finding integral of $ \sqrt {50-50\cos(t)} $

Question

I have a cycloide

$ {x(t)=5 (t-\sin(t)), y(t)=5(1-\cos(t))} $ and

$ {x(t)=5 (t-\sin(t)), y(t)=-5(1-\cos(t))} $

and the aim is to find the circumference and to find it I have to show and take the differentiate integral:

$$ \int \sqrt{50-50\cos(t)}\ dt $$

so I need help with it

Is the right anwser $ \frac{-20}{\sqrt{tan^2( \frac{t}{2})+1}} $ ??

Answer 1

Hint:

$$\sqrt{50-50\cos(t)}=\sqrt{50}\cdot \sqrt{1-\cos(t)}$$

and

$$\sqrt{1-\cos(t)}=\sqrt{2}\cdot|\sin(t/2)| $$