This assignment is an investigation of polar equations. The polar equation that will be investigated is r = b cos (kθ).

The polar coordinates for a point P is
written as the ordered pair (r,
θ), where r
is the distance from the origin, and θ is the
angle in radians measured counter clock-wise from the horizontal axis.
See Figure 1 below.

Figure 1

Figure 1

Part I

Lets take a look at the equation r = b cos (kθ). In this equation r is a function of theta.

Let k = 2, and let θ range from 0 to 2π. See Figure 2 below.

This graph
was created by Graphing
Calculator 3.2.

Figure 2

Figure 2

The
curve above is called a four-leaved
rose. In this example the number of leaves is twice the value of
k. Note that as b increases the size of the "roses"
increases. This can also be seen in the table below.

If you look at the row for θ = π, as b increases across the row, the value for r increases. You will also notice some negative values for r. See Figure 3 below.

Figure 3

In the figure above P' represents a point with angle θ and a negative r.

θ | r | b = 1 | b = 2 | b = 3 | b = 4 |

0 | 1.0000 | 1.0000 | 2.0000 | 3.0000 | 4.0000 |

π/8 | 0.7071 | 0.7071 | 1.4142 | 2.1213 | 2.8284 |

π/4 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

3π/8 | -0.7071 | -0.7071 | -1.4142 | -2.1213 | -2.8284 |

π/2 | -1.0000 | -1.0000 | -2.0000 | -3.0000 | -4.0000 |

5π/8 | -0.7071 | -0.7071 | -1.4142 | -2.1213 | -2.8284 |

3π/4 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

7π/8 | 0.7071 | 0.7071 | 1.4142 | 2.1213 | 2.8284 |

π | 1.0000 | 1.0000 | 2.0000 | 3.0000 | 4.0000 |

9π/8 | 0.7071 | 0.7071 | 1.4142 | 2.1213 | 2.8284 |

5π/4 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

11π/8 | -0.7071 | -0.7071 | -1.4142 | -2.1213 | -2.8284 |

3π/2 | -1.0000 | -1.0000 | -2.0000 | -3.0000 | -4.0000 |

13π/8 | -0.7071 | -0.7071 | -1.4142 | -2.1213 | -2.8284 |

7π/4 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

15π/8 | 0.7071 | 0.7071 | 1.4142 | 2.1213 | 2.8284 |

2π | 1.0000 | 1.0000 | 2.0000 | 3.0000 | 4.0000 |

If you look at the row for θ = π, as b increases across the row, the value for r increases. You will also notice some negative values for r. See Figure 3 below.

Figure 3

In the figure above P' represents a point with angle θ and a negative r.

Part II

Now lets take a look at the equation r = b cos (kθ) again, but let k = 3, and let θ range from 0 to 2π. See Figure 4 below.

θ | r | b = 1 | b = 2 | b = 3 | b = 4 |

0 | 1.0000 | 1.0000 | 2.0000 | 3.0000 | 4.0000 |

π/12 | 0.7071 | 0.7071 | 1.4142 | 2.1213 | 2.8284 |

π/6 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

π/4 | -0.7071 | -0.7071 | -1.4142 | -2.1213 | -2.8284 |

π/3 | -1.0000 | -1.0000 | -2.0000 | -3.0000 | -4.0000 |

5π/12 | -0.7071 | -0.7071 | -1.4142 | -2.1213 | -2.8284 |

π/2 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

7π/12 | 0.7071 | 0.7071 | 1.4142 | 2.1213 | 2.8284 |

2π/3 | 1.0000 | 1.0000 | 2.0000 | 3.0000 | 4.0000 |

3π/4 | 0.7071 | 0.7071 | 1.4142 | 2.1213 | 2.8284 |

5π/6 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

11π/12 | -0.7071 | -0.7071 | -1.4142 | -2.1213 | -2.8284 |

π | -1.0000 | -1.0000 | -2.0000 | -3.0000 | -4.0000 |

13π/12 | -0.7071 | -0.7071 | -1.4142 | -2.1213 | -2.8284 |

7π/6 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

5π/4 | 0.7071 | 0.7071 | 1.4142 | 2.1213 | 2.8284 |

4π/3 | 1.0000 | 1.0000 | 2.0000 | 3.0000 | 4.0000 |

17π/12 | 0.7071 | 0.7071 | 1.4142 | 2.1213 | 2.8284 |

3π/2 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

19π/12 | -0.7071 | -0.7071 | -1.4142 | -2.1213 | -2.8284 |

5π/3 | -1.0000 | -1.0000 | -2.0000 | -3.0000 | -4.0000 |

7π/4 | -0.7071 | -0.7071 | -1.4142 | -2.1213 | -2.8284 |

11π/6 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

23π/12 | 0.7071 | 0.7071 | 1.4142 | 2.1213 | 2.8284 |

2π | 1.0000 | 1.0000 | 2.0000 | 3.0000 | 4.0000 |

Table 2

Let's take a look at the points for θ = 5π/12 [P = (-0.7071, 5π/12)] and θ = 17π/12 [P' = (0.7071, 17π/12)]. The figure below shows the two points on a graph.

Figure 5

As it turns out, the two polar coordinates above represent the same
point. Polar coordinates differ from rectangular coordinates in that
any point has more than one representation in polar coordinates. For
example, the polar coordinates (r,
θ) and (-r, θ + π) represent the same point P, as shown in Figure 5. More
generally, this same point P
has the polar coordinates (r, θ + nπ) for any even integer n, as well as (-r, θ + nπ) for any odd integer n. It becomes obvious that there
are an infinite number of polar coordinates that can represent point P. As it turns out the three-leaved rose in Figure 4 has created when θ reached π. For θ from π to 2π a second "rose" was created on top of the first one.

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