Give 4 different pairs of polar coordinates that correspond to the Cartesian Coordinates (2,2)

DefinitionA Polar Coordinate System is a method of locating a point (r, ) in a plane where r is a distance from a point called the pole directed at an angle θ measured counterclockwise from a polar axis.

Example

3

10,3

Polar Grid

Pole

Polar Axis

r=10

(r,θ )= 10,3

3

r=10

r= ̶ 10

Negative radius means go in the opposite directions

10,3

Remember to consider the rotation FIRST

( , ) ( , 180 )r r

Example

35,4

(r,θ )=

Find the polar coordinates:

Example

35,

4

(r,θ )=

Find the polar coordinates:

Plot (8, 105 )

Name the point with three other polar coordinates using 360 360

(8,255 )

( 8,75 )

( 8, 285 )

Important Idea

cosx r siny r

tany

x

2 2r x y

Polar coordinates are related to rectangular coordinates as follows:

( , )r

( , )x y

θr y

x

ExampleGraph:r=6

What aboutr = 1 6?

ExampleGraph:θ=

6

What about

?6

Some Classic Polar Graphs

Example

4 6sinr

limacon

A Classic Polar GraphExampl

e3

2r

Spiral of Archimedes

Some Classic Polar Graphs

Example

6sin(4 )r

8 petal rose

Use your graphing calculator to graph2 2sinr

MODE POL

GRAPH

Y= 1\ 2 2sinr

CARDIOID

Polar Roses

( ) 8cos(2 )r

Important Idea

84 petals

Polar Roses

( ) 10cos(4 )r

Important Idea

10 8 petals

Polar Roses

( ) 10cos(4 )r

Try This Name the angle in rads?

4

Important Idea

( ) 8cos(5 )r ( ) 8cos(2 )r

What do you observe about the number of petals? The period?

Some CalculusFind the slope of 1 sin r at . 4

We can write polar equations parametricallycos

sin

x r

y r

r f

cos sin

sin cos

drrdy dy d d

drdx dx d rd

where