I am dealing with the book Differential geometry of curves and surfaces, by Manfredo P. do Carmo (see Spanish version here).
The exercise number 1.a of page 36 ask:
Given the parameterized curve (propeller) $$\alpha(s)=\bigg(a\cos \dfrac{s}{c}, a\sin\dfrac{s}{c},b\dfrac{s}{c}\bigg),\qquad s \in \mathbb{R},$$ where $c^2=a^2+b^2,$
a) Show that the parameter $s$ is the arc length.
Well, I know that the parametrisation is by arc length iff the tangent has norm $1$... right?
But:
$$|\alpha'(s)|^2=a^2\bigg(\cos^2\dfrac{s}{c}+\sin^2\dfrac{s}{c}\bigg)+b^2\dfrac{s^2}{c^2}=a^2+b^2\dfrac{s^2}{c^2}=a^2\bigg(1-\dfrac{s^2}{c^2}\bigg)+1.$$
I believe that is easy, but I am stucked.
Many thanks!