Polar Coordinates and Graphs

To graph equations we have been using the Cartesian rectangular coordinate system
Now we introduce the use of the polar coordinate system

     See Example 1, page 614, of the textbook


          Polar Form – Rectangular Form Conversion

Sometimes it it easier to draw the graph of an equation
when the equation is written using the coordinates of
another coordinate system

               
The conversion equations between polar coordinates and rectangular coordinates are

                                             
                               or               
                             or                
                            or                 

The signs of    and    determine in which quadrant is the terminal side of the angle
By convention, The angle
                                                             
is chosen so that
                                                        (radians)
     or
                                       (degrees).


     See Examples 2 – 4, pages 615 – 617, of the textbook



               Graphing Polar Equations

The graph of an equation written in polar coordinates
is the set of all the points in the polar coordinate plane
that satisfy the polar equation

For example, following is the graph of the polar equation
                with           in radian measure


     
Transforming to Cartesian rectangular coordinates, the equation of the graph is

          
which is a circle with center at (4,0) and radius 4



               Point-by-Point Plotting
Make a table of values that satisfy the equation
Plot these points
Join the plotted with a smooth curve

     See Example 5, page s 617 – 618, of the textbook


               Rapid Polar Sketching
If the polar equation involves only sines or cosines and only a rough sketch is required,
we can use how these functions numerically vary to sketch the graph relatively quickly
The sine and the cosine functions both continuously vary from -1 to +1

     


     See Examples 6 – 7, pages 619 – 620, of the textbook

     Using a Graphing Utility – see Example 8, page 621, of the textbook




                                        Standard Polar Graphs

                         








                         







                         





                    




                                                            


In rectangular coordinates the graph of      is a vertical line
                                               the graph of     is a horizontal line
These are simple equations
In polar coordinates the equations of straight lines are not always so simple
Look at graphs (a) – (c) above

The graph of a circle can have a simple equation in polar coordinates
Look at graph (d) above


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