13-3: Intersections of Polar Curves

Hi Guys! Sorry it took so long for this post to get online, but I wasn't 100% sure if it was my turn or not. I'm just going to assume it is. As usual, a bland, colorless blog...

The main focus of this chapter is on determining whether or not an intersection of two graphs in polar coordinates is a true intersection.

Here we see a graph with the polar equations

3 + 3 cos (theta)
and
5 sin (2 theta)



There appear to be eight intersections, as shown by the black dots (two are kind of smushed together), but not all are TRUE intersections. In order for an intersection to be true, the two points must meet at the same theta value. In order to find out which values are true, go to the "Mode" function on your calculator, and change the standard "Sequential" setting to "Simul" (which is Latin for "at the same time"- just a little fun fact.) If you visit your graph once again, the two equations will be graphed simultaneously (hence the "simul" setting). Watch closely for the intersections- the true intersections of the graphs will meet at the same point at the same time. After graphing our equations using "simul", we find that only these points are true intersections.



In order to find the coordinates of these points, we must leave polar mode and switch back to "function" mode. Plug in the exact same points in function mode, but using "x" as opposed to "theta". Find the intersections between the two graphs using the "calc" function; these points represent to coordinates on the polar graph.

Here is the graph in function (and radian) mode...



Here's a Problem!

Graph the following equations, and indicate the true intersections. calculate the values.

r= 2 + 8cos(theta)
r= 12 sin (3theta)

Solution

Start by graphing the two equations in polar, simultaneous mode. Make sure your window is big enough to see the intersections. Your graphs should look something like this...



By using the "SIMUL" function, you should end up with about these points as "true" intersections.



Plugging the values into the "function" mode gives you a graph like this-



The values for these points are (17.760, 9.619), (47.267, 7.431), (117.329, -1.673)



Additional Help




Debby's Up Next!




Personalization:
There's a Sketch Comedy group called The Whitest Kids You Know who recently got a show on fuse- these guys are extremely talented and have a lot of creative, original ideas. Here are some of their videos.














The Girl in this video is my cousin Sarah- this was the first sketch of the first episode of the show.



This Video is by a comedy group called "Derrick"