9.6 – POLAR COORDINATES I N THIS SECTION, YOU WILL LEARN TO plot points in the polar coordinate...

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9.6 – POLAR COORDINATES

IN THIS SECTION, YOU WILL LEARN TO

plot points in the polar coordinate system

convert points from rectangular to polar form and vice versa

convert equations from rectangular to polar form and vice versa

POLAR COORDINATE SYSTEM: So far, you have been working in

the rectangular coordinate system, where

(x, y) represented the directed distances from the coordinate axes.

You will now be working in the polar coordinate system.

POLAR COORDINATE SYSTEM:

a) Definition: A point P in the plane has polar coordinates if the line segment OP has length and the angle that OP makes with the positive axis is (measured in a counter clockwise direction). The fixed point O is called a pole and initial ray from O is called the polar axis.

,r r

POLAR COORDINATE SYSTEM:

Polar Coordinate

Directed angle Pole Polar Axis

,r

directed dis: the from Otance to Pr

: the directed angle measured counterclockwise from the polar  axis to segment OP.

POLAR COORDINATE SYSTEM:

,r

,r

If point Q has the coordinate   , , then

point P has the coordinate , if it lies

on the straight line containing OQ and has the same distance from the pole.  

r

r

POLAR COORDINATE SYSTEM:

The coordinate , has multiple

representations by using positive andnegative as well as   . In general,  ,

can be represented as , 2 or

, 2 1 where is any integer.   

r

r r

r n

r n n

In rectangular coordinates, each point can be expressed in one unique representation of  ( , ).  This is not the case for polar coordinates.  

x y

POLAR COORDINATE SYSTEM:

1) Graph   2,30 and find multiple

representation of this polar coordinate.

2,210

2, 330

2, 150

POLAR COORDINATE SYSTEM:

2) Graph   3,300 and find multiple

representation of this polar coordinate.

3,120

3, 60

3, 240

POLAR COORDINATE SYSTEM:53) Graph   6,  and find multiple4

representation of this polar coordinate.

POLAR COORDINATE SYSTEM:

6,4

36,4

76,4

56,4

POLAR COORDINATE SYSTEM:24) Graph   5,  and find multiple3

representation of this polar coordinate.

POLAR COORDINATE SYSTEM:

55,3

5,3

45,3

25,3

COORDINATE CONVERSION: Rectangular coordinates can be

converted to polar coordinates and vice versa.

Then the polar coordinates and the cartesian coordinates (x,y) of the same point are related as follows:

,r

POLAR TO RECTANGULAR COORDINATES: To convert between polar and

rectangular coordinates, we make a right triangle to the point (x,y) like this:

siny r

cosx r

sin yr

cos xr

POLAR TO RECTANGULAR COORDINATES:

Therefore, the polar point   ,  can be

converted to rectangular coordinates ,

by : cos , sin ,

r

x y

r r x y

RECTANGULAR TO POLAR COORDINATES:

Rectangular coordinates   ,   are related to

the  polar coordinates   , by :

x y

r

1tan tany yx x

2 2 2r x y

COORDINATE CONVERSION:

Example #1: Convert   3,   to rectangular 4

coordinates. 

COORDINATE CONVERSION:Example #1: Convert   3,   to rectangular

4coordinates. 

siny r cosx r

3 2 3 2, 2.1, 2.12 2

2 3 232 2

x

3cos4

x

3sin4

y

2 3 232 2

y

COORDINATE CONVERSION:Example #1: Convert   3,   to rectangular

4coordinates. 

1 2 3 4 5 6 7 8 9-1-2-3-4-5-6-7-8-9123456789

-1-2-3-4-5-6-7-8-9

x

y

COORDINATE CONVERSION: Example #2 : Convert    2, 2   to polar

coordinates. 

COORDINATE CONVERSION: Example #2 : Convert    2, 2   to polar

coordinates. 

2tan tan 12

COORDINATE CONVERSION:

2, 2 is in the IV Quadrant

Example #2 : Convert    2, 2   to polar

coordinates. 

7 3 52 2, , 2 2, , 2 2, , 2 2,4 4 4 4

COORDINATE CONVERSION: Example #2 : Convert    2, 2   to polar

coordinates. 

1 2 3 4 5 6 7 8 9-1-2-3-4-5-6-7-8-9123456789

-1-2-3-4-5-6-7-8-9

x

y

COORDINATE CONVERSION: Example #3: Convert    1, 3   to polar

coordinates. 

CONVERTING POLAR EQUATIONS TO RECTANGULAR EQUATIONS: To convert a rectangular equation to polar

form, replace by cos and by sin .Solve the equation in terms of .

x r y r

Example #1: Convert  3 1 0  to polar form.

x y

CONVERTING POLAR EQUATIONS TO RECTANGULAR EQUATIONS:

3 1 0 3 cos sin 1 0x y r r

Example #1: Convert  3 1 0  to polar form.

x y

3 cos sin 1r r

3cos sin 1r

13cos sin

r

CONVERTING RECTANGULAR EQUATIONS TO POLAR EQUATIONS:

When you graph this on the polar system,

it is a circle with radius 3. Therefore, the

rectangular equation should also reflect a

circle with radius 3.

Example #1: 3r

2 23 3r r 2 2 2Substitute withr x y

2 2 2 2 23 9x y x y

CONVERTING RECTANGULAR EQUATIONS TO POLAR EQUATIONS:

When you graph this on the polar system,

it is line at this angle.

2Example #2 :3

23

2tan3

yx

3 yx

3y x

CONVERTING RECTANGULAR EQUATIONS TO POLAR EQUATIONS:

Example #3: cscr

cscr

1sin

r

sin 1r

1y