Explanation Result used: The slope of a tangent line to a curve x = f ( t ) , y = g ( t ) is given by d y d x = d y d t d x d t evaluated at a point. Calculation: The given equation of the curve is x = t − sin t , y = 1 − cos t . Using the above result the slope of the tangent to the curve x = t − sin t , y = 1 − cos t at t = π 6 is d y d x | t = π 6 and at t = 2 π 3 is d y d x | t = 2 π 3 . First find d y d x as follows. d y d x = d d t ( 1 − cos t ) d d t ( t − sin t ) = sin t 1 − cos t Therefore, slope of the tangent at point t = π 6 is calculated as follows. d y d x | t = π 6 = sin ( π 6 ) 1 − cos ( π 6 ) = 1 2 1 − 3 2 = 1 2 − 3 Hence, the slope of the tangent to the curve x = t − sin t , y = 1 − cos t at t = π 6 is d y d x | t = π 6 = 1 2 − 3 . Similarly, the slope of the tangent at point t = 2 π 3 is calculated as follows. d y d x | t = 2 π 3 = sin ( 2 π 3 ) 1 − cos ( 2 π 3 ) = 3 2 1 − ( − 1 2 ) = 3 3 = 1 3 Hence, the slope of the tangent to the curve x = t − sin t , y = 1 − cos t at t = 2 π 3 is d y d x | t = π 6 = 1 3 . The point ( x , y ) at t = π 6 is ( π 6 − sin ( π 6 ) , 1 − cos ( π 6 ) ) = ( π − 3 6 , 2 − 3 2 )

MyLab Math with Pearson eText -- 24 Month Access -- for Calculus with Integrated Review

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ISBN: 9780135243435

Author: Bill Briggs

Publisher: Pearson Education (US)

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Chapter 12, Problem 13RE

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To find: The equation of the tangent line to the curve x=t−sint,y=1−cost at the points t=π6 and t=2π3.

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Chapter 12, Problem 12RE

Chapter 12, Problem 14RE

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MyLab Math with Pearson eText -- 24 Month Access -- for Calculus with Integrated Review

Ch. 12.1 - Identify the graph generated by the parametric...Ch. 12.1 - Prob. 2QC Ch. 12.1 - Prob. 3QC Ch. 12.1 - Prob. 4QC Ch. 12.1 - Prob. 5QC Ch. 12.1 - Prob. 6QC Ch. 12.1 - Explain how a pair of parametric equations...Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E

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The equation of the tangent line to the curve x = t − sin t , y = 1 − cos t at the points t = π 6 and t = 2 π 3 .

Solution Summary: The author calculates the slope of a tangent line to the curve x=t-mathrmsint,y=1-

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To find: The equation of the tangent line to the curve x=t−sint,y=1−cost at the points t=π6 and t=2π3.

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Chapter 12 Solutions