1. **State the problem:** We need to find the value of $\cos L$ in a right triangle $\triangle LMN$ where the right angle is at vertex $M$. The side opposite angle $L$ is $MN = 4$, and the hypotenuse $LN = 8$. 2. **Recall the definition of cosine in a right triangle:** $$\cos L = \frac{\text{adjacent side to } L}{\text{hypotenuse}}$$ 3. **Identify the adjacent side to angle $L$:** Since $MN$ is opposite $L$ and $LN$ is the hypotenuse, the remaining side $LM$ is adjacent to $L$. We need to find $LM$ using the Pythagorean theorem: $$LM = \sqrt{LN^2 - MN^2} = \sqrt{8^2 - 4^2} = \sqrt{64 - 16} = \sqrt{48}$$ 4. **Simplify $\sqrt{48}$:** $$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$$ 5. **Calculate $\cos L$:** $$\cos L = \frac{LM}{LN} = \frac{4\sqrt{3}}{8} = \frac{\cancel{4}\sqrt{3}}{\cancel{8}2} = \frac{\sqrt{3}}{2}$$ 6. **Evaluate the decimal value:** $$\cos L \approx \frac{1.732}{2} = 0.866$$ 7. **Round to the nearest hundredth:** $$\cos L \approx 0.87$$ **Final answer:** $\boxed{0.87}$