Polar Coordinates

and GraphingSection 10.5a

Polar CoordinatesTo define polar coordinates, we first fix an origin O (calledthe pole) and an initial ray from O. Then each point P canbe located by assigning to it a polar coordinate pair : ,r

,P r Directed distancefrom O to P

Directed angle frominitial ray to ray OP

OOrigin (Pole)

Initial ray

,P r

r

Polar CoordinatesImportant note: The polar coordinates for any given pointare not unique!!!

O

Origin(Pole)

Initial ray

2,6

P

62

Example: Plot the given point, and find two other sets ofpolar coordinates that define the point.

2,6

P

112,

6P

72,6

P

Guided PracticePlot the points with the given polar coordinates and findtheir Cartesian coordinates.

53,6

x

yx

y3

5 53cos , 3sin

6 6

3 13 , 3

2 2

3 3 3,

2 2

6

Cartesian coordinates:

Guided PracticePlot the points with the given polar coordinates and findtheir Cartesian coordinates.

1 45, tan3

x

y

3x

4y 5

5cos ,5sin

3 45 ,55 5

3,4

Cartesian coordinates:

Guided PracticePlot the points with the given Cartesian coordinates andfind two sets of polar coordinates for each.

0, 2

x

y

2

32,2

2,2

Polar coordinates:

Guided PracticePlot the points with the given Cartesian coordinates andfind two sets of polar coordinates for each.

3, 1

72,6

2,6

Polar coordinates:

x

y

3

r

1

2 2

3 1r 2

6

Polar GraphingEquationr a

Polar Graph

Circle of radius centered ata O Line through making an angle with the

initial rayO

1r and are equations for the circle of radius 11r centered at :O

Examples:

O x

1

1r

Polar GraphingEquationr a

Polar Graph

Circle of radius centered ata O Line through making an angle with the

initial rayO

6 , , and are equations7 6 for the line:

Examples:

Ox0

6

5 6

7 6 6

Guided PracticeGraph the set of points whose polar coordinates satisfythe given conditions.

0 3r

x

y3

3

Guided PracticeGraph the set of points whose polar coordinates satisfythe given conditions.

1 2r and 02

x

y

1 2

Guided PracticeGraph the set of points whose polar coordinates satisfythe given conditions.

3 2r and4

x

y

3

24

Guided PracticeGraph the set of points whose polar coordinates satisfythe given conditions.

0r and4

x

y

4

Guided PracticeGraph the set of points whose polar coordinates satisfythe given conditions.

2 5

3 6

x

y