Polar Coordinates
It is often useful to use a polar coordinate system to describe certain curves because
the polar equations greatly simplify some curves as we shall see. We begin with a
description as well as a method to convert between the rectangular system and the
polar system. This can be understood from a sketch.
Polar coordinates are generally written in the form (r,θ
θ) where r is the distance from the
origin (the pole) to the point and θ is the angle from the positive x axis (the polar axis)
to the ray connecting the point and the pole.
By this definition and drawing, the method for conversion should be easily seen:
2
2
Rectangular to polar: r = x + y , θ = tan
− 1 y 
 x
 
Polar to rectangular x = r⋅ cos( θ ) , y = r⋅ sin( θ )
There are a few small troubles with rectangular to polar, but since representation in
rectangular coordinates is unique, the conversion from polar to rectangular is trouble
free.
Trouble spots for converting rectangular to polar:
1) we have to add π if x<0.
π
π
2) what if x=0? If y>o then θ = , if y<0 then θ = −
2
2
3) Representation of a point in polar coordinates is not unique for two reasons:
a) Coterminal angles
b) possibility that r<0
Ex 1 convert the following polar points to rectangular coordinates and plot on a graph.
1) (5,π)
x=5cos(π) = -5 , y=5sin(π) =0
(-5,0)
π
3 1


2)  1 ,  x=1cos(π/6), y=1sin(π/6) 
, 
 6
 2 2
 4π  x=5cos(4π/3),y=5sin(4π/3)  − 5 , − 5 3 



2 
 3
 2
3)  5 ,
4) (-5,0)
(-5,0)
π

5)  4 , −  (0,-4)
 2
 5π   4 , 4  or ( 2 2 , 2 2 )
6)  −4 ,


4   2

2
Ex 2 Give two different conversions to polar coordinates and plot on a graph:
3π 
3⋅ π 
11⋅ π 
7π  
2
2

1) (-4,4) r = ( −4 ) + 4 = 4 2 , θ =
 4⋅ 2 ,
 and  −4 2 ,   4⋅ 2 ,

4 
4 
4 
4  

 3π  and  −6 , π 
2) (0,-6)  6 ,



 2 
 2
π
− 1 1  −π


 = tan  −  = 6 so  10 , − 6 


5 3
 3
2
π
2
−1

  2π 
4) ( −10 , 10 3 ) r = ( 10⋅ 3 ) + ( −10) = 20 and θ = tan ( − 3 )  20 , − + π  or  20 ,

3
3 

 
2
2
−1 8
−1 4
−1 4
5) (6,8) r = 6 + 8 = 10 θ = tan    10 , tan    and  10 , tan   + 2π 
6
3
  
 

3

− 1 4 


6) (-6,-8) r = 10  10 , tan   + π 

3

3) ( 5 3 , −5 ) r =
(5⋅ 3)2 + ( −5) 2 = 10 , θ = tan− 1
−5
Graphs in polar coordinates
Polar equations are often in the form r = f(θ)
Ex 3 sketch by plotting points (ugh!):
r = 4cos(θ)
 θ r = f (θ ) 
 0

4


 π 2 3 , 3.5 
 6



 π 2 2 , 2.8 
 4

 π

2
 3



 π

0
 2



 2π
−2 
 3

The following polar equations will generate circles:
r = a⋅ sin( θ ) r = a⋅ cos( θ ) and r = a
Ex 4 Show why 1) r = a*sin(θ) and 2) r = a generate circles by
converting to rectangular equations
r = a⋅ sin( θ )
x + y = a⋅ sin tan
2
2
− 1 y


2
x
y
2
x + y = a⋅
2
2
x +y
2
2
2
2
x + y = a⋅ y
x + y − a⋅ y = 0
x +  y −
2

a
2

=
2
a
2
4
circle centered at  0 ,
a
a
 with radius 2 .
 2
r=a
2
2
x + y =a
2
2
x + y =a
2
circle centered at origin radius a
Ex 5 Graph on a calculator or software after thinking about it first:
r = 4 cos( θ )
90
120
60
150
4 cos( θ )
30
180
0
0
2
4
210
330
240
300
270
θ
The following polar curves are called the limaçon family:
r = a + b ⋅ cos( θ ) and r = a + b ⋅ sin( θ )
Ex 6 Use a calculator to sketch the limaçon r = 4 − 4 sin( θ )
90
120
60
150
4− 4⋅ sin( θ )
30
180
0
02 4 6 8
210
330
240
300
270
θ
Notes about the limaçon polar curves r = a + b ⋅ sin( θ ) and r = a + b ⋅ cos( θ )
If a = b it is called a cardioid (heart shape like Ex 6)
If b > a then it is a limaçon with a 'loop'
If a > b then it is a limaçon with a 'dimple'
Ex 7 Graph the following polar curves to observe the limaçon family:
1) r = 3 − 5 cos( θ )
2) r = 6 + 4 sin( θ )
3) r = 4 + 4 cos( θ )
90
120
60
150
30
3− 5⋅ cos( θ )
6+ 4⋅ sin( θ )
4+ 4⋅ cos( θ )
180
0
0
5
10
210
330
240
300
270
θ
The following polar curves are called rose curves:
r = a⋅ cos( n ⋅ θ ) and r = a⋅ sin( n ⋅ θ )
Ex 8 Draw the following rose curves with a calculator or software:
1) r = 5 ⋅ cos( 4θ )
2) r = 3 ⋅ sin( 5θ )
3) r = 4 cos( 2θ )
90
120
60
150
30
5⋅ cos( 4θ )
3⋅ sin( 3⋅ θ )
5⋅ cos( 2⋅ θ )
180
0
0
2
4
210
6
330
240
300
270
θ
Notes about the rose curves r = a⋅ cos( n ⋅ θ ) and r = a⋅ sin( n ⋅ θ ):
1) The length of each petal is a
2) The number of petals is n if n is odd
3) The number of petals is 2n if n is even
Ex 9 Draw the rose curve r = 5 cos( 2θ ) without a calculator
The following graphs are in the spiral family:
r = θ (spiral)
r=e
a⋅ θ
(logarithmic spiral)
Ex 10 Graph on a calculator or computer:
1) r = θ
θ
2) r = e
2
90
120
θ
60
150
30
θ
e
180
2
0
0 2
4
210
6
330
240
300
270
θ
Due to software limitations, I can only graph with 0 ≤ θ ≤ 2π but in
the drawing below I 'fooled' it into graphing r = θ on 0 ≤ θ ≤ 6π
90
120
θ
60
150
30
θ + 2π
θ + 4π
180
0
0 5 10 15
210
330
240
300
270
θ
θ
In this one, I try to make clearer the logartihmic spiral r = e
90
120
60
2
e
θ
e
150
θ
2
2
−π
30
180
0
0
10
210
20
330
240
300
270
θ
To convert equations from polar to rectangular coordinates and from rectangular
to polar coordinates, use the formulas given at the beginning of the lesson, or
else draw the triangles the formulas are derived from and proceed from there.
Ex 11 Convert the rectangular equation to a polar equation:
2
2
x − y = 16
r cos ( θ ) − r sin ( θ ) = 16
2
2
2
2
r  cos ( θ ) − sin ( θ )  = 16
2
2
2
r ⋅ cos( 2θ ) = 16
2
2
r =
16
cos( 2θ )
90
r = 16 sec ( 2θ )
2
120
60
150
r = 4 sec( 2θ ) , r = −4 sec( 2θ )
4 sec( 2θ )
30
180
0
0
10
210
20
330
240
300
270
θ
Ex 12 convert the polar equations into rectangular equations.
1) r = 3
2) r = 4 sec( θ )
3) r( 3 cos( θ ) − 4 sin( θ ) ) = 12
2
2
2
x + y =3
x + y = 4 sectan
2
2
x + y =9
2
− 1 y 
 x 
 

2
2
2
x + y =4
1=
2
x +y
x
4
x
x=4
r( 3 cos( θ ) − 4 sin( θ ) ) = 12
3r⋅ cos( θ ) − 4 ⋅ r⋅ sin( θ ) = 12
3x − 4y = 12