Polar Form and Complex
Numbers
In a rectangular coordinate system,
There is an x and a y-axis.
In polar coordinates, there is one
axis, called the polar axis, and its
vertex is called the pole.
While Cartesian Coordinates depend
on x and y values, Polar Coordinates
depend on r and 
For any point plotted, it can be
represented by 4 different polar
coordinates.
For example: let’s plot (4,60 )
0
Now, plot the ordered pair (3, -210 degrees)
Next, write three other ordered pairs that represent the same point
Distance Formula in the Polar Plane
How can we find the DISTANCE between two points
Defined in the Polar Plane? Well, we can use the Law of
Cosines…
Distance Formula in the Polar Plane
If P1 r1 , 1  and P(r2 ,  2 ) are two points in
the polar plane, then...
P1P2  r1  r2 - 2r1r2 cos( 2  1 )
2
2
Polar vs. Rectangular Forms
The following relationships exist between Polar
Coordinates (r, ) and Rectangular Coordinates (x, y):
y
tan  
x
x y r
2
2
8
6
( r , )
2
4
(x, y)
r
2
y  r sin 
x  r cos
y

x
-5
-2
5
10
Polar vs. Rectangular Forms
Rewrite the following polar coordinates in
rectangular form: (4,120o )
Polar vs. Rectangular Forms
Now, rewrite the following rectangular
coordinates in polar form: (5, 5).
An equation whose variables are
polar coordinates is called a polar
equation. The graph of a polar
equation consists of all points
whose polar coordinates satisfy the
equation.
Identify and graph the equation: r = 2
r2
r 4
2
x y 4
2
2
Circle with center at the pole and radius
2.
105
90
75
120
60
135
45
150
30
165
15
180
0
0
1
2
3
195
4
345
210
330
225
315
240
300
255
270
285
Identify and graph the equation:  =


tan   tan 
 3
3

1
y
3

x 1
y  3x

3
105
90
75
120
60
135


3
45
150
30
165
15
180
0
0
1
2
3
195
4
345
210
330
225
315
240
300
255
270
285
Identify and graph the equation: r sin   2
y
sin  
 y  r sin 
r
y  2
120
105 90 75
60
135
45
150
30
165
15
180
0
0 1
195
2
210
3
4
345
330
225
315
240
255 270 285
300
r sin   a
is a horizontal line a units above the
pole if a > 0 and a units below the
pole if a < 0.
r cos  a
is a vertical line a units to the right of
the pole if a > 0 and a units to the left
of the pole if a < 0.
4
r cos  3
2
5
5
2
4
Identify and graph the equation: r  4 cos
r  4r cos
2
x  y  4x
2
2
x  4x  y  0
2
2
x  4x  4  y  4
2
2
 x  2  y  4
2
2
120
105 90 75
60
135
45
150
30
165
15
180
0
0 1
195
2
210
3
4
345
330
225
315
240
255 270 285
300
r  2a cos
r  2a cos
Circle: radius a; center
at (a, 0) in rectangular
coordinates.
Circle: radius a; center
at (-a, 0) in rectangular
coordinates.
r  2a sin 
r  2a sin 
Circle: radius a; center
at (0, a) in rectangular
coordinates.
Circle: radius a; center
at (0, -a) in rectangular
coordinates.
r  4 sin 
4
2
5
5
2
r  6 sin 
2
5
5
2
4
6
8
In order to use your graphing calculator to graph Polar
Equations, change your MODE to POLAR (instead of
Function). Also, change your viewing window as follows…
For DEGREES:
For RADIANS:
min = 0
max = 360
step = 10
Xmin = -8
Xmax = 8
Xscl = 1
Ymin = -8
Ymax = 8
Yscl = 1
min = 0
max = 2
step = /18
Xmin = -8
Xmax = 8
Xscl = 1
Ymin = -8
Ymax = 8
Yscl = 1
Now that you have your graphing calculator set up to
graph Polar Equations, graph the following equations and
see if you can identify the shape and how the numbers
affect the graph itself…
r = 2 + 2sin
r = 2 + 2cos
r = 1 + sin
r = -2 + -2cos
r = 3 + 3sin
r = 3 + 3cos
r  a  a cos
or
r  a  a sin 
6
6
4
4
2
2
5
5
10
5
5
2
2
4
4
6
6
Is the graph of a CARDIOID (heart)
shape, symmetric to either the x axis
(for cosine) or y axis (for sine)
10
Now graph the following equations and see if you can
identify the shape and how the numbers affect the graph
itself…
r = 2 + 3sin
r = 1 + 2cos
r = 1 + 4sin
r = 3 + 2cos
r = 2 + sin
r = 4 + 2cos
r  a  b cos
r  a  b sin 
or
6
2
4
1
2
4
2
2
4
5
5
1
2
2
4
6
3
Is the graph of a Limacon (pronounced “leema-sahn”) shape, symmetric to either the x
axis (for cosine) or y axis (for sine)
r  2  4 cos
r  3 2 sin 
6
2
4
1
2
4
2
2
4
5
5
1
2
2
a<b
4
a>b
6
3
Notice how the graph of a limacon changes
depending on whether a > b or a < b
Now graph the following equations and see if you can
identify the shape and how the numbers affect the graph
itself…
r = 3sin2
r = 2cos4
r = 4sin3
r = 5cos2
r = 3sin
r = -3cos3
r  a cos b
r  a sin b
or
3
4
3
2
2
1
1
6
4
2
2
4
6
4
2
2
4
1
1
2
2
3
4
3
Is the graph of a ROSE shape, symmetric to
either the x axis (for cosine) or y axis (for
sine)
Below are the graphs of the roses for
r  3 cos 2
r  4 sin 3
and
3
4
3
2
2
1
1
6
4
2
2
4
6
4
2
2
4
1
1
2
2
3
4
3
Notice how the ‘b’ value affects the graph: if
b is even, then there are ‘2*b’ number of
rose petals (loops); if ‘b’ is odd, there are ‘b’
number of petals
The next type of graph we are going to look at
involves the following formats for the equation:
r  a cos 2
2
and
r  a sin 2
2
However, with the graphing calculator, we cannot
Type the equations in this fashion.
Instead, we take the square root of both sides of the
Equation and type that equation into the calculator.
For example:
r  9 cos 2
2
is typed in as
r  9 cos 2
Now graph the following equations and see if you can
identify the shape and how the numbers affect the graph
itself…
r  9 cos 2
2
r  4 sin 2
2
r  16 cos 2
2
r  5 sin 2
2
r  a cos 2
2
r  a sin 2
2
or
4
4
3
3
2
2
1
1
6
6
4
2
2
4
4
2
2
4
6
1
1
2
3
4
2
3
4
Is the graph of a lemniscate (pronounced
“lem-nah-scut”) shape, symmetric to either
the x axis (for cosine) or the line y = x (for
sine)
6
The next type of graph we are going to look at
involves the following format for the equation:
r  a
However, with the graphing calculator, we will not be
able to see much of the graph if we work with degrees,
because r keeps increasing as the angle measure does.
So switch to RADIAN MODE and be sure to modify
the X and Y values in WINDOW to accommodate
each graph.
Now graph the following equations and see if you can
identify the shape and how the numbers affect the graph
itself…
r 
r  3
r  2
r  a
12
10
8
6
4
2
15
10
5
5
10
15
20
2
4
6
8
10
12
Is the graph of a Spiral of Archimedes
(pronounced “Ar-cah-mee-dees”) shape.