Section 12.5 - The Polar Coordinate System

12.1

The keys……

2 2 2

x r cos

y r sin

x y r

ytan

x

56,

3

74,

54.5,

4

53

5,0

,

,6

4

,

23

r

Four Translations

Point - Polar to Rectangular

Point - Rectangular to Polar

Equation - Polar to Rectangular

Equation - Rectangular to Polar

Point – Polar to Rectangular

34,

2

3x 4cos

23

y 4sin2

x 0

y 4

0, 4

r,

x,y

53,

6

5x 3cos

65

y 3sin6

3 3x

23

y2

3 3 3,

2 2

32,

4

3x 2cos

43

y 2sin4

x 2

y 2

2, 2

Point – Rectangular to Polar

2,6

3cos

21

sin2

r 2

6

3,1

r,

x,y

52 2,

4

2cos

2 22

sin2 2

r 2 2

5

4

2, 2

510,

3

5cos

10

5 3sin

10

r 10

5

3

5, 5 3

22 23 1 r 2 2 22 2 r 22 25 5 3 r

Equation - Polar to Rectangular

r csc 2

r2

sin

2r

2r sin

2 2x y 2y

2 2r 2r cos sin 1

2r 2 r cos r sin 1 2 2x y 2xy 1

r 4 tan sec

2

4sinr

cos

2r cos 4sin

2 2r cos 4r sin 2x 4y

Equation - Rectangular to Polar

2 22x 2y 5

2 22 x y 5 22r 5

2y 2x

2r sin 2r cos

2 2r sin 2r cos 2r sin 2cos

2xy 1

r 2cot csc

2 5r

2

2 r cos r sin 1 2r sin2 1

2r csc 2

Shapes of Polar Curves

Graphing Polar Curves on Calculator

Finding Points of Intersection(Boundaries of Integration)

The Shapes

Lines

Circles

Cardioid

Lemniscate

Sprial

Rose Curves

The Shapes - Linesr cos 3

Vertical Line

r sin 3 Horizontal Line

r 2cos 5sin 3 0 General Line

The Shapes - Circles

r 5

r 2cos

r 4sin

The Shapes – Cardioids(Hearts)

r

r 2 4sin

4 2cos

r

r a bsin

a bcos

Note: Extra loopOnly if b > a

The Shapes – Lemniscates(Propellers)

2 2r a cos2 2 2r a sin2

r 4cos2 r 5sin2

The Shapes - Spirals

r 2 0,2

r a

r 0.3 0,12

The Shapes – Rose Curves

r 5cos3 r 6sin4

r asin n

r bsin n

n even – 2n pedalsn odd – n pedals

Graphing Polar Curves on Calculator

1. Change mode to Polar

2. Hit y = (you’ll see r = indicating polar mode)

3. Enter the equation 3

r 5cos2

4. Graphing once should give you a sense of how to change the x, y and theta constraints.

5. Now change the constraints in WINDOW

min 2

max 2

Xmin 5

Xmax 5

Xscl 1

Ymin 5

Ymax 5

Yscl 1

Finding Points of Intersection(Boundaries of Integration)

2 2r sin and r cos

sin cos

tan 1

5,

4 4

2 2r

2

4 42 8r

22

4 4 4 48 8 8 5 8 5, , , , , , ,

2 4 2 4 2 4 2 4

r 1and r 2sin2

1sin2

2

5 13 172 , , ,

6 6 6 6

5 13 17, , ,

12 12 12 12

5 13 171, , 1, , 1, , 1,

12 12 12 12

5 5 11 13 17 19 231, , 1, , 1, , 1, , 1, , 1, , 1, , 1,

12 12 12 12 12 12 12 12

WRONG ANSWER

r 1 cos and r 1 sin2 2

tan 12

3 7 3,

2 4 4 2

r 1 sin2

3r 1 sin

4

2r 1

2

2 2 31 , , 1 ,

2 2 2 2