1. **State the problem:** We need to find the value of $\cos L$ in the right triangle $\triangle LKJ$ where $LK=5$, $KJ=7$, and the hypotenuse $LJ=\sqrt{74}$. The right angle is at vertex $K$. 2. **Recall the cosine definition:** In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. 3. **Identify sides relative to angle $L$:** - The hypotenuse is $LJ = \sqrt{74}$. - The side adjacent to angle $L$ is $LK = 5$. 4. **Write the formula:** $$\cos L = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{LK}{LJ} = \frac{5}{\sqrt{74}}$$ 5. **Simplify the expression:** $$\cos L = \frac{5}{\sqrt{74}} = \frac{5}{\sqrt{74}} \times \frac{\sqrt{74}}{\sqrt{74}} = \frac{5\sqrt{74}}{74}$$ 6. **Calculate the decimal value:** First, approximate $\sqrt{74} \approx 8.6023$. Then, $$\cos L \approx \frac{5 \times 8.6023}{74} = \frac{43.0115}{74} \approx 0.5815$$ 7. **Round to the nearest hundredth:** $$\cos L \approx 0.58$$ **Final answer:** $$\boxed{0.58}$$