1. **Problem statement:** Given two sides of a triangle with lengths 5.7 cm and 7.5 cm, and the angle opposite the side of length 7.5 cm is 105 degrees, find the unknowns of the triangle. 2. **Formula used:** We use the Law of Sines which states: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a$, $b$, and $c$ are sides opposite angles $A$, $B$, and $C$ respectively. 3. **Assign known values:** Let side $b = 7.5$ cm, angle $B = 105^\circ$, and side $a = 5.7$ cm. We want to find angle $A$ opposite side $a$. 4. **Apply Law of Sines:** $$\frac{a}{\sin A} = \frac{b}{\sin B} \implies \sin A = \frac{a \sin B}{b}$$ 5. **Calculate $\sin A$:** $$\sin A = \frac{5.7 \times \sin 105^\circ}{7.5}$$ Calculate $\sin 105^\circ \approx 0.9659$: $$\sin A = \frac{5.7 \times 0.9659}{7.5} = \frac{5.5056}{7.5} = 0.7341$$ 6. **Find angle $A$:** $$A = \sin^{-1}(0.7341) \approx 47.3^\circ$$ 7. **Find angle $C$:** Sum of angles in a triangle is $180^\circ$: $$C = 180^\circ - A - B = 180^\circ - 47.3^\circ - 105^\circ = 27.7^\circ$$ 8. **Find side $c$ opposite angle $C$ using Law of Sines:** $$\frac{c}{\sin C} = \frac{b}{\sin B} \implies c = \frac{b \sin C}{\sin B} = \frac{7.5 \times \sin 27.7^\circ}{\sin 105^\circ}$$ Calculate $\sin 27.7^\circ \approx 0.4646$: $$c = \frac{7.5 \times 0.4646}{0.9659} = \frac{3.4845}{0.9659} = 3.61 \text{ cm}$$ **Final answers:** - Angle $A \approx 47.3^\circ$ - Angle $C \approx 27.7^\circ$ - Side $c \approx 3.61$ cm