   Chapter 10.3, Problem 76E

Chapter
Section
Textbook Problem

Investigate the family of polar curves r = 1 + cos n θ where n is a positive integer. How does the shape change as n increases? What happens as n becomes large? Explain the shape for large n by considering the graph of r as a function of θ in Cartesian coordinates.

To determine

To investigate: The polar curve r=1+cosnθ and how does the shape changes when n increases.

Explanation

Given:

The polar equation is r=1+cosnθ

Calculation:

The equation for the variable x is given below,

x=rcosθ (1)

The equation for the variable y is given below,

y=rsinθ (2)

The polar equations r=1+cosnθ (3)

Substitute (2) for n in equation (3),

r=1+cos2θ

Substitute (1+cos2θ) for in equation (1),

x=rcosθ=(1+cos2θ)cosθ (3)

Substitute (1+cos2θ) for r in equation (2),

y=rsinθ=(1+cos2θ)sinθ (4)

Substitute (0) for θ in equation (3),

x=(1+cos2θ)cosθ=(1+cos2(0))cos(0)=2

Substitute (0) for θ in equation (4),

y=(1+cos2θ)sinθ=(1+cos2(0))sin(0)=0

The values of x and y for each value of θ is tabulated in the below table.

 θ (degrees) θ (rad) x y 0 0 2 0 10 0.174611111 1.9398668 0.342207249 20 0.349222222 1.7692669 0.644273817 30 0.523833333 1.5151633 0.875253837 40 0.698444444 1.2150228 1.020172919

Substitute (10) for n in equation (3),

r=1+cos10θ

Substitute (1+cos10θ) for in equation (1),

x=rcosθ=(1+cos10θ)cosθ (3)

Substitute (1+cos10θ) for r in equation (2),

y=rsinθ=(1+cos10θ)sinθ (4)

Substitute (0) for θ in equation (3),

x=(1+cos10θ)cosθ=(1+cos10(0))cos(0)=2

Substitute (0) for θ in equation (4),

y=(1+cos10θ)sinθ=(1+cos10(0))sin(0)=0

The values of x and y for each value of θ is tabulated in the below table.

 θ (degrees) θ (rad) x y 0 0 2 0 10 0.174611111 1.8296842 0.322770192 20 0.349222222 1.4438033 0.525757135 30 0.523833333 1.071114 0.618742983 40 0.698444444 0.8189978 0.687657363

Substitute (1000) for n in equation (3),

r=1+cos1000θ

Substitute (1+cos1000θ) for in equation (1),

x=rcosθ=(1+cos1000θ)cosθ (5)

Substitute (1+cos1000θ) for r in equation (2),

y=rsinθ=(1+cos1000θ)sinθ (6)

Substitute (0) for θ in equation (5),

x=(1+cos1000θ)cosθ=(1+cos1000(0))cos(0)=2

Substitute (0) for θ in equation (6),

y=(1+cos1000θ)sinθ=(1+cos1000(0))sin(0)=0

The values of x and y for each value of θ is tabulated in the below table.

 θ (degrees) θ (rad) x y 0 0 2 0 10 0.174611111 0.9847944 0.173725214 20 0.349222222 0.9396391 0.342167081 30 0.523833333 0.8659081 0.500203119 40 0.698444444 0.7658434 0.643027154

In polar curve r=1+cosnθ where n is the even positive integer. Figure 1 shows that the curve is in peanut shape when n value is 2 . When n is increased the curve ends are squeezed, when n value is large the curve looks like unit circle with spikes to the points (2,0) and (2,π) .

In figure 2 for r as a function of θ in Cartesian coordinates for the same values of n .

When the n value is large the graph is similar to the graph of y=1 with spikes y=2 for x=0,π and 2π .

Substitute (1) for n in equation (3),

r=1+cos1θ

Substitute (1+cosθ) for in equation (1),

x=rcosθ=(1+cosθ)cosθ (7)

Substitute (1+cosθ) for r in equation (2),

y=rsinθ=(1+cosθ)sinθ (8)

Substitute (0) for θ in equation (7),

x=(1+cosθ)cosθ=(1+cos(0))cos(0)=2

Substitute (0) for θ in equation (8),

y=(1+cosθ)sinθ=(1+cos(0))sin(0)=0

The values of x and y for each value of θ is tabulated in the below table

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