ASSIGNMENT 11
Presented by Godfried Lawson

With the graphing calculator 2.7 I will investigate the graph of:

* The investigation will focus on how different values of a, b, and k affect the shape of the graph..
* I will compare the above equation with
* Then I will observe what happen when cos(k-theta ) is replaced with sin(k-theta ).

DEFINITION OF POLAR COORDINATE SYSTEM
The polar coordinate system is a system of coordinate in which a point in the plane is identified by its distance r along a ray from a fixed point (the pole)  and by the angle theta  between a fixed line ( the polar axis ) and the ray
Every point P, other than the pole , can be represented by an ordered pair of real numbers ( r, theta).
Before we observe the graph of :

Let observe the table and the graph of:

 Theta in degree cos(theta) 1 + cos(theta) 1 + cos(2 theta) 1 + 3cos(2 theta) 0 1 2 3 4 30 .15 1.15 .05 -1.9 60 -.95 .05 1.81 3.44 90 -.45 .55 .40 -.80 120 .81 1.81 1.32 1.98 150 .70 1.70 .98 .93 180 -.60 .40 .72 .150 210 -.88 .12 1.56 2.69 240 .33 1.33 .21 -1.34 270 .98 1.98 1.94 3.81 300 -.02 .98 0.00 -2.00 330 -.99 .01 1.97 3.90 360 -.28 .72 .166 -1.51

Polar graphs can be classified in families. The graphs of cos(theta) ( in light blue ) and sin( theta) ( not shown on the on graph above ) are the the parents graphs of polar graphs.
Both graphs are circles with a diameter of 1 unit and both pass through the origin. As with many families of graphs, you can alter the position and shape of the graph by multiplying the function by a number or by adding to it. You can also multiply theta by a number or add a number to it to alter the graph.
Notice that the graph of the equation.    ( in  dark blue ) is symmetric with respect to the polar axis, as is its parent graph .
However, the appearance of a small loop inside the outside graph, which is no a circle, does not resemble the parent  graph.
This graph is called a Limacon.
Polar equations of the form make graphs that are called Rose.  K is a positive integer.
Look at he graph of the equations below to see the pattern in the amplitudes and in k-theta.

( in green )

( in blue )

( in purple )

What happen when cosine is replace by sine?
( dark blue )
(  green )
( light blue )
(red )

The difference between r = a + bcos (theta) and  = a + bsin (theta) is due to fact that
for any angle in standard position with measure theta , a point p(x, y) on its terminal side , and ,
the trigonometric functions of theta are as follow;

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