1. **Problem Statement:** Given a right triangle with vertices A, B, and C, where angle C is the right angle, find $\sin \angle B$ and $\cos \angle B$. 2. **Recall definitions:** In a right triangle, for an angle $\theta$: - $\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}$ - $\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}$ 3. **Identify sides relative to $\angle B$:** - Opposite side to $\angle B$ is side $b$ (vertical side AC). - Adjacent side to $\angle B$ is side $a$ (horizontal side CB). - Hypotenuse is side $c$ (side AB). 4. **Apply formulas:** $$\sin \angle B = \frac{b}{c}$$ $$\cos \angle B = \frac{a}{c}$$ 5. **Explanation:** Since $\angle C$ is the right angle, sides $a$ and $b$ are the legs of the triangle, and $c$ is the hypotenuse. The sine of angle $B$ is the ratio of the length of the side opposite to $B$ over the hypotenuse, and the cosine of angle $B$ is the ratio of the length of the side adjacent to $B$ over the hypotenuse. **Final answers:** $$\sin \angle B = \frac{b}{c}$$ $$\cos \angle B = \frac{a}{c}$$