1. **State the problem:** Find the exact values of $\cos 210^\circ$ and $\sin 315^\circ$ without using a calculator. 2. **Recall the unit circle and reference angles:** - The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. - For angles in different quadrants, the signs of sine and cosine change. 3. **Find $\cos 210^\circ$:** - $210^\circ$ is in the third quadrant (between 180° and 270°). - Reference angle: $210^\circ - 180^\circ = 30^\circ$. - In the third quadrant, cosine is negative. - $\cos 210^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}$. 4. **Find $\sin 315^\circ$:** - $315^\circ$ is in the fourth quadrant (between 270° and 360°). - Reference angle: $360^\circ - 315^\circ = 45^\circ$. - In the fourth quadrant, sine is negative. - $\sin 315^\circ = -\sin 45^\circ = -\frac{\sqrt{2}}{2}$. **Final answers:** $$\cos 210^\circ = -\frac{\sqrt{3}}{2}$$ $$\sin 315^\circ = -\frac{\sqrt{2}}{2}$$