I was trying to derive the time it takes a particle to reach the bottom of a cycloid. I know the result is π√(a/g) where g is the acceleration due to gravity and a is the constant in the following parametric equations that give the cycloidal path x(θ) = a(θ-sin[θ]) y(θ)= a(1-cos[θ]) but I am having trouble with the derivation. Here's what I have: The time is found by t = ∫ds/v where ds is the arc length and v is the speed. The speed is easily found using energy conservation v = √(2g(y(0)-y(t)) In terms of the parameter θ, we have v = √(2ga(cos[ψ]-cos[θ]) where ψ gives the initial position of the particle. The arc length ds = √(dx^2+dy^2), but since the curve is parametrized in terms of θ we can write ds = (ds/dθ)dθ = √((dx/dθ)^2+(dy/dθ)^2)dθ We have dx/dθ = a(1-cos[θ]) dy/dθ=a sin[θ] After some simplification (squaring dx/dθ and dy/dθ, adding the squares and simplifying) ds = a √(1-2 cos[θ]) Finally we get t = ∫a√(1-2 cos[θ])/√(2ga(cos[ψ]-cos[θ]) with limits from ψ to π. Not sure how to approach this integral. I did find an online homework assignment that suggests to write ds = 2 R sin[θ/2] where I assume R is what I was calling 'a', but I don't see how to apply any identity(ies) to get this result.