   Chapter 10.1, Problem 52E

Chapter
Section
Textbook Problem

# Investigate the family of curves defined by the parametric equations x = cos t, y = sin t − sin ct, where c > 0. Start by letting c be a positive integer and see what happens to the shape as c increases. Then explore some of the possibilities that occur when c is a fraction.

To determine

To plot: The graph for members of family of curves using the parametric equations x=cost and y=sintsinct.

Explanation

Given data:

The parametric equation for the variable x is as below.

x=cost (1)

The parametric equation for the variable y is as below.

y=sintsinct (2)

Calculation:

The family of curve for the different values of c for the variable x is as below.

The parametric equation for the variable x is as below.

x=cost

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

The parametric equation for the variable y is as below.

y=sintsinct

Substitute the value of 2 for c in equation (2).

y=sintsinct=sintsin(2)ty=sintsin2t

Substitute 3 for c in equation (2).

y=sintsinct=sintsin(3)ty=sintsin3t

Substitute 4 for c in equation (2).

y=sintsinct=sintsin(4)ty=sintsin4t

Substitute 5 for c in equation (2).

y=sintsinct=sintsin(5)ty=sintsin5t

Substitute 12 for c in equation (2).

y=sintsinct=sintsin(12)ty=sintsin0.5t

Substitute 34 for c in equation (2).

y=sintsinct=sintsin(34)ty=sintsin0.75t

Substitute 25 for c in equation (2).

y=sintsinct=sintsin(75)ty=sintsin1.4t

Substitute 2.3 for c in equation (2).

y=sintsinct=sintsin(2.3)ty=sintsin2.3t

Substitute 4.7 for c in equation (2).

y=sintsinct=sintsin(4.7)ty=sintsin4.7t

Substitute 2.359 for c in equation (2).

y=sintsinct=sintsin(2.359)ty=sintsin2.359t

Equations x=cost and y=sintsin2t are used to obtain the values of x and y respectively.

Substitute 0 for t in equation x=cost.

x=cost=cos(0)x=1

Substitute 0 for t in equation y=sintsin2t.

y=sintsin2t=sin(0)sin2(0)=0

Repeat the calculation of x and y values using parametric equations x=cost and y=sintsin2t for values of t ranging from 0 degree till 360 degrees.

Form a set of 360 coordinate points for the curve and plot the curve as below.

Graph:

The graph is plotted for values of c=2.

Equations x=cost and y=sintsin3t are used to obtain the values of x and y respectively.

Substitute 0 for t in equation x=cost.

x=cost=cos(0)x=1

Substitute 0 for t in equation y=sintsin3t.

y=sintsin3t=sin(0)sin3(0)=0

Repeat the calculation of x and y values using parametric equations x=cost and y=sintsin3t for values of t ranging from 0 degree till 360 degrees.

Form a set of 360 coordinate points for the curve and plot the curve as below.

Graph:

The graph is plotted for values of c=3.

Equations x=cost and y=sintsin4t are used to obtain the value of x and y respectively

Substitute 0 for t in equation x=cost.

x=cost=cos(0)x=1

Substitute 0 for t in equation y=sintsin4t.

y=sintsin4t=sin(0)sin4(0)y=0

Repeat the calculation of x and y values using parametric equations x=cost and y=sintsin4t for values of t ranging from 0 degree till 360 degrees.

Form a set of 360 coordinate points for the curve and plot the curve as below.

Graph:

The graph is plotted for values of c=4.

Equations x=cost and y=sintsin5t are used to obtain the value of x and y respectively

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