1. **Problem Statement:** Find the values of $\sin 120^\circ$ and $\cos 225^\circ$ using the unit circle. 2. **Recall the Unit Circle:** The unit circle has radius 1. The coordinates of a point on the unit circle at an angle $\theta$ are $(\cos \theta, \sin \theta)$. 3. **Find $\sin 120^\circ$:** - $120^\circ$ is in the second quadrant where sine is positive. - Reference angle is $180^\circ - 120^\circ = 60^\circ$. - $\sin 120^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2}$. 4. **Find $\cos 225^\circ$:** - $225^\circ$ is in the third quadrant where cosine is negative. - Reference angle is $225^\circ - 180^\circ = 45^\circ$. - $\cos 225^\circ = -\cos 45^\circ = -\frac{\sqrt{2}}{2}$. 5. **Final answers:** $$\sin 120^\circ = \frac{\sqrt{3}}{2}$$ $$\cos 225^\circ = -\frac{\sqrt{2}}{2}$$