114,12
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