Polar Coordinates

1. 11.1 Polar Coordinates and Graphs
Objective
1) To graph polar equations.
2) To convert polar to rectangular
3) To convert rectangular to polar

2. One way to give someone directions is to tell them to go
three blocks East and five blocks South. This is like x-y
Cartesian graphing.
Another way to give directions is to point and say “Go a half
mile in that direction.”
Polar graphing is like the second method of giving directions.
Each point is determined by a distance and an angle.

Initial ray
r A polar coordinate pair
determines the location of
a point.
 
,
r 
O

3. The center of the graph is
called the pole.
Angles are measured from
the positive x-axis.
Points are
represented by a
radius and an angle
(r, )
To plot the point






4
,
5

First find the angle
Then move out along
the terminal side 5

4. A negative angle would be measured clockwise like usual.
To plot a point with
a negative radius,
find the terminal
side of the angle
but then measure
from the pole in the
negative direction
of the terminal side.







4
3
,
3








3
2
,
4


5. 





2
,
7
 







2
,
7







2
5
,
7








2
3
,
7

Therefore unlike in
the rectangular
coordinate system,
there are many ways
to express the same
point.

6. Let's take a point in the rectangular coordinate system and
convert it to the polar coordinate system.
(3, 4)
r

Based on the trig you
know can you see how
to find r and ?
4
3
r = 5
2
2
2
4
3 r


3
4
tan 

93
.
0
3
4
tan 1







 

We'll find  in radians
(5, 0.93)
polar coordinates are:
Convert Cartesian Coordinates to Polar Coordinates

7. Let's generalize this to find formulas for converting from
rectangular to polar coordinates.
(x, y)
r

y
x
2
2
2
r
y
x 

x
y


tan
2 2
r x y
  






 
x
y
1
tan

x = r cos, y = r sin

8. (b) (–1, 1) lies in quadrant II.
Since one possible value for  is
135º. Also,
Therefore, two possible pairs of polar coordinates
are
,
1
tan 1
1


 

.
2
1
)
1
( 2
2
2
2





 y
x
r
).
225
,
2
(
and
)
135
,
2
( 


Giving Alternative Forms for Coordinates of a Point

9. Now let's go the other way, from polar to rectangular
coordinates.
4cos ,
4
x


rectangular coordinates are:






4
,
4

4 y
x
4

2
2
2
2
4 









x
4sin ,
4
y

 2
2
2
2
4 









y
Convert Polar Coordinates to Cartesian Coordinates

10. Let's generalize the conversion from polar to rectangular
coordinates.
r
x


cos
 

,
r
r y
x

r
y


sin

cos
r
x 

sin
r
y 
Convert Polar Coordinates to Cartesian Coordinates

11. Graphs of Polar Equations
• Equations such as
r = 3 sin , r = 2 + cos , or r = ,
are examples of polar equations where r and  are
the variables.
• The simplest equation for many types of curves turns
out to be a polar equation.
• Evaluate r in terms of  until a pattern appears.

12. Find a rectangular equation for r = 4 cos θ

13. y
x 4
2


cos
r
x 

sin
r
y 
   

 sin
4
cos
2
r
r 

 sin
4
cos2
2
r
r 
substitute in for
x and y
We wouldn't recognize what this equation looked like in
polar coordinates but looking at the rectangular equation
we'd know it was a parabola.
What are the polar conversions
we found for x and y?
Converting a Cartesian Equation to a Polar Equation
2
4sin
cos
r


 4 tan sec
r  


14. (b) Solve the rectangular equation for y to get
(c)
.
2
2
3


 x
y

15. Convert a Cartesian Equation to a Polar Equation
3x + 2y = 4
Let x = r cos  and y = r sin  to get
.
sin
2
cos
3
4
or
4
sin
2
cos
3







 r
r
r
Cartesian Equation Polar Equation

16. Convert r = 5 cos to rectangular equation.
Since cos = x/r, substitute for cos.
5
x
r
r
 
Multiply both sides by r, we have
r2 = 5x
Substitute for r2 by x2 + y2, then
This represents a circle centered at (5/2, 0) and of
radius 5/2 in the Cartesian system.
x² + y² = 5x

17. 2
r
r
y
 
Now you try: Convert r = 2 csc to rectangular form.
Since csc = r/y, substitute for csc.
Multiply both sides by y/r.
Simplify, we have (a horizontal line) is
the rectangular form.
y = 2
2
y r y
r
r y r
   

18. For the polar equation
(a) convert to a rectangular equation,
(b) use a graphing calculator to graph the polar
equation for 0    2, and
(c) use a graphing calculator to graph the rectangular
equation.
(a) Multiply both sides by the denominator.
,
sin
1
4



r
4
sin 4
1 sin
r r r 

   

sin 4 4
r r r y

    
2 2
4 (4 )
r y r y
    
2 2 2
(4 )
x y y
  

19. Convert to a rectangular equation:
Multiply both sides by the denominator.
,
sin
1
4



r
4
sin 4
1 sin
r r r 

   

sin 4 4
r r r y

    
2 2
4 (4 )
r y r y
    
2 2 2
(4 )
x y y
  

20. 2 2 2
2
2
16 8
8 16
8( 2)
x y y y
x y
x y
   
 
 
Square both sides.
rectangular equation
It is a parabola vertex at (0, 2) opening down and p = –2,
focusing at (0, 0), and with diretrix at y = 4.
(b) The figure shows (c) Solving x2 = –8(y – 2)
a graph with polar for y, we obtain
coordinates.
.
2 2
8
1
x
y 


21. Theorem Tests for Symmetry
Symmetry with Respect to the Polar
Axis (x-axis):

22. Theorem Tests for Symmetry

23. Theorem Tests for Symmetry
Symmetry with Respect to the Pole
(Origin):

24. The tests for symmetry just presented are
sufficient conditions for symmetry, but
not necessary.
In class, an instructor might say a student
will pass provided he/she has perfect
attendance. Thus, perfect attendance is
sufficient for passing, but not necessary.

25. Identify points on the graph:

26. Polar axis:
Symmetric with respect to the polar
axis.
Check Symmetry of:

27. The test fails so the graph may or may not be
symmetric with respect to the above line.

28. The pole:
The test fails, so the graph may or may not be
symmetric with respect to the pole.

29. Cardioids (a heart-shaped curves)
are given by an equation of the form
r  a(1  cos) r  a(1 sin)
r  a(1 cos ) r  a(1 sin)
where a > 0. The graph of cardioid passes
through the pole.

30. Graphing a Polar Equation (Cardioid)
Example 3 Graph r = 1 + cos .
Analytic Solution Find some ordered pairs until a pattern is
found.
 r = 1 + cos   r = 1 + cos 
0º 2 135º .3
30º 1.9 150º .1
45º 1.7 180º 0
60º 1.5 270º 1
90º 1 315º 1.7
120º .5 360º 2
The curve has been
graphed on a polar
grid.

31. Limacons without the inner loop
are given by equations of the form
where a > 0, b > 0, and a > b. The graph
of limacon without an inner loop does not
pass through the pole.

32. 
0   5
1
2
3 

6

73
.
4
2
3
2
3 









Let's let each unit be 1.
3

4
2
1
2
3 







2
   3
0
2
3 

3
2 2
2
1
2
3 








6
5
27
.
1
2
3
2
3 










   1
1
2
3 


Since r is an even function of ,
let's plot the symmetric points.
This type of graph is called a
limacon without an inner loop.

cos
2
3

r
Graph r = 3 + 2cos

33. Limacons with an inner loop
are given by equations of the form
where a > 0, b > 0, and a < b. The graph
of limacon with an inner loop will pass
through the pole twice. Ex: r = 1 – 2cosθ

34. Lemniscates
are given by equations of the form
and have graphs that are propeller shaped.
Ex: r = 2
3 sin2

35. Graphing a Polar Equation (Lemniscate)
Graph r2 = cos 2.
Solution Complete a table of ordered pairs.

0º ±1
30º ±.7
45º 0
135º 0
150º ±.7
180º ±1

2
cos


r
Values of  for 45º <  < 135º are not
included because corresponding values of
cos 2 are negative and do not have real
square roots.

36. Rose curves
are given by equations of the form
and have graphs that are rose shaped. If n
is even and not equal to zero, the rose has
2n petals; if n is odd not equal to +1, the
rose has n petals. Ex: r = 2sin(3θ) and
r = 2sin(4θ)

37. Assignment
P. 400 #1 – 11 odd ( a and b is enough but
can do all if want more practice)

38. 330
315
300
270
240
225
210
180
150
135
120
0
90
60
30
45
Polar coordinates can also be given with the angle in
degrees.
(8, 210°)
(6, -120°)
(-5, 300°)
(-3, 540°)

39. Give three other pairs of polar coordinates for the point
P(3, 140º).
(3, –220º)
(–3, 320º)
(–3, –40º)

40. Since r is –4, Q is 4 units in the negative direction from the
pole on an extension of the ray.
The rectangular coordinates:
3
2
2
2
1
4
3
2
cos
4












x
3
2
2
3
4
3
2
sin
4













y
).
3
2
,
2
( 
Plot each point by hand in the polar coordinate
system. Then determine the rectangular coordinates of each
point.
 
2
3
4,
Q 


41. Graphing a polar Equation Using a
Graphing Utility
• Solve the equation for r in terms of θ.
• Select a viewing window in POLar mode.
The polar mode requires setting max and
min and step values for θ. Use a square
window.
• Enter the expression from Step1.
• Graph.