1. **Problem statement:** Given a right triangle with hypotenuse $\sqrt{34}$, vertical side 3, and horizontal side 5, find $\sin X$ and $\tan Y$ where $X$ and $Y$ are angles at vertices $X$ and $Y$ respectively. 2. **Recall definitions:** - $\sin$ of an angle in a right triangle is the ratio of the length of the side opposite the angle to the hypotenuse. - $\tan$ of an angle is the ratio of the length of the side opposite the angle to the side adjacent to the angle. 3. **Identify sides relative to angles:** - For angle $X$, the side opposite is $YZ = 3$. - For angle $Y$, the side opposite is $XZ = 5$. 4. **Calculate $\sin X$:** $$\sin X = \frac{\text{opposite to } X}{\text{hypotenuse}} = \frac{3}{\sqrt{34}}$$ 5. **Calculate $\tan Y$:** $$\tan Y = \frac{\text{opposite to } Y}{\text{adjacent to } Y} = \frac{5}{3}$$ 6. **Final answer:** $$\sin X = \frac{3}{\sqrt{34}}, \quad \tan Y = \frac{5}{3}$$