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Calculus / Calculus: Early Transcendentals 1 / Chapter 10 / Problem 7RE

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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Solution: Tangent lines Find an equation of the line tangent

Chapter 8, Problem 7RE

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QUESTION:

Tangent lines Find an equation of the line tangent to the cycloid x = t − sin t, y = 1 − cos t at the points corresponding to \(t = \pi/6\) and \(t = 2 \pi /3\).

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QUESTION:

Tangent lines Find an equation of the line tangent to the cycloid x = t − sin t, y = 1 − cos t at the points corresponding to \(t = \pi/6\) and \(t = 2 \pi /3\).

ANSWER:

Solution 7RE
Step 1:

The slope of line tangent to any curve is $m=\frac{dy}{dy}$
Here y and x are given in parametric forms, so first we find $\frac{dy}{dt}$ and $\frac{dx}{dt}$ such that $m=\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ .
$\frac{dy}{dt}=\frac{d\left({1-cos\ t}\right)}{dt}=sin\ t$
$\frac{dx}{dt}=\frac{d\left({1-sin\ t}\right)}{dt}=-cos\ t$
Hence $m=\frac{sin\ t}{-cos\ t}=-tan\ t$
The equation of tangent line at point ( ${x}_{1},{y}_{1}$ ) with slope m is

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