Yeah, it's a silly thing to be excited about, but I get it! I get it! I actually understand what the hell polar form is! Yay!

So the context here is that I'm putting together a trig class. And I'm looking over the required elements from various states (California and West Virginia require the exact same things out of a trig class, which made it easy) and one of these is to "express complex numbers in polar form."

And I read this, and I think to myself, "What the hell is polar form?" It's always disheartening to look at the outline of a course you want to teach and realize you don't get all the terminology...

In basic terms, the concept is easy! You have a point in the coordinate plane: x, y. And you want to switch this so that instead you know 1) the distance to the point, which we call r, and 2) the angle a line would need to be at to go from the center of the coordinate plane to the point. (Put another way, if this is a point on a circle, what's the radius of the circle and where is this point on the circle? Just plug the numbers in to the equation for a circle, get r, and then use simple trig functions [SOHCAHTOA anyone?] to get the angle!)

Say a point is (1, 1). You can use the distance formula to find out the distance from 0 to this point is SQRT(2).

And we know from geometry that a triangle with sides 1 and 1 and a hypoteneuse of SQRT(2) is a 45-45-90 triangle. So I'll cheat by knowing that to realize the angle here is going to be 45 degrees. (Actually I also used this to know the distance would be SQRT(2), but I digress.)

So in polar form, the point is (SQRT(2), 45 degrees).

Yeah I know, that explanation probably didn't make it sound simple, but it is! And I get it! w00t!

Heck, I made the students create a triangle from a point in the coordinate plane in the geometry class, just cause I thought it was neat that it could be done. I was priming them for this without even realizing it...

Now I go bouncing off...