1. **State the problem:** We need to find the cosine of angle $S$ in a right triangle $TUS$ where the right angle is at vertex $U$. 2. **Identify the sides relative to angle $S$:** - The side opposite angle $S$ is $TU$ with length 77. - The side adjacent to angle $S$ is $US$ with length 36. - The hypotenuse is the side opposite the right angle $U$, which is $TS$. 3. **Use the Pythagorean theorem to find the hypotenuse $TS$:** $$TS = \sqrt{TU^2 + US^2} = \sqrt{77^2 + 36^2}$$ Calculate the squares: $$77^2 = 5929, \quad 36^2 = 1296$$ Sum: $$5929 + 1296 = 7225$$ So, $$TS = \sqrt{7225} = 85$$ 4. **Recall the cosine definition:** $$\cos(S) = \frac{\text{adjacent side}}{\text{hypotenuse}}$$ For angle $S$, the adjacent side is $US = 36$ and the hypotenuse is $TS = 85$. 5. **Calculate cosine:** $$\cos(S) = \frac{36}{85}$$ 6. **Final answer:** $$\boxed{\cos(S) = \frac{36}{85}}$$