8. Curves in Polar Coordinates
r = sin (2θ) − 1.7
This is a real graph using polar coordinates.
Okay, I admit to adding the
eyes and smile. :-)
We will plot the graphs in this section using a computer, since it doesn't make a lot of sense to plot millions of points and join them. However, you are encouraged to try to sketch them on paper first because it gives you a much better idea of how they work.
Don't worry about all the difficult-looking algebra on this page - it is just there to demonstrate that polar coordinates are much simpler than rectangular coordinates for certain graphs. The conversions are achieved using what we learned in Polar Coordinates.
Examples
Sketch and convert each to an equation in rectangular coordinates.
(1) r = 2
(2) r = 3 cos 2θ
First, let's see this in LiveMath.
Now for the normal answer:
(3) r = 2 + 3 sin θ
(This polar graph is called a limacon from the Latin word for "snail". See also Equiangular Spiral.)
(4) r = sin θ - 1
(This one is called a cardioid because it is heart-shaped. It is a special case of the limacon.)
See also Polar coordinates and cardioid microphones for an interesting application of graphs using polar coordinates.
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