 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/14/a/solution

Differentiate the cycloid parametrization:

\frac{d x}{d t} = a (1 - cos t), \frac{d y}{d t} = a sin t, \frac{d y}{d x} = \frac{sin t}{1 - cos t} = cot \frac{t}{2} .

The half-angle simplification is valid throughout

0 < t < 2 π

, where

1 - cos t > 0

. In particular the horizontal coordinate increases strictly along the arch.

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