1. The problem asks to evaluate $\sin\left(\frac{5\pi}{6}\right)$ without using a calculator. 2. Recall the sine function properties and the unit circle: $\sin(\theta)$ is positive in the second quadrant, and $\sin(\pi - \theta) = \sin(\theta)$. 3. Note that $\frac{5\pi}{6} = \pi - \frac{\pi}{6}$, so $$\sin\left(\frac{5\pi}{6}\right) = \sin\left(\pi - \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right).$$ 4. We know $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$. 5. Therefore, $$\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}.$$