Exploring Polar Equations

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Objective: To explore differnt forms of polar equations.

Polar equations are represented with 'r' and theta.

We begin by investigating the equation:

1st Case: Let a > b, k varying from -5 to 5

Observations

1. The length of the petal is always a +b

2. The maximum amount of petals is the maximum value of k

3. The petals do not start from the center

4. The vertex of one petal goes through the horizontal axis

Click here to investigate k as integers and here for k as real numbers

2nd Case: Let a < b, k varying from -5 to 5

Observations

1. The length of the petal is always a +b

2. The maximum amount of petals is the maximum value of k

3. There are two sets of petals

Click here to investigate k as integers and here for k as real numbers

3rd Case : Let a = b, k varying from -5 to 5

Observations

1. The length of the petal is always a +b

2. The maximum amount of petals is the maximum value of k

3. There is only one set of petals starting from the center

Click here to investigate k as integers and here for k as real numbers

Next, let's investigate the equation

Observations

1. The length of the petal is always b

2. The petals double when k is an even integer but doesn't when k is odd

3. There are two set of petals overlaying each other.

Click here to investigate k as integers and here for k as real numbers

Next, let's investigate the equation

1st Case: Let a > b, k varying from -5 to 5

Observations

1. The length of the petal is always a +b

2. The maximum amount of petals is the maximum value of k

3. The petals do not start from the center

4. Unlike the cosine case, the vertex of one petal goes through the vertical axis

Click here to investigate k as integers and here for k as real numbers

2nd Case: Let a < b, k varying from -5 to 5

Observations

1. The length of the petal is always a +b

2. The maximum amount of petals is the maximum value of k

3. There are two sets of petals

Click here to investigate k as integers and here for k as real numbers

3rd Case : Let a = b, k varying from -5 to 5

Observations

1. The length of the petal is always a +b

2. The maximum amount of petals is the maximum value of k

3. There is only one set of petals starting from the center

Click here to investigate k as integers and here for k as real numbers

Next, let's investigate the equation

Observations

1. The length of the petal is always b

2. The petals double when k is an even integer but doesn't when k is odd

3. There are two set of petals overlaying each other.

Click here to investigate k as integers and here for k as real numbers

In conclusion, the difference between the sine and cosine polar functions is the axis in which the vertex of one of the petals pass through. The number of petals is a funcition of k. If k isan integer, then it is k petals. If I is a rational number k = a/b then it is b sets of a petals. The figure here is 2 sets of 9 petals. The number of revolutions for theta needs to be at least b.