Just as in the case of rectangular coordinates, it is possible to use relations between the
polar coordinates to specify points in the plane. The process of sketching their graphs is
very similar to that used to sketch graphs of functions in rectangular coordinates. I
will only consider the case where the relation between the polar coordinates is
of the form, r = f

(θ)

. To graph such a relation, you can make a table of the
form

and then graph the resulting points and connect them up with a curve. The following
picture illustrates how to begin this process.

PICT

To obtain the point in the plane which goes with the pair

(θ,f (θ))

, you draw the ray
through the origin which makes an angle of θ with the positive x axis. Then you move
along this ray a distance of f

(θ)

to obtain the point. As in the case with rectangular
coordinates, this process is tedious and is best done by a computer algebra
system.

Example 12.1.1Graph the polar equation r = 1 + cosθ.

Using a computer algebra system, here is the graph of this cardioid.

PICT

You can also see just from your knowledge of the trig. functions that the graph should
look something like this. When θ = 0,r = 2 and then as θ increases to π∕2, you
see that cosθ decreases to 0. Thus the line from the origin to the point on the
curve should get shorter as θ goes from 0 to π∕2. Then from π∕2 to π, cosθ gets
negative eventually equaling −1 at θ = π. Thus r = 0 at this point. Viewing the
graph, you see this is exactly what happens. The above function is called acardioid.

Here is another example. This is the graph obtained from r = 3 + sin

(7θ)
6-

.

Example 12.1.2Graph r = 3 + sin

( )
76θ

for θ ∈

[0,14π]

.

PICT

In polar coordinates people sometimes allow r to be negative. When this
happens, it means that to obtain the point in the plane, you go in the opposite
direction along the ray which starts at the origin and makes an angle of θ with the
positive x axis. I do not believe the fussiness occasioned by this extra generality
is justified by any sufficiently interesting application so no more will be said
about this. It is mainly a fun way to obtain pretty pictures. Here is such an
example.