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().|
We make a table of values by selecting several values of
and computing values of r. To
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.
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.
|Cardioids and Limaçons|
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 x-axis, origin, and y-axis, respectively, in the Cartesian plane.)
The graph will be symmetric about the polar axis if
whenever (r,) lies on the graph so
(r,−). This means that
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().|
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.
|Example:||Find a polar equation of the circle with radius |a| centered at (a,0).|
coordinates, the equation of this circle is
(x − a)2 + y2 = a2
(rcos() − a)2 + (rsin())2 = a2
r2cos2() − 2arcos() + a2 + r2sin2() = a2
r2(cos2() + sin2()) − 2arcos() = 0
+ sin2() = 1, we have:
Now plot some values of r and and check that you indeed get points on the circle shown in the sketch.
Converting a polar equation to Cartesian is not so straightforward in general. There are some polar equations in which we can algebraically make r2 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 r2 = 49 which is x2 + y2 = 49, a circle of radius 7.|