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The Distance Formula in Polar Coordinates - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

In order to calculate the distance from two points in polar coordinates, we use the polar coordinates distance formula. In order to derive the polar coordinates distance formula, we use the law of cosines. We can also use the polar coordinates distance formula to help us come up with the polar equation for a circle centered at the origin.

I want to derive the distance formula in polar coordinates and to do this I'm going to need to recall the law of cosines. If you have a triangle, it's not necessarily a right triangle and you know three, two sides and the angle between them, you can find the third side using this formula, c squared equals a squared plus b squared minus 2ab cosine theta. And again a and b are the 2 known sides, theta's the angle between them.
Alright. Let's look at our picture here. I've graphed 2 points in polar coordinates. One is r1 theta 1, the other's r2 theta 2. I want to fi nd the distance between these points. Now in order to do this I need the length of these 2 sides and I need this angle. Now r2 theta 2 is r2 away from the origin, so that gives me this length as r2. And r1 theta 1 is r1 away from the origin. Theta 2 represents the angle that this point makes with the positive x axis and theta 1 is the angle this would make with the positive x axis. So this angle between them is going to be theta 2 minus theta 1.
So I have 2 sides and the angle between them and I'm ready to use the law of cosines to derive the distance formula. So the distance formula looks like this. d squared equals r1 squared plus r2 squared minus twice the product r1 and r2. 2 r1 r2 times the cosine of the angle between them which is theta 2 minus theta 1. That's it, that's the distance formula. Now obviously you'll take the square root over the result but it's basically in the form of the law of cosines and we'll use this to find the distance between 2 points in polar coordinates.

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