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 rectangular equations to polar equations. Let's do a harder example. Convert this to polar form, x² minus 6x plus y² plus 4y equals 0. You may or may not recognize, that this is the graph of a circle. Let's see what it looks like in polar.
First, I'm going to move the x² plus y² together, because those two are going to add up to r². So we get r² here minus, x is our cosine theta. So this is minus 6r cosine theta, plus 4 times, and y is our sine theta, r sine theta. So let me factor out r.
I get r times the quantity r minus 6 cosine theta, plus 4 sine theta equals 0. That means that either r equal 0, or r equals 6 cosine theta minus 4 sine theta. Now I want to check to see if it's actually necessary to say r equals. because if r ever equals 0 as part of this equation, I don't need this part. It becomes redundant. So let me direct your attention over here. All we need to know is does 6 cosine theta equal 4 sine theta ever? It does.
If you look at the graph of y equals 6 cosine theta, and y equals 4 sine theta, they actually cross each other infinitely many times. So there'll be lots of places where the two functions have equal values, and thus this minus this will be 0. So I don't need this r equals 0, it's redundant.
That means my final answer is r equals 6 cosine theta minus 4 sine theta. That's the equation for this circle in polar coordinates.