Mathematics

Problem

Consider the curve defined by the parametric equations

x = t + \sin t, \qquad y = 1 - \cos t, \qquad t > 0.

(i) Obtain

\dfrac{dy}{dx}

in terms of

t

. (ii) Find all positive values of

t

for which the slope of the tangent equals

\dfrac13

. (iii) For the least positive value of

t

found in part (ii), determine the equation of the tangent line to the curve.