1. **State the problem:** We need to apply Rolle's Theorem to the function $f(x) = \sin x$ on the interval $(0, 4\pi)$.\n\n2. **Recall Rolle's Theorem:** If a function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a,b)$ such that $f'(c) = 0$.\n\n3. **Check conditions:**\n- $f(x) = \sin x$ is continuous and differentiable everywhere.\n- Evaluate $f(0) = \sin 0 = 0$ and $f(4\pi) = \sin 4\pi = 0$. So, $f(0) = f(4\pi)$.\n\n4. **Apply Rolle's Theorem:** There exists at least one $c \in (0, 4\pi)$ such that $f'(c) = 0$.\n\n5. **Find $f'(x)$:** $f'(x) = \cos x$.\n\n6. **Solve $f'(c) = 0$:**\n$$\cos c = 0$$\nThe solutions for $\cos c = 0$ in $(0, 4\pi)$ are:\n$$c = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \frac{7\pi}{2}$$\n\n7. **Conclusion:** There are four points $c$ in $(0, 4\pi)$ where $f'(c) = 0$, satisfying Rolle's Theorem.