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Converting from Polar Coordinates to Rectangular - 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.

It can often be useful to convert polar coordinates to rectangular coordinates. Sometimes operations are easier in one coordinate system than the other. The conversion from polar coordinates to rectangular coordinates involves using the sine and cosine functions to find x and y. It is also important to understand how to convert from rectangular to polar coordinates.

We're converting polar equations to rectangular. Let's start with some simple examples. r=10. Now let's remember that we have these equations to work with. We have x squared plus y squared equals r squared, tangent theta equals y over x. x=r cosine theta and y equals y sine theta.
So my first thought is square both sides and I'll get r squared. So I get r squared equals 100 and I replace the r squared with x squared plus y squared. And then I can see that r=10 is a circle centered at the origin radius 10. It's much simpler in polar coordinates than in rectangular but there it is that's the rectangular version of this equation. And then for b, theta equals minus pi over 4. Here I remember that I have this equation tangent theta equals y over x. So let me take the tangent of both sides and I get tangent theta equals tangent of negative pi over 4. And the tangent of negative pi over 4 is -1. And so remember that tangent theta is y over x so y over x equals -1 and you multiply both sides by x you get y equals -x. So the equation theta equals negative pi over 4 is the same as the equation y equals -x in rectangular.
What about this one? r=5 secant theta. Well, I know that r cosine theta is x and I can get our cosine theta together if I multiply both sides by cosine. Remember the secant is 1 over cosine theta. So I multiply those sides by cosine and I get r cosine theta equals 5. This is x, x=5. and so what I have here is a vertical line x=5. That's what this equation represents.

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