1. **State the problem:** We need to find the value of $\cos X$ in a right triangle $\triangle ZYX$ with a right angle at vertex $Y$. 2. **Identify the sides:** The triangle has vertices $Z$ (bottom-left), $Y$ (top, right angle), and $X$ (bottom-right). - Side $YX = \sqrt{3}$ - Side $ZX = 6$ 3. **Recall the cosine definition:** In a right triangle, $\cos$ of an angle is the ratio of the adjacent side to the hypotenuse. 4. **Determine the hypotenuse:** The hypotenuse is the side opposite the right angle, which is $ZX = 6$. 5. **Find the side adjacent to angle $X$:** Angle $X$ is at vertex $X$, so the side adjacent to $X$ (other than the hypotenuse) is $YX = \sqrt{3}$. 6. **Apply the cosine formula:** $$\cos X = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{YX}{ZX} = \frac{\sqrt{3}}{6}$$ 7. **Simplify the fraction:** $$\frac{\sqrt{3}}{6} = \frac{\sqrt{3}}{6}$$ This fraction is already in simplest form. **Final answer:** $$\boxed{\frac{\sqrt{3}}{6}}$$