1. **State the problem:** We are given a triangle LMN with sides $\ell=15$, $m=36$, and $n=39$. We need to find $\sin L$, $\cos L$, $\tan L$, $\sin M$, $\cos M$, and $\tan M$. 2. **Recall definitions:** For angle $L$, opposite side is $\ell=15$, adjacent side is $m=36$, and hypotenuse is $n=39$. 3. **Formulas:** $$\sin L = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\ell}{n}$$ $$\cos L = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{m}{n}$$ $$\tan L = \frac{\text{opposite}}{\text{adjacent}} = \frac{\ell}{m}$$ 4. **Calculate for angle L:** $$\sin L = \frac{15}{39} = \frac{5 \times 3}{13 \times 3} = \frac{5}{13} \approx 0.38$$ $$\cos L = \frac{36}{39} = \frac{12 \times 3}{13 \times 3} = \frac{12}{13} \approx 0.92$$ $$\tan L = \frac{15}{36} = \frac{5 \times 3}{12 \times 3} = \frac{5}{12} \approx 0.42$$ 5. **For angle M:** Opposite side is $m=36$, adjacent side is $\ell=15$, hypotenuse is $n=39$. 6. **Formulas for angle M:** $$\sin M = \frac{m}{n}$$ $$\cos M = \frac{\ell}{n}$$ $$\tan M = \frac{m}{\ell}$$ 7. **Calculate for angle M:** $$\sin M = \frac{36}{39} = \frac{12}{13} \approx 0.92$$ $$\cos M = \frac{15}{39} = \frac{5}{13} \approx 0.38$$ $$\tan M = \frac{36}{15} = \frac{12}{5} = 2.4$$ **Final answers:** - $\sin L = \frac{5}{13}$ - $\cos L = \frac{12}{13}$ - $\tan L = \frac{5}{12}$ - $\sin M = \frac{12}{13}$ - $\cos M = \frac{5}{13}$ - $\tan M = \frac{12}{5}$