Question

A cycloid is the path traced out by a point on a circle rolling on a flat surface (think the light on the rim of a moving bicycle wheel). The cycloid generated by a circle of radius a is given by the parametric equations x(t)=a(t-\sin (t)) \text { and } y(t)=a(1-\cos (t)) The parameter range 0 <t < 2n produces one arch of a cycloid. Determine the length of one arch of a cycloid, using the following half-angle formula: \sin (\theta / 2)=\sqrt{\frac{1-\cos (\theta)}{2}}

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