1. **State the problem:** We need to find the unknown side of a triangle given one side length and two angles. The side length is 11.9 cm, and the angles are 51 degrees and 52 degrees (assuming 522 degrees was a typo). 2. **Identify the law to use:** The Law of Sines is used to find unknown sides or angles in any triangle when we know one side and two angles or two sides and one angle. It 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. **Calculate the third angle:** The sum of angles in a triangle is 180 degrees. $$C = 180^\circ - 51^\circ - 52^\circ = 77^\circ$$ 4. **Assign known values:** Let side $a = 11.9$ cm opposite angle $A = 51^\circ$, angle $B = 52^\circ$, and angle $C = 77^\circ$. 5. **Find side $b$ opposite angle $B$ using Law of Sines:** $$\frac{a}{\sin A} = \frac{b}{\sin B} \Rightarrow b = \frac{a \sin B}{\sin A}$$ 6. **Substitute values:** $$b = \frac{11.9 \times \sin 52^\circ}{\sin 51^\circ}$$ 7. **Calculate sine values:** $$\sin 51^\circ \approx 0.7771, \quad \sin 52^\circ \approx 0.7880$$ 8. **Calculate $b$:** $$b = \frac{11.9 \times 0.7880}{0.7771} = \frac{9.3772}{0.7771} \approx 12.06 \text{ cm}$$ **Final answer:** The unknown side opposite the 52-degree angle is approximately $12.06$ cm.