1. **Problem Statement:** We are given a triangle with one unknown side length and two unknown angles. We need to find the unknown side length rounded to the nearest tenth and the two unknown angles rounded to the nearest tenth of a degree. 2. **Formula and Rules:** - Use the Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a,b,c$ are sides opposite angles $A,B,C$ respectively. - The sum of angles in a triangle is always $$180^\circ$$. 3. **Step-by-step Solution:** - Identify the known sides and angles. - Use the Law of Sines to find the unknown side or angle. - Calculate the remaining angle using $$C = 180^\circ - A - B$$. - Round all answers to the nearest tenth. 4. **Example:** Suppose side $a=7$, angle $A=30^\circ$, and side $b=10$ are known. 5. Use Law of Sines to find angle $B$: $$\frac{7}{\sin 30^\circ} = \frac{10}{\sin B}$$ 6. Simplify: $$\frac{7}{0.5} = \frac{10}{\sin B}$$ $$14 = \frac{10}{\sin B}$$ 7. Cross multiply: $$14 \sin B = 10$$ 8. Solve for $\sin B$: $$\sin B = \frac{10}{14} = \frac{\cancel{10}}{\cancel{14}} = 0.7143$$ 9. Find angle $B$: $$B = \sin^{-1}(0.7143) \approx 45.6^\circ$$ 10. Find angle $C$: $$C = 180^\circ - 30^\circ - 45.6^\circ = 104.4^\circ$$ 11. Use Law of Sines to find side $c$: $$\frac{c}{\sin 104.4^\circ} = \frac{7}{\sin 30^\circ} = 14$$ 12. Solve for $c$: $$c = 14 \times \sin 104.4^\circ \approx 14 \times 0.9703 = 13.6$$ **Final answers:** - Unknown side $c \approx 13.6$ (nearest tenth) - Unknown angles $B \approx 45.6^\circ$, $C \approx 104.4^\circ$ (nearest tenth)