Polar equations are composed of the polar coordinateswhere r is the directed distance of a point P on the polar curve to the origin; and is the angle measure, in radians, of the point P from the xaxis. Therefore, the coordinates determine the location of the point P. It is possible to rewrite an equation that is in polar coordinates as rectangular coordinates using the following coversion identities:
The graph appears to be an ellipse. But, how can we know for sure? If it is an ellipse, how can we determine, algebraically, information about the center and the major and minor axes? Merely, by observation, we can estimate the following information :
R Major Axis: 5
R Minor Axis: 4
R Center: (3,0)
We could, therefore, guess that the equation for this figure, in rectangular coordinates is: .This information, however, would be more difficult to observe for various figures.
As you will notice, this equation is very similar to the previous one. The only difference between the two is the angular measure of which the cosine is being taken for each .
In rectangular coordinates, cos(x2) is the translation of the function cos(x) 2 units to the right. Similarly, in polar coordinates, is the rotation of about the orgin by 45 degrees in the positive direction.
Let's take this knowledge about the cosine function as we look at the polar equation of real interest: and how it compares with: .
Again, it seems as though the blue graph is a rotation of the red about the origin. Does it hold that the rotation angle is by 45 degrees as we saw in the graphs of cosine?










By looking at this table of values, you can see that indeed is the a or 45 degrees rotation of . Click here for a Graphing Calculator 2.2 animation demonstrating how this fact.
THIRDLY, let's investigate the function .
The following is a graph of the function:
Why is it that we get an assymptote at x=2 and y=2? It is much easier to answer these questions by looking at the equation in its rectangular coordinates.
is equivalent to in rectangular coordinates. Click here to verify this.
It is easy to see that has a vertical assymptote at x=(2) because this value of x gives a zero in the denominator of the function.
The horizontal assymptote can be found by taking the limit, L, of as x approaches infinity. Since the highest power of x in the numerator is equivalent to the highest power of x in the denominator, L = (2/1) = 2. Therefore, a horizontal assymptote occurs at y=2.
Therfore, because the vertical assymptote is determined by the "2" in the original polar equation, we see that whatever constant, c, appears in the numerator x= c is the vertical assymptote. To test this out, let's try graphing
Notice, however, that by altering the numerator, you also alter the location of the horizontal asssymptote as well. By reverting to our rectangular coordinates, we will find that the new horizontal assymptote lies as y =  5.