The main reason for using polar coordinates is that they can be used to simply describe regions in the plane that would be very difficult to describe using Cartesian coordinates. For example, graphing the circle in Cartesian coordinates requires two functions - one for the upper half and one for the lower half. In polar coordinates, the same circle has the very simple representation .

Name | Equation |

cardioid | or |

limaçon | or |

rose | or |

- A point in the plane can have more than one representation in polar coordinates. For example, , is the same point as , . In general a point in the plane can have an infinite number of representations in polar coordinates, just by adding multiples of to . Even if you restrict a point in the plane can have several different representations.
- The origin is determined by . The angle can have any value.

>plot(cos(2*theta),theta=0..2*Pi,coords=polar);Don't forget the option ,coords=polar! This graph is a four-leafed rose. Polar graphs can be hard to understand. Animating the graph as the angle increases will help.

>with(CalcP7): >ParamPlot([cos(2*t),t],t=0..2*Pi,coords=polar);When you run the ParamPlot command, you first get a set of axes with no curves drawn, and you think that there is something wrong. What you need to do to see the curves is first click on the graph. A box appears around the graph and a set of controls appears in the context bar just below the menu shortcut buttons at the top of the main Maple window. The set of controls works like those on a VCR. To see the animated graph, click on the play button. The other controls in the context bar allow you to slow down or speed up the animation, step through the animation one frame at a time, stop the animation, and even run the animation in reverse. I suggest you play with them until you feel comfortable. To find where two graphs intersect you set the functions equal to each other as they both equal the radius and then solve for the angle.

>plot({2-sin(theta),3*sin(3*theta)},theta=0..Pi,coords=Polar);As discussed above there can be infinite solutions so use the fsolve command and choose a range of angles values in which the intersection point occurs.

>a:=fsolve(2-sin(theta)=3*sin(3*theta),theta=0..0.5); >b:=fsolve(2-sin(theta)=3*sin(3*theta),theta=0.5..1.5); >c:=fsolve(2-sin(theta)=3*sin(3*theta),theta=1.5..2.5); >d:=fsolve(2-sin(theta)=3*sin(3*theta),theta=2.5..3.5);The following commands find the radius vaue for each angle.

>2-sin(a); >2-sin(b); >2-sin(c); >2-sin(d);

- For each of the following polar equations, plot the graph in polar coordinates using the
`plot`command and identify the graph as a cardioid, limaçon, or rose.- A)
- B)
- C)

- Find all points of intersection for each pair of curves in polar
coordinates.
- A)
- and for .
- B)
- and for .
- C)
- and for .

2011-09-13