The general idea behind graphing a function in polar coordinates is the same as graphing a function in rectangular coordinates. In a similar fashion, we can graph a curve that is generated by a function r = f ( θ ).
In the rectangular coordinate system, we can graph a function y = f ( x ) y = f ( x ) and create a curve in the Cartesian plane.
Now that we know how to plot points in the polar coordinate system, we can discuss how to plot curves. In the polar coordinate system, each point also has two values associated with it: r r and θ. Note that every point in the Cartesian plane has two values (hence the term ordered pair) associated with it. This correspondence is the basis of the polar coordinate system. This observation suggests a natural correspondence between the coordinate pair ( x, y ) ( x, y ) and the values r r and θ. The angle between the positive x x-axis and the line segment has measure θ. The line segment connecting the origin to the point P P measures the distance from the origin to P P and has length r. The point P P has Cartesian coordinates ( x, y ). To find the coordinates of a point in the polar coordinate system, consider Figure 7.27. In this section we see that in some circumstances, polar coordinates can be more useful than rectangular coordinates. The polar coordinate system provides an alternative method of mapping points to ordered pairs.
This is called a one-to-one mapping from points in the plane to ordered pairs. The rectangular coordinate system (or Cartesian plane) provides a means of mapping points to ordered pairs and ordered pairs to points.
The polar coordinate system looks like a series of concentric circles representing r centered at the origin. If we graph on the polar axis, we graph in terms of r and theta (the angle measure). If you plot y=sinx or y=cosx on the Cartesian (x,y) plane, you get the familiar waves that are anchored on the x-axis. When we talk about sine, cosine, and tangent functions and their reciprocals, we are talking about what are called periodic functions, meaning that they repeat values at regular intervals. So for anyone with a little precalc background, let's talk polar roses and their cousins. So I've been teaching her some topics that she hasn't yet gotten to in class (and I'm nervous that with all that's going on, she WON'T get to at all), and one of these topics has been things involving trig and polar equations of all sorts. One of my students is prepping for the SAT, and that prep includes prep for the Math SAT II. However, with all the doom and gloom we have in the news, and because while we've been self isolating in our houses, spring has arrived, I figured I'd change gears today and head back to math activities. My original plan was to publish another long form rant, this time about the murder of Thomas Becket in 1170 (I have a soft spot for everything related to Henry II of England).