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)|
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.
What happen when cosine is replace by sine?
( dark blue )
( green )
( light blue )
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;