   Chapter 10.1, Problem 50E

Chapter
Section
Textbook Problem

# Graph several members of the family of curves x = sin t + sin nt, y = cos t + cos nt, where n is a positive integer. What features do the curves have in common? What happens as n increases?

To determine

To plot: The graph members of family using the parametric equations. x=sint+sinnt and y=cost+cosnt.

Explanation

Given data:

The parametric equation for the variable x is as follows.

x=sint+sinnt (1)

The parametric equation for the variable y is as follows.

y=cost+cosnt (2)

Calculation:

Initially, the family of curve is determined for the different values of c for the variable x.

Substitute n=1 in equation (1),

x=sint+sinnt=sint+sin(1)t

Substitute n=2 in equation (1),

x=sint+sinnt=sint+sin(2)t

Substitute n=3 in equation (1),

x=sint+sinnt=sint+sin(3)t

x=sint+sin3t

Substitute n=5 in equation (1),

x=sint+sinnt=sint+sin(5)t

The family of curve for the different values of c for the variable y.

The parametric equation for the variable y is as follows.

y=cost+cosnt

Substitute n=1 in equation (2),

y=cost+cosnt=cost+cos(1)t

Substitute n=2 in equation (2),

y=cost+cosntcost+cos(2)t

Substitute n=3 in equation (2),

y=cost+cosntcost+cos(3)t

Substitute n=5 in equation (2),

y=cost+cosntcost+cos(5)t

The value of t is increased from 2 to 2 with a step value of 1 and substituted in the parametric Equations x=sint+sint and y=cost+cost to obtain the value of x and y respectively.

Substitute 1 for t in equation (1),

x=sint+sint=sin(1)+sin(1)x=1.68

Substitute π for t in equation (2),

y=cost+cost=cos(1)+cos(1)y=1.080

For the parametric equations x=sint+sint and y=cost+cost The values of x and y for each step value of t is tabulated in the below table.

 t 0.5 1 1.5 2 x 1.54 0.58 0.01 0.34 y 1.147 1.08 0.88 0.66

Graph:

For n=1 the graph is plotted as shown below in Figure 1.

The value of t is increased from 1 to 4 with a step value of 1 and substituted in the parametric Equations x=sint+sin2t and y=cost+cos2t to obtain the value of x and y respectively.

Substitute 1 for t in equation (1),

x=sint+sin2t=sin(1)+sin2(1)x=1.75

Substitute 1 for t in equation (2),

y=cost+cos2t=cos(1)+cos2(1)y=0

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