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.
We're plotting points in a polar coordinate system, and changing from polar coordinates to rectangular. Here is my first example. Let's plot the point A which is 4, 2 pi over 3.
Now the first thing I want to do is, face the direction 2 pi over 3. So imagine you're standing on the pole, and you want to turn into the direction 2 pi over 3, which is this direction. You want to walk forward four units 1, 2, 3, 4 and that's your point A; 4, 2 pi over 3. I'll also draw a little segment connecting it to the pole. That makes it easier to see the angle 2 pi over 3.
Let's find the coordinates of this point. Now the rectangular coordinates, I can find them using these two formulas here; x equals r cosine theta, and y equals r sine theta. If I know r and theta, I can get x and y using these. So first the x coordinate. X equals r, which is 4, cosine theta, which is 2 pi over 3. Now the cosine of 2 pi over 3 is -1/2. So it's 4 times -1/2, which is -2.
The y coordinate is r, which is 4, times the sine of theta which is 2 pi over 3. The sine of 2 pi over 3 is root 3 over 2. So this is 4 times root 3 over 2 which is 2 root 3. So I have -2, and 2 root 3. These are my rectangular coordinates. So polar coordinates, rectangular coordinates.
Let's try point B. Point B has coordinates r is 4, and theta is negative pi over 3. So going to my polar graph, I face the direction negative pi over 3 which is negative pi over 6, 2 pi over 6, negative pi over 3 is this direction. I want to go 4 in that direction, so 1, 2, 3, 4. So this is my point B; 4, negative pi over 3.
Now you may notice that this point is a reflection of this point around the origin. It's a 180 degree around the origin from point A. That means that its x and y coordinates, are going to be the opposites of this guy. If this guy has an x coordinate of -2, B is going to have an x coordinate of +2 by symmetry. This guy has a y coordinate of 2, root 3, so this guy has a y coordinate of -2, root 3 by symmetry. So our coordinates are 2, -2 root 3. Those are our rectangular coordinates.
Now let's do point C. Point C is -4, pi over 3. So r is -4, theta is pi over 3. Pi over 3, that's pi over 6, that's pi over 3. So I have to go -4 in this direction which means walk backwards -1, -2, -3, -4. This is point C; -4, pi over 3. Again, you see that there is symmetry with this point, and the others.
Suppose I use the symmetry with point B, how are the rectangular coordinates of these two points going to be related? It looks like they are on the same y value, but opposite x value. So let me use that. The x value for B was 2, so the x value for this guy since it's opposite will be -2. The y value was -2, root 3, and this guy should have the same y value. So the rectangular coordinates are -2, -2 root 3.
That's it. You can often use symmetry to get rectangular coordinates from polar coordinates, as long as your points are symmetrical. In the future, we'll show you some tricks for finding the symmetries between points that are given polar coordinates.