Up to this point we’ve dealt exclusively with the Cartesian
(or Rectangular, or xy) coordinate
system. However, as we will see, this is
not always the easiest coordinate system to work in. So, in this section we will start looking at the
polar coordinate system.
Coordinate systems are really nothing more than a way to
define a point in space. For instance in
the Cartesian coordinate system at point is given the coordinates (x,y) and we use this to define the point
by starting at the origin and then moving x
units horizontally followed by y
units vertically. This is shown in the
sketch below.
This is not, however, the only way to define a point in two
dimensional space. Instead of moving
vertically and horizontally from the origin to get to the point we could
instead go straight out of the origin until we hit the point and then determine
the angle this line makes with the positive xaxis. We could then use the distance of the point
from the origin and the amount we needed to rotate from the positive xaxis as the coordinates of the
point. This is shown in the sketch
below.
Coordinates in this form are called polar coordinates.
The above discussion may lead one to think that r must be a positive number. However, we also allow r to be negative. Below is a
sketch of the two points and .
From this sketch we can see that if r is positive the point will be in the same quadrant as θ. On the other hand if r is negative the point will end up in the quadrant exactly
opposite θ.
Notice as well that the coordinates describe the same point as the coordinates do. The
coordinates tells us to rotate an angle of from the positive xaxis, this would put us on the dashed line in the sketch above,
and then move out a distance of 2.
This leads to an important difference between Cartesian
coordinates and polar coordinates. In
Cartesian coordinates there is exactly one set of coordinates for any given
point. With polar coordinates this isn’t
true. In polar coordinates there is
literally an infinite number of coordinates for a given point. For instance, the following four points are
all coordinates for the same point.
Here is a sketch of the angles used in these four sets of
coordinates.
In the second coordinate pair we rotated in a clockwise
direction to get to the point. We shouldn’t
forget about rotating in the clockwise direction. Sometimes it’s what we have to do.
The last two coordinate pairs use the fact that if we end up
in the opposite quadrant from the point we can use a negative r to get back to the point and of course
there is both a counter clockwise and a clockwise rotation to get to the
angle.
These four points only represent the coordinates of the
point without rotating around the system more than once. If we allow the angle to make as many
complete rotations about the axis system as we want then there are an infinite
number of coordinates for the same point.
In fact the point can be represented by any of the following
coordinate pairs.
Next we should talk about the origin of the coordinate
system. In polar coordinates the origin
is often called the pole. Because we aren’t actually moving away from the
origin/pole we know that . However, we can still rotate around the
system by any angle we want and so the coordinates of the origin/pole are .
Now that we’ve got a grasp on polar coordinates we need to
think about converting between the two coordinate systems. Well start out with the following sketch
reminding us how both coordinate systems work.
Note that we’ve got a right triangle above and with that we
can get the following equations that will convert polar coordinates into Cartesian coordinates.
Polar to Cartesian
Conversion Formulas
Converting from Cartesian is almost as easy. Let’s first notice the following.
This is a very useful formula that we should remember,
however we are after an equation for r
so let’s take the square root of both sides.
This gives,
Note that technically we should have a plus or minus in
front of the root since we know that r
can be either positive or negative. We
will run with the convention of positive r
here.
Getting an equation for θ is almost as simple. We’ll start with,
Taking the inverse tangent of both sides gives,
We will need to be careful with this because inverse
tangents only return values in the range . Recall that there is a second possible angle
and that the second angle is given by .
Summarizing then gives the following formulas for converting
from Cartesian coordinates to polar coordinates.
Cartesian to Polar
Conversion Formulas
Let’s work a quick example.
Example 1 Convert
each of the following points into the given coordinate system.
(a) into Cartesian coordinates. [Solution]
(b) (1,1)
into polar coordinates. [Solution]
Solution
(a) Convert into Cartesian
coordinates.
This conversion is easy enough. All we need to do is plug the points into
the formulas.
So, in Cartesian coordinates this point is .
[Return to Problems]
(b) Convert (1,1) into polar coordinates.
Let’s first get r.
Now, let’s get θ.
This is not the correct angle however. This value of θ is in the first quadrant and the point we’ve
been given is in the third quadrant.
As noted above we can get the correct angle by adding π onto this.
Therefore, the actual angle is,
So, in polar coordinates the point is . Note as well that we could have used the
first θ that we got by using a negative r.
In this case the point could also be written in polar coordinates as .
[Return to Problems]

We can also use the above formulas to convert equations from
one coordinate system to the other.
Example 2 Convert
each of the following into an equation in the given coordinate system.
(a) Convert
into polar coordinates. [Solution]
(b) Convert
into Cartesian coordinates. [Solution]
Solution
(a) Convert into polar coordinates.
In this case there really isn’t much to do other than
plugging in the formulas for x and y (i.e.
the Cartesian coordinates) in terms of r
and (i.e.
the polar coordinates).
[Return to Problems]
(b) Convert into Cartesian coordinates.
This one is a little trickier, but not by much. First notice that we could substitute
straight for the r. However, there is no straight substitution
for the cosine that will give us only Cartesian coordinates. If we had an r on the right along with the cosine then we could do a direct
substitution. So, if an r on the
right side would be convenient let’s put one there, just don’t forget to put one on the left side as well.
We can now make some substitutions that will convert this
into Cartesian coordinates.
[Return to Problems]

Before moving on to the next subject let’s do a little more
work on the second part of the previous example.
The equation given in the second part is actually a fairly
well known graph; it just isn’t in a form that most people will quickly recognize. To identify it let’s take the Cartesian
coordinate equation and do a little rearranging.
Now, complete the square on the x portion of the equation.
So, this was a circle of radius 4 and center (4,0).
This leads us into the final topic of this section.
Common Polar
Coordinate Graphs
Let’s identify a few of the more common graphs in polar
coordinates. We’ll also take a look at a
couple of special polar graphs.
Lines
Some lines have fairly simple equations in polar
coordinates.
 .
We can see that this is a line by converting to Cartesian coordinates as
follows
This is a line that goes through
the origin and makes an angle of β with the positive xaxis. Or, in other words
it is a line through the origin with slope of .
This is easy enough to convert to Cartesian coordinates to . So, this is a vertical line.
Likewise, this converts to and so is a horizontal line.
Circles
Let’s take a look at the equations of circles in polar
coordinates.
 .
This equation is saying that no matter what angle we’ve got the distance
from the origin must be a. If you think about it that is exactly
the definition of a circle of radius a
centered at the origin.
So, this is a circle of radius a
centered at the origin. This is
also one of the reasons why we might want to work in polar
coordinates. The equation of a
circle centered at the origin has a very nice equation, unlike the corresponding
equation in Cartesian coordinates.
 .
We looked at a specific example of one of these when we were converting
equations to Cartesian coordinates.
This is a circle of radius and center . Note that a might be negative (as it was in our example above) and so
the absolute value bars are required on the radius. They should not be used however on the
center.
 .
This is similar to the previous one.
It is a circle of radius and center .
 .
This is a combination of the previous two and by completing the square
twice it can be shown that this is a circle of radius and center . In other words, this is the general
equation of a circle that isn’t centered at the origin.
Note that it takes a range of for a complete graph of and it only takes a range of to graph the other circles given here.
Cardioids and Limacons
These can be broken up into the following three cases.
 Cardioids
: and .
These have a graph that is vaguely heart shaped and always contain the
origin.
 Limacons
with an inner loop : and with .
These will have an inner loop and will always contain the origin.
 Limacons
without an inner loop : and with .
These do not have an inner loop and do not contain the origin.
Example 5 Graph
,
,
and .
Solution
These will all graph out once in the range . Here is a table of values for each followed
by graphs of each.




0

5

1

6


0

7

2


5

13

2


10

7

2


5

1

6


There is one final thing that we need to do in this
section. In the third graph in the
previous example we had an inner loop.
We will, on occasion, need to know the value of θ for which the graph will pass through the
origin. To find these all we need to do
is set the equation equal to zero and solve as follows,