1. **Problem Statement:** Find the sine, cosine, and tangent of angle $S$ in the right triangle $TUS$ where $US=16$, $TS=34$, and angle $U$ is the right angle. 2. **Recall definitions of trigonometric ratios for angle $S$:** - $\sin(S) = \frac{\text{opposite side to } S}{\text{hypotenuse}}$ - $\cos(S) = \frac{\text{adjacent side to } S}{\text{hypotenuse}}$ - $\tan(S) = \frac{\text{opposite side to } S}{\text{adjacent side to } S}$ 3. **Identify sides relative to angle $S$:** - Opposite side to $S$ is $U S = 16$ - Hypotenuse is $T S = 34$ - Adjacent side to $S$ is $T U$ (unknown, find it using Pythagoras theorem) 4. **Find adjacent side $T U$ using Pythagoras theorem:** $$T U = \sqrt{T S^2 - U S^2} = \sqrt{34^2 - 16^2} = \sqrt{1156 - 256} = \sqrt{900} = 30$$ 5. **Calculate the trigonometric ratios:** - $$\sin(S) = \frac{16}{34} = \frac{8}{17}$$ - $$\cos(S) = \frac{30}{34} = \frac{15}{17}$$ - $$\tan(S) = \frac{16}{30} = \frac{8}{15}$$ **Final answers:** - $\sin(S) = \frac{8}{17}$ - $\cos(S) = \frac{15}{17}$ - $\tan(S) = \frac{8}{15}$