= Solution
{c}
Consider real roots. Every zero at $\epsilon=0$ is $k\pi$, $k\in\mathbb Z$, and its <derivative> $\cos(k\pi)=(-1)^k$ is nonzero. For a <regular perturbation>, write $x=k\pi+\delta$. The <Taylor series>
$$
\sin(k\pi+\delta)=(-1)^k\left(\delta-\frac{\delta^3}{6}+O(\delta^5)\right)
$$
gives $\delta=(-1)^k\epsilon+O(\epsilon^3)$. Thus, through the requested order,
$$
\boxed{x_k=k\pi+(-1)^k\epsilon+O(\epsilon^3),\qquad k\in\mathbb Z}.
$$
There is no quadratic term. The exact parametrization $x_k=k\pi+(-1)^k\arcsin\epsilon$ also verifies completeness and the remainder order for $|\epsilon|<1$.
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