10.6 Polar Coordinates

The Basics

So far we have used the Cartesian coordinate plane with rectangular coordinates. This section is about polar coordinates, which is simply another way to locate points.
Instead of using (x,y), the format is:  

Rectangular coordinates are plotted horizontally and then vertically. Polar coordinates on the other hand represent a radius (r) and a rotation (theta). 

A polar coordinate plane looks like this:

**This plane uses radians, but degrees may also be used to measure the amount of rotation

And this picture shows how polar coordinates are graphed:

As you can see, the radius is being rotated theta degrees.

Different Ways to Plot Polar Coordinates

The same point can have different representations by changing the sign of the radius, and the direction of rotation.

For example:

 

is equal to,,,  and

Converting Rectangular to Polar

Converting between the two is pretty intuitive, especially if you imagine the coordinate as a triangle. Simply drop down an altitude from the point. 

Using p. theorem, we know that 

And from basic trig, we know that 

From this we can create a formula for polar coordinates using only the x's and y's of rectangular ones:

We also know with basic trig that  and 

Therefore, the formula for rectangular coordinates from polar is: 

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