Explanation Given: The parametric equation for the variable x is as below. x = cos t (1) The parametric equation for the variable y is as below. y = sin t − sin c t (2) Calculation: The parametric equation for the variable x is, x = cos t . 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 = sin t − sin c t When c = 1, the y value becomes 0. Substitute the value of 2 for c in equation (2), y = sin t − sin c t = sin t − sin ( 2 ) t y = sin t − sin 2 t Substitute 3 for c in equation (2), y = sin t − sin c t = sin t − sin ( 3 ) t y = sin t − sin 3 t Substitute 4 for c in equation (2), y = sin t − sin c t = sin t − sin ( 4 ) t y = sin t − sin 4 t Substitute 5 for c in equation (2), y = sin t − sin c t = sin t − sin ( 5 ) t y = sin t − sin 5 t Substitute 1 2 for c in equation (2), y = sin t − sin c t = sin t − sin ( 1 2 ) t y = sin t − sin 0.5 t Substitute 3 4 for c in equation (2), y = sin t − sin c t = sin t − sin ( 3 4 ) t y = sin t − sin 0.75 t Substitute 2 5 for c in equation (2), y = sin t − sin c t = sin t − sin ( 7 5 ) t y = sin t − sin 1.4 t Substitute 2.3 for c in equation (2), y = sin t − sin c t = sin t − sin ( 2.3 ) t y = sin t − sin 2.3 t Substitute 4.7 for c in equation (2), y = sin t − sin c t = sin t − sin ( 4.7 ) t y = sin t − sin 4.7 t Substitute 2.359 for c in equation (2). y = sin t − sin c t = sin t − sin ( 2.359 ) t y = sin t − sin 2.359 t Equations x = cos t and y = sin t − sin 2 t are used to obtain the values of x and y respectively. Substitute 0 for t in equation x = cos t , x = cos t = cos ( 0 ) x = 1 Substitute 0 for t in equation y = sin t − sin 2 t , y = sin t − sin 2 t = sin ( 0 ) − sin 2 ( 0 ) = 0 Repeat the calculation of x and y values using parametric equations x = cos t and y = sin t − sin 2 t for values of t ranging from 0 degree till 360 degrees. Graph: The graph is plotted for values of c = 2 . Equations x = cos t and y = sin t − sin 3 t are used to obtain the values of x and y respectively. Substitute 0 for t in equation x = cos t . x = cos t = cos ( 0 ) x = 1 Substitute 0 for t in equation y = sin t − sin 3 t . y = sin t − sin 3 t = sin ( 0 ) − sin 3 ( 0 ) = 0 Repeat the calculation of x and y values using parametric equations x = cos t and y = sin t − sin 3 t for values of t ranges from 0 degree till 360 degrees. Graph: The graph is plotted for values of c = 3 . Equations x = cos t and y = sin t − sin 4 t are used to obtain the value of x and y respectively Substitute 0 for t in equation x = cos t . x = cos t = cos ( 0 ) x = 1 Substitute 0 for t in equation y = sin t − sin 4 t . y = sin t − sin 4 t = sin ( 0 ) − sin 4 ( 0 ) y = 0 Repeat the calculation of x and y values using parametric equations x = cos t and y = sin t − sin 4 t for values of t ranging from 0 degree till 360 degrees. Graph: The graph is plotted for values of c = 4 . Equations x = cos t and y = sin t − sin 5 t are used to obtain the value of x and y respectively

8th Edition

ISBN: 9781285741550

Author: James Stewart

Publisher: Cengage Learning

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Chapter 10.1, Problem 52E

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

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Chapter 10.1, Problem 51E

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Calculus: Early Transcendentals

Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E

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To plot: The graph for members of family of curves using the parametric equations x = cos t and y = sin t − sin c t .

Solution Summary: The author explains the parametric equations for the variables x and y.

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

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