Given the parametrized curve $\alpha(s)=\left(a \cos\frac{s}{c}, a \sin\frac{s}{c}, b\frac{s}{c}\right), s \in \mathbb R, $ where $c^2=a^2+b^2$, show that the tangent lines to $\alpha$ make a constant angle with the $z$ axis.
The tangent lines are given by $$\alpha(s)+t\alpha'(s)=\left(a \cos\frac{s}{c}, a \sin\frac{s}{c}, b\frac{s}{c}\right)+t\left( -\frac{a}{c} \sin\frac{s}{c}, \frac{a}{c} \cos\frac{s}{c}, \frac{b}{c}\right).$$
For the $z$ axis, we can use the vector $(0,0,1)$.
I am trying to compute $$\cos^{-1}\left(\frac{(\alpha(s)+t\alpha'(s))\cdot (0,0,1)}{|\alpha(s)+t\alpha'(s)||(0,0,1)|}\right)$$ but I am not getting a constant angle.
Am I doing something wrong?