Rectangular coordinates allow us to describe a point
in the plane in a different way, namely
where is any real number and is an angle.
Polar coordinates are extremely useful, especially
when thinking about complex numbers. Note, however,
representation of a point is
First, is not determined by the point. You could add
to it and get the same point:
Also that can be negative introduces further non-uniqueness:
Think about this as follows: facing in the direction and
backing up 1 meter gets you to the same point as looking in the
direction and walking forward 1 meter.
We can convert back and forth between cartesian and polar
coordinates using that
and in the other direction
is a circle sitting on top the
We plug in points for one period of the function we are
graphing--in this case :
Notice it is nice to allow
to be negative, so we don't have to
restrict the input. BUT it is really painful to draw this
graph by hand.
To more accurately draw the graph, let's try converting the equation to
one involving polar coordinates. This is easier if we multiply both
sides by :
Note that the new equation has the extra solution
we have to be careful not to include this point.
Now convert to cartesian coordinates using (4.1.1
to obtain (4.1.3
The graph of (4.1.5
is the same as that of
. To confirm
this we complete the square:
Thus the graph of (4.1.5
is a circle of radius
Actually any polar graph of the form
is a circle, as you will see in homework problem 67
by generalizing what we just did.