1. **Problem:** Find the value of $\sin 210^\circ$ without using a calculator. 2. **Formula and rules:** The sine function for angles greater than $180^\circ$ can be found using the reference angle and the unit circle. The reference angle for $210^\circ$ is $210^\circ - 180^\circ = 30^\circ$. Since $210^\circ$ is in the third quadrant, where sine is negative, we have: $$\sin 210^\circ = -\sin 30^\circ$$ 3. **Intermediate work:** We know from standard trigonometric values that: $$\sin 30^\circ = \frac{1}{2}$$ Therefore: $$\sin 210^\circ = -\frac{1}{2}$$ 4. **Explanation:** The sine of an angle in the third quadrant is negative because the y-coordinate on the unit circle is negative there. The reference angle helps us find the sine value by relating it to a known angle in the first quadrant. **Final answer:** $$\sin 210^\circ = -\frac{1}{2}$$