Suppose $\gamma$ is a real number with $|\gamma|\ll1$. The function $$ \theta(s)=s-\frac{\sin \left( \sqrt{1+\gamma} \, k \, \pi \, s \right)}{\sin \left( \sqrt{1+\gamma}\, k \, \pi \right)}, \qquad k = 1,2,3\dots $$ is negative on $[0,1]$ when $\gamma>0$ is small, and positive on $[0,1]$ when $\gamma<0$ is small.
Doesn't look difficult, but I don't see it yet. I'm trying to Taylor the function but do not find an answer.
Edit: Your claim is not true. Suppose $\gamma>0$ is small and $k=2$. Then \begin{align*} \theta(s)&=s-\frac{\sin\left(\sqrt{1+\gamma} \, 2\pi s\right)}{\sin\left(\sqrt{1+\gamma} \, 2\pi\right)},\\ \theta\left(\frac1{4\sqrt{1+\gamma}}\right)&=\frac1{4\sqrt{1+\gamma}}-\frac{1}{\sin\left(\sqrt{1+\gamma} \, 2\pi\right)} < 0,\\ \theta\left(\frac3{4\sqrt{1+\gamma}}\right)&=\frac3{4\sqrt{1+\gamma}}+\frac{1}{\sin\left(\sqrt{1+\gamma} \, 2\pi\right)} > 0.\\ \end{align*} Hence $\theta$ has both positive values and negative values on $[0,1]$.
