11.7 Polar Equations
By now you've seen, studied, and graphed many functions and equations  perhaps
all of them in Cartesian coordinates. Sometimes it is more convenient to
use polar equations: perhaps the nature of the graph is
better described that way, or the equation is much simpler.
Examples of polar equations are:
Graphing a Polar Equation
The graph of a polar equation is the set of all points in the plane whose
polar coordinates (at least one representation) satisfy the equation.
The graph of the polar equation r = 1 consists of those points in
the plane whose distance from the pole is 1. That is the circle of radius
1 centered at the pole.
The graph of
= /4
is the set of all points which are at an angle of
/4
to the polar axis. In other words, it is a straight line passing through
the pole at an angle of
/4
to the polar axis. However, not all the graphs of polar equations are so
easy to describe.
We will restrict our discussion to polar equations of the form
r = f(), i.e. r is a
function of .
Our main goal is to write polar equations for conic sections in Section 11.8.
In this section we will take a brief look at graphing in general and desmonstrate
with a few specific examples.
One method we use to sketch the
graph of a polar equation is to plot points. In graphing a polar equation
of the form
r = f()
we treat
as the independent variable and r as
the dependent variable. We select several values of
, calculate the corresponding value of r,
then plot the points
(r,). Through these points we draw a
smooth curve. Let's work an example.
Example: 
Sketch the graph of r = 1 + sin().

Solution: 
We make a table of values by selecting several values of
and computing values of r. To
aid graphing
we use approximate values of r most of the time. Also note that
1 + sin() is periodic of period
2 so that we only need choose values of the
angle in one revolution.

0 
/6 
/4 
/3 
/2 
2/3 
3/4 
5/6 

5/4 
3/2 
7/4 
r 
1 
1.5 
1.71 
1.87 
2 
1.87 
1.71 
1.5 
1 
0.29 
0 
0.29 
Our next step is to plot these points in the polar plane.
To help us place points, we have drawn in
several rays from the pole at intervals of /4.
We have also drawn four concentric circles centered at the pole of diameters
1 through 4, respectively.
After plotting the twelve points from our table of values, we obtain the
following picture.
Finally, through these points we draw a curve. This graph is called
a cardioid.

There are some other techniques we can use to help sketch polar graphs. One
is to recognize certain forms of polar equations and the corresponding
graphs. As in the above example, the graph of
r = a
(1 f()),
where f is the sine or cosine function, and a ≠ 0,
is a cardioid. The value of a scales the curve, and the choice
of f determines the orientation.
Cardioids are special cases of curves called limaçons (pronounced leemason)
which are equations of the form
r = a
bf()),
a, b ≠ 0.
In the following demonstration you can investigate cardioids and limaçons.
You will be observing the graph of
r = a + bsin()
or
r = a + bcos(),
where a and b are nonzero real numbers.
You can select the values of a and b within certain ranges.
The resulting graph will be displayed as you change the values.
Initially, the polar equation uses the sine function. Press the button marked
"sin" to change to the cosine function. Press the button labeled "b > 0" to
change to negative values of b.
Another technique used to help sketch the graph of a polar equation is to
look for symmetry. There are three types of symmetry we look for; symmetry
about the polar axis, symmetry about the pole, and symmetry about the line
= /2. (These are
analogous to symmetry about the xaxis, origin, and yaxis, respectively, in the Cartesian plane.)
The graph will be symmetric about the polar axis if
whenever (r,) lies on the graph so
too does
(r,−). This means that
f()
= f(−).
For example, consider the polar equation
r = cos(). Now,
f(−) = cos(−) = cos()
= f(). So its graph is symmetric
about the polar axis.


The graph will be symmetric about the pole if
whenever (r,) lies on the graph so
too does
(r,+). This means that
f()
= f(+).
For example, consider the polar equation
r = sin(2). Now,
f(+) =
sin(2(+)) =
sin(2+2) =
sin(2) =
f().


The graph will be symmetric about the line
= /2 if
whenever (r,) lies on the graph so
too does
(r, − ). This means that
f()
= f( − ).
For example, consider the polar equation
r = 2 + sin(). Now,
f( − ) =
2 + sin( − ) =
2 + sin()cos() −
cos()sin() =
2 + (0)cos() −
(−1)sin() =
2 + sin() =
f().


Cartesian Equations and Polar Equations
When we want to reference points in a plane with both Cartesian coordinates and
polar coordinates, we superimpose the planes so that the polar axis
coincides with the positive direction of the xaxis, and the pole
corresponds to the origin. This allows us to more easily rewrite a Cartesian
equation as a polar equation and vice versa.
Our first example shows how to convert a Cartesian equation to a polar equation.
Conversion in this direction is fairly straightforward as we merely have to
substitute for x and y as demonstrated below. There is no guarantee,
however, that the resulting polar equation can be easily simplified.
What would the polar equation be if the circle had been centered at
(0,a)? Carry out a similar calculation, and then click
here to check your answer.
Converting a polar equation to Cartesian is not so straightforward in general. There are some
polar equations in which we can algebraically make r^{2} show up and that can
be an advantage. The next example shows such a case.
Example:

Convert r = 7 to a Cartesian equation.

Solution: 
Squaring both sides produces r^{2} = 49 which is
x^{2} + y^{2} = 49, a circle of radius 7.

It is true this example worked out very quickly but consider an equation
such as r = 15 + 9 + 22cos().
It is likely that converting this to Cartesian form would
produce something quite complicated and no easier to understand than its current
form. Fortunately, as we will see, the polar forms
of conics sections presented in Section 11.8 can be converted easily to Cartesian equations,
thus verifying the equivalence of the two versions of the conic sections.