Angle L 014Bd6

1. **Problem Statement:** We are given a triangle $JKL$ with $m\angle K = 38^\circ$, side $JL = 18$, and side $JK = 8$. We need to find the measure of angle $L$ to the nearest tenth. 2. **Relevant Formula:** We can use the Law of Cosines or Law of Sines to find the unknown angle. Since we know two sides and the included angle, the Law of Cosines is appropriate: $$c^2 = a^2 + b^2 - 2ab \cos(C)$$ where $C$ is the angle opposite side $c$. 3. **Identify sides and angles:** - $m\angle K = 38^\circ$ - Side opposite $K$ is $JL = 18$ - Side $JK = 8$ - Side $KL$ is unknown, denote as $x$ 4. **Find side $KL$ using Law of Cosines:** $$JL^2 = JK^2 + KL^2 - 2 \times JK \times KL \times \cos(m\angle K)$$ Substitute known values: $$18^2 = 8^2 + x^2 - 2 \times 8 \times x \times \cos(38^\circ)$$ 5. **Simplify:** $$324 = 64 + x^2 - 16x \cos(38^\circ)$$ Rearranged: $$x^2 - 16x \cos(38^\circ) + 64 - 324 = 0$$ $$x^2 - 16x \cos(38^\circ) - 260 = 0$$ 6. **Calculate $\cos(38^\circ)$:** $$\cos(38^\circ) \approx 0.7880$$ 7. **Substitute and write quadratic:** $$x^2 - 16 \times 0.7880 \times x - 260 = 0$$ $$x^2 - 12.608x - 260 = 0$$ 8. **Solve quadratic equation:** Using quadratic formula: $$x = \frac{12.608 \pm \sqrt{(-12.608)^2 - 4 \times 1 \times (-260)}}{2}$$ Calculate discriminant: $$\sqrt{158.82 + 1040} = \sqrt{1198.82} \approx 34.62$$ 9. **Find roots:** $$x = \frac{12.608 \pm 34.62}{2}$$ Two solutions: - $$x = \frac{12.608 + 34.62}{2} = 23.61$$ - $$x = \frac{12.608 - 34.62}{2} = -11.01$$ (discard negative length) So, $KL \approx 23.61$. 10. **Find $m\angle L$ using Law of Sines:** $$\frac{\sin(m\angle L)}{JK} = \frac{\sin(m\angle K)}{KL}$$ Substitute known values: $$\frac{\sin(m\angle L)}{8} = \frac{\sin(38^\circ)}{23.61}$$ 11. **Calculate $\sin(38^\circ)$:** $$\sin(38^\circ) \approx 0.6157$$ 12. **Solve for $\sin(m\angle L)$:** $$\sin(m\angle L) = \frac{8 \times 0.6157}{23.61} = \frac{4.9256}{23.61} \approx 0.2087$$ 13. **Find $m\angle L$:** $$m\angle L = \arcsin(0.2087) \approx 12.0^\circ$$ 14. **Final answer:** The measure of angle $L$ is approximately $\boxed{12.0^\circ}$ to the nearest tenth.