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 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.