*Replacing cos (k theta) with sin
(k theta)*

### Marianne Parsons

What would happen to our graphs if we replace
cos (k theta) from our original equation with sin (k theta) to
form a new equation. Let's investigate different values of a,
b, and k with this new equation:

### Values of a

To investigate different values of **a**,
let b=1 and k=1. Let's see what happens to our graph when **a**
changes.

Clearly, this equation generates similar graphs
as seen by our original equation containing cos (k theta). The
difference here, however, seems to be that our graphs have rotated
90 degrees counter-wise about the origin, and are now symmetrical
about the y-axis.

### Values of b

To investigate different values of **b**,
let a=1 and k=1. Let's see what happens to our graph when **b**
changes.

This new equation generates similar graphs
for varying values of b as well. Compare these with our original
graphs for the equation containing cos (k theta). The difference
here also seems to be that our graphs have rotated 90 degrees
counter-wise about the origin, and are now symmetrical about the
y-axis.

### Values of k

To investigate different values of **k**,
let a=1 and b=1. Let's see what happens to our graph when **k**
changes. It is easier to view the effect of a changing k value
if these graphs are viewed individually.

These graphs generated by our new equation
are similar to those generated by the equation containing cos
(k theta). For example, the number of pedals in our "n-leaf
rose" is still the value of k. The difference here however,
concerns the orientations of the graphs. When k=1 for example,
this graph is rotated 90 degrees clock-wise about the origin.
Notice when k=2 that the graph is not simply rotated 90 degrees.
Similarly with the graph when k=3 and k=4. Do you notice a pattern?
View the animation of both the original and new equations below
to see the difference in the orientation of these graphs.

### Some interesting animations

View the animation as the value of **k** goes from 0 to 25 and watch
as more "pedals" are generated with this new equation
containing sin (k theta).

Watch what happens when the values of **a**, **b**, and **k** all change together from 0
to 10. See how the "n-leaf rose"
is generated this way.

How does this compare to our original equation
containing cos (k theta)? Watch the animation as both of our equations
are graphed with their **k** values going from 0 to
25.

*Please note the ***BLACK** graph
represents the graph of our original equation: ,

and the **RED** graph represents the new equation: where
a=1, and b=1.

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11