## 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.