1. The problem is to find the value of $\sin 10^\circ$. 2. The sine function gives the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle. 3. Since $10^\circ$ is not one of the special angles with exact sine values, we use a calculator or approximation methods. 4. Using a calculator, $\sin 10^\circ \approx 0.1736$. 5. Therefore, the value of $\sin 10^\circ$ is approximately $0.1736$.