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 using the distance formula, and polar coordinates. My problem asks me to plot these two points on the polar coordinates system, and then find the distance between them. So let me start what with 3, 7 pi over6.
I first have to identify the angle 7 pi over 6. It's pi over 6 past pi, so 30 degrees past pi. This is pi, so 30 degrees past is this direction, so we're going in this direction. The first coordinate r is 3. So I need to go three units 1, 2, 3, and that's my first point; 3, 7 pi over 6. I'm going to actually draw a little line to the origin, to the pole.
My second point; 5, pi over 2. Pi over 2 is 90 degrees from the positive x axis. So 5, pi over 2 is 1, 2, 3, 4, 5 right here. Now I need to measure the distance between these two guys. I want to find this distance right here. I'll mark it in purple, that length. So I use distance formula.
Recall the distance formula in polar coordinates is d² equals r1² plus r2² minus 2r1r2 times cosine of theta 2 minus theta 1. R and theta here are the coordinates of our two points. So we can make this r1,theta 1 and r2,theta 2. So d², is r1² which is 9, plus r2² which is 25, minus 2 times r1, times r2. So 2 times 3 times 5, times the cosine of theta 2 minus theta 1, so that's pi over 2 minus 7 pi over 6. So distance squared is 9 plus 25, 34, minus 2 times 3, 6, times 5 is 30. Now to do the subtraction, I'll have to convert this to 6, so that's 3 pi over 6, so minus 7 pi over 6, or -4 pi over 6. That's -2 pi over 3.
Now the cosine of -2 pi over 3 is the same as the cosine of +2 pi over 3, because cosine is even. The cosine of 2 pi over 3 is -1/2. So this is 34 minus 30 times -1/2. So -30 times -1/2, 15. 34 plus 15 and I get 49. It's nice that it's a perfect square. So d is 7, and that's the distance between my two points. This distance is 7.