Finding arc length parametrization of a cycloid

Question

Find an arc length parametrization of the cycloid with parametrization r(t)= . I took the derivative and found the speed to be sqrt(2(1-cost))but now I'm unsure how to integrate that to get s. How do you integrate sqrt(2(1-cost))?

Answer 1

Hint: If ${\bf r}: I \to \Bbb R^n$ is a curve, it's arclength function is given by: $$s(t) = \int_{t_0}^t \|{\bf r}'(\xi)\|\,{\rm d}\xi.$$ If $0 \in I$, we usually take $t_0 = 0$, for simplicity. Solve the integral, and you will have $s$ as a function of $t$. Solve for $t$ in terms of $s$.

Answer 2

Try this one

$(1-\cos t) = 2 \sin ^2(\frac{t}{2})$