Polar Coordinates and Complex Numbers
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
I have a couple of polar equations here and I want to ask a question; what will the graphs be without actually graphing them? What will the graph of r equals 8 sine 5, times the quantity theta minus pi over 5 look like?
Well first of all, it looks like it’s some kind of translation of r equals 8 sine of 5 theta. Now this is a rose, symmetric about the y axis because it’s a sine rose. And it’s going to be a 5 leaf rose. So what does replacing theta by a theta minus pi over 5 do to the graph?
If this were a rectangular graph, this would mean a shift to the right pi over 5. A shift in the positive direction. But theta is not a horizontal coordinate any more, it’s now an angle. And so shifting pi over 5 in the positive direction, means rotating pi over 5 in the positive direction, which is counter clockwise. So this is a 5 leaf rose rotated, pi over 5 or 36 degrees counter clockwise.
Now let’s take a look at this one. R equals 8 sine of 4 time the quantity theta plus pi over 4. This is a transformation of r equal 8 sine of 4 theta. Now recall the difference between the roses that have an even value of b, to the roses that have an odd value. The odd value of b indicates exactly that number of leaves on the rose. But with an even value you get twice as many leaves. So this is going to be an 8 leaf rose. And this graph is going to have some kind of rotation, just like the other one did. Only now, if this were a rectangular graph this will indicate a shift in the negative direction pi over 4.
The same thing is going to happen here, except it’s going to be rotation in the negative direction which is clockwise pi over 4. So this is rotated pi over 4 clockwise. Now one of the things you should know, is that if you have an 8-leaf rose, let me just draw a quick sketch 1, 2, 3, 4, 5, 6, 7, 8. Each of these leaves is going to be 45 degrees apart. You have 8 of them and one complete circle is 360 degrees. So 360 divide by 8 is 45 degrees or pi over 4. So if you rotate the graph, pi over 4 in either direction, it’s going to coincide with itself. And so, this graph will actually end up looking exactly like this graph.
So the rotation doesn’t actually have a visible effect. It is rotated, that’s with this transformation indicates, but you’ll get the same graph. And what’s interesting about that is, it's one of the things about polar graphs, a lot of different equations can give you the same final result.
So my leaf rose rotated pi over 4 clockwise.