1. **State the problem:** We need to find the cosecant of angle $W$ in right triangle $V X W$ where $\angle X$ is the right angle. 2. **Identify sides relative to $\angle W$:** - Opposite side to $W$ is $V X$ (vertical leg). - Adjacent side to $W$ is $X W$ (horizontal leg) labeled $3\sqrt{6}$. - Hypotenuse is $V W$ labeled $2\sqrt{22}$. 3. **Recall the definition of cosecant:** $$\csc(W) = \frac{1}{\sin(W)} = \frac{\text{hypotenuse}}{\text{opposite}}$$ 4. **Find the length of the opposite side $V X$ using the Pythagorean theorem:** $$V W^2 = V X^2 + X W^2$$ $$\Rightarrow V X^2 = V W^2 - X W^2 = (2\sqrt{22})^2 - (3\sqrt{6})^2$$ $$= 4 \times 22 - 9 \times 6 = 88 - 54 = 34$$ $$\Rightarrow V X = \sqrt{34}$$ 5. **Calculate $\csc(W)$:** $$\csc(W) = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{2\sqrt{22}}{\sqrt{34}}$$ 6. **Simplify and rationalize the denominator:** $$\csc(W) = \frac{2\sqrt{22}}{\sqrt{34}} \times \frac{\sqrt{34}}{\sqrt{34}} = \frac{2\sqrt{22} \times \sqrt{34}}{34} = \frac{2\sqrt{748}}{34}$$ 7. **Simplify $\sqrt{748}$:** $$748 = 4 \times 187 \Rightarrow \sqrt{748} = \sqrt{4 \times 187} = 2\sqrt{187}$$ 8. **Substitute back:** $$\csc(W) = \frac{2 \times 2 \sqrt{187}}{34} = \frac{4\sqrt{187}}{34}$$ 9. **Simplify the fraction:** $$\csc(W) = \frac{\cancel{4}^{2} \sqrt{187}}{\cancel{34}^{17}} = \frac{2\sqrt{187}}{17}$$ **Final answer:** $$\boxed{\csc(W) = \frac{2\sqrt{187}}{17}}$$