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 polar coordinates, and then converting from polar coordinates to rectangular. Here are three examples and we'll start with A; the point 6, negative pi over 4. So let's plot that first. 6, negative pi over 4.
So I first have to find the direction negative pi over 4, it's -45 degrees. So it's this direction. Remember that negative is clockwise in Mathematics, it's this direction. I have to go 6 in this direction, so 1, 2, 3, 4, 5, 6. So that's my point A; 6, negative pi over 4. I'll draw a line to the pole. So this is my angle negative pi over 4.
Let's do the x and y coordinates for this guy. First, the x coordinate. Remember the formula; x equals r cosine theta. So it's r, which is 6, cosine negative pi over 4, that's our theta. Now the cosine of negative pi over 4, because cosine is even, is the same as the cosine of pi over 4, which is root 2 over 2. So 6 times root 2 over 2. 6 and 2 cancel leaving 3. So this is just 3 root 2.
Now the y coordinate is 6 times the sine of theta, which is negative pi over 4. Now the sine of negative pi over 4, is the same as the opposite of the sine of pi over 4, because sine is odd. So that's going to be minus root 2 over 2. 6 times negative root 2 over 2, and that's minus 3 root 2. So our rectangular coordinates are 3 root 2, minus 3 root 2. That's our answer.
Let's try point B. 6 units away from the origin, an angle of 7 pi over 4. Now if I just count by pi over 4's, I'll be able to find that easily enough. Now notice that this polar coordinate system is divided up into six sectors per quadrant. So each of these is going to be pi over 12. This is pi over 6 and this is pi over 4, 45 degrees. That's 1 pi over 4, 2 pi over 4, 3 pi over 4, 4, 5, 6, 7. 7 pi over 4 is the exact same direction as negative pi over 4. I have to go six units in this direction. So guess what? I end up at exactly the same point; B is 6, 7 pi over 4. It gives me the same point, only this time, the angle that got me there was 7 pi over 4, this angle.
What about coordinates? Well it's the same point, so it should have the same rectangular coordinates. It might have different polar coordinates, but rectangular polar coordinates are unique. So it's exactly the same rectangular coordinates.
Now let's try part C. Point C is -6, 3 pi over 4. So let me find the direction 3 pi over. That's 1 pi over 4, 2 pi over 4, 3 pi over 4. I have to go -6 in this direction, which means backwards 1, 2, 3, 4, 5, 6, and here I am again. Just so you can see all of these, I'll draw the line next to it there. That's point C; -6, 3 pi over 4. Three sets of polar coordinates that give me the same point in the plane. And again I can use that.
The rectangular coordinates will be exactly the same. Rectangular coordinates are always unique, but polar coordinates are very much not. You can always find infinitely many polar coordinates to define a single point. That's what this problem is a really good illustration of. Any point in the plane can be described with only one set of rectangular coordinates, but infinitely many polar coordinates.