3 3

cos3sin2y sin

3sin2

cos sin 3cos6

3sin2 sin cos 3cos6

1 2/3 2/1 6 0

2 1 2/1 2/3 6 0 1

Area in Polar CoordinatesThe area of the region between the origin and the curve r = f () for

≤ ≤ is

2 2

1drA .

2

1

2

df

Example Finding Area

5. Find the area of the region in the plane enclosed by the cardioid

. cos12 r

dA cos122

1 22

0

d coscos2122

0

2

2cos2

1

2

1cos2

d 32

0 cos4 2cos

2

0

3

sin4

2

2sin

6 0 0 0 0 0 6

Area Between Polar CurvesThe area of the region between r

1( ) and r

2( ) for ≤ ≤ is

21

22

2

1

2

1drdrA

. 2

1

21

22

drr

Example Finding Area Between Curves

6. Find the area of the region that lies inside the circle r = cos and outside the cardioid r = 1 – cos .

cos1cos 1cos2

2

1cos

3

3

,

d 2

13

3

2cos 2cos1

Find points of intersection. The outer curve is r = cos The inner curve is r = 1 – cos

Example Finding Area Between Curves

6. Find the area of the region that lies inside the circle r = cos and outside the cardioid r = 1 – cos .

d cos2

13

3

2

3

3

2

1

3sin2

32

1

3

3

d 2

13

3

2cos 2cos1

1 cos2 2cos

sin2

3sin2

3

685.0

Pg. 557, 10.1 #1-59 odd