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Converting from Rectangular Coordinates to Polar - Problem 1
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
We're converting from rectangular coordinates to polar coordinates. This time I want to try to extend this approach to equations. So I've got three rectangular equations. I want to convert them to polar form, to see what they look like in polar.
Now these are pretty simple ones like y equals 5 is a horizontal line. Remember how we convert back and forth between polar and rectangular. We use these conversion equations.
For example, for y equals 5, I could use this equation; y is the same as r sine theta. So r sine theta equals 5, is actually a pretty good equation for this. Well usually, if it's possible we like to write r as a function of theta, so we solve this for r. We write 5 over sine theta, which is the same as r equals 5 cosecant theta. This would be your equation in polar coordinates for the horizontal line y equals 5.
Part b; x² plus y² equals 25. This is a circle of radius 5 centered at the origin. There is a really easy conversion for x² plus y², that's actually r². So I just make that replacement r² equals 25.
Now we want to simplify this if we can. Normally, we'd write r equals plus or minus 5, but it turns out that the entire circle can be described just with the equation r equals 5. So that's our final equation. You don't need the r equals -5. It gives you exactly the same circle.
Then for y equal x, we use our equation for the tangent of theta. If you take a look, tangent theta equals y over x. How do we use that? Well let me pull the x over to the left side. I get y over x equals 1. So that's the tangent of theta. Tangent theta equals 1. You have to think about what angle has a tangent equal to one? Well theta equals pi over 4.
So this is interesting. The line y equals x, the line that makes 45 degree angle with the x axis, and passes through the origin, its equation in polar is theta equals pi over 4.
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