Polar equations

                          By Na Young Kwon


We will see the Polar equations in this page.

LetÕs  investigate the graph of r = a + b cos (kq).

First, we think the case of a=1, b=1 and k=1.





To see more graphs, we put k=0,1,2,3 and fix a=1 and b=1.





We can find as the value of k is increasing, the number of leaf is also increasing.

When a and b are equal and k is an integer, it is called the Ò n-leaf roseÓ.

Next, we see when a=0, that is,  r = b cos(kq).

At first, fix b=1 and investigate graphs of various k values.





Here the value of r says a distance from original point (0,0) to some point.

We check when r=1, the graph is a circle, when the value of k is even,

the number of leaf is two times of k value and when the value of k is odd,

the number of leaf is equal to the k value.


Next, we change the value of b (b=0,1,2,3).




What we know here is the value of b says a diameter of a circle.

For example, the value of b is 2, then the graph of r = 2 cos q is a circle

which is with a diameter 2.


Next, we will see if the value of k is changed then which graph of sin(mq)

is similar to the graph of cos(kq).

See the following graphs.




We can see the graph of sin(kq) is similar to the graph of cos(kq).

For example, the graph of sin(q) is rotated the graph of cos(q) by 90 degrees.

We know that cos(q)=sin(p/2 -q). Then we change the graph expression

and see the graph.





Generally, we can say the graph of  is equal to .



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