Explanation The parametric equation for the variable x is as follows. x = a sin n t (1) The parametric equation for the variable y is as follows. y = b cos t (2) The parametric equation for the variable x is as follows. x = a sin n t Substitute a = b = 1 and n = 1 in equation ( 1 ) , x = a sin n t = 1 sin ( 1 ) t = sin t Substitute a = b = 1 and n = 1 in equation ( 2 ) , y = b cos t = 1 cos t y = cos t That is, for a = b = 1 and n = 1 , the parametric equation becomes x = sin t and y = cos t . That is, the graph is a unit circle. Similarly, substitute a = b = 1 and n = 2 in equation ( 1 ) , x = a sin n t = ( 1 ) sin ( 2 ) t = sin 2 t Substitute a = b = 1 and n = 2 in equation ( 2 ) , y = b cos t = ( 1 ) cos t = cos t That is, for a = b = 1 and n = 2 , the parametric equation becomes x = sin 2 t and y = cos t . Substitute a = b = 1 and n = 3 in equation ( 1 ) , x = a sin n t = ( 1 ) sin ( 3 ) t = sin 3 t Substitute a = b = 1 and n = 3 in equation ( 2 ) , y = b cos t = ( 1 ) cos t = cos t That is, as a = b = 1 and n = 3 , the parametric equation becomes x = sin 3 t and y = cos t . Obtain the value of the parametric equations x = sin t and y = cos t for the different values of t. Substitute 1 for t in equation x = sin t , x = sin t = sin ( 1 ) = 0.01 Substitute 1 for t in equation y = cos t , y = cos t = cos ( 1 ) = 0.999 Similarly, the values of x and y for each some values of t is tabulated in the below table 1. t − 1 − 0.5 0 0.5 1 x − 0.8414 − 0.479 0 0.479 0.8414 y 0.999 0.999 1 0.66 0.999 Table 1 Obtain the value of parametric equations x = sin 2 t and y = cos t for the different values of t. Substitute 1 for t in equation x = sin 2 t , x = sin 2 t = sin 2 ( 1 ) = 0.034 Substitute 1 for t in equation y = cos t , y = cos t = cos ( 1 ) = 0.999 Similarly, the values of x and y for each some values of t is tabulated in the below table 2. t − 1 − 0.5 0 0.5 1 x − 0.034 − 0.017 0 0.017 0.034 y 0.999 0.999 1 0.99 0.999 Table 2 Obtain the value of parametric equations x = sin 3 t and y = cos t for the different values of t. Substitute 1 for t in equation x = sin 3 t , x = sin 3 t = sin 3 ( 1 ) = 0.052 Substitute 1 for t in equation y = cos t , y = cos t = cos ( 1 ) = 0.999 Similarly, the values of x and y for each some values of t is tabulated in the below table 3. t − 1 − 0.5 0 0.5 1 x − 0.052 − 0.026 0 0.026 0.052 y 0.999 0.999 1 0.99 0.999 Table 3 Graph: For the values a = b = 1 and n = 1 , n = 2 and n = 3 , the graph of x = a sin n t and y = b cos t is plotted as shown below in Figure 1. Substitute a = 1 2 , b = 2 and n = 2 in equation (1), x = a sin n t = ( 1 2 ) sin ( 2 ) t x = sin 2 t 2 Substitute a = 1 2 , b = 2 and n = 2 in equation ( 2 ) , y = b cos t = ( 2 ) cos t = 2 cos t That is, for a = 1 2 , b = 2 and n = 2 , the parametric equation becomes x = sin 2 t 2 and y = 2 cos t . Substitute a = 2 , b = 1 and n = 2 in equation ( 1 ) , x = a sin n t = ( 2 ) sin ( 2 ) t = 2 sin 2 t Substitute a = 2 , b = 1 and n = 2 in equation ( 2 ) , y = b cos t = ( 1 ) cos t = cos t That is, for a = 2 , b = 1 and n = 2 , the parametric equation becomes 2 sin 2 t and y = cos t

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Author: James Stewart

Publisher: Cengage Learning

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Chapter 10.1, Problem 51E

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To explain: How the curves x=asinnt and y=bcost varies when a,b and n varies.

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Chapter 10.1, Problem 50E

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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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How the curves x = a sin n t and y = b cos t varies when a , b and n varies.

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