# BY

# MUHAMMET ARICAN

## Assignment 11: Polar Equations

We call r = a + b cos(kθ) polar function. In this assignment I am going to show some examples for polar functions given values a, b, k, and θ.

If r = 1 + cos(θ), here a = 1, b = 1, k = 1, and 0 ≤θ ≤Π/ 2, then our graph looks like:

If a = 1, b= 1, and k = 0.5 then graph looks like below: Here argument θ takes the same values as above graph

If 0 ≤θ ≤Π then our graph looks like:

As you see above graph if θ is between 0 and Π then the graph getting thicker.

If 0 ≤θ ≤ 2Π then our graph looks like :

Here as you see our function getting more thicker then previous functions.

Now if r = 2 + cos(θ), here a = 2, b=1, k = 0, and0 ≤θ ≤ Π then our graph looks like:

As you see from above graph this is a circle centered 0 with radius 3.

If r=1+ cos(2θ) then our function will have two leaves and graph looks like:

If r = 1+ cos(6θ) then our graph will have 6 leaves and graph looks like:

Instead of giving integer values for k lets take it k=2.5 then our function will have two and a half leaves. Here is a graph for this:

What happens if we give values for k from -4 to 4 in the polar function r = a + b cos(kθ)? Here is a GSP file for this:

GSP File

If we don't include value a in to our function then it looks like: r= bcos(kθ)

Here are two animations for this function:

1. Animation: " r= 2cos(kθ) here k takes values from -4 to 4, b=2, and 0 ≤θ ≤ Π "

2. Animation:" r= 4cos(kθ) here k takes values from -4 to 4, b=4, and 0 ≤θ ≤ Π "

As we understand from animations when we change the value of b then the radius of our polar function changes. For eample: for animation 1 the radius of our function is 2 and for animation 2 the radius of our function is 4.

Now what happens if we change cosine function to sine function:

r=a+bsin(kθ)

If r=1+sin(θ) then our graph looks like: here a=b=k=1 and 0 ≤θ ≤ Π

If r=1+sin(1/2 θ), here we give the value 0.5 for k, then graph looks like:

If r=1+sin(2θ), here k= 2 and a=b=1, then our graph looks like:

As you can see from above graphs when we change cosine to sine then our graphs shift 90 degrees to the right.