1. **State the problem:** Calculate the angles $\beta$ and $\gamma$ in a triangle where angle $\alpha = 32^\circ$, the side opposite $\alpha$ is 14 cm, and the side adjacent to $\alpha$ is 11 cm. 2. **Recall the Law of Sines:** $$\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}$$ where $a$, $b$, and $c$ are the sides opposite angles $\alpha$, $\beta$, and $\gamma$ respectively. 3. **Assign known values:** - $a = 14$ cm (opposite $\alpha$) - $b = 11$ cm (adjacent to $\alpha$, opposite $\beta$) - $\alpha = 32^\circ$ 4. **Find $\beta$ using Law of Sines:** $$\frac{14}{\sin 32^\circ} = \frac{11}{\sin \beta}$$ 5. **Solve for $\sin \beta$:** $$\sin \beta = \frac{11 \sin 32^\circ}{14}$$ Calculate $\sin 32^\circ \approx 0.5299$: $$\sin \beta = \frac{11 \times 0.5299}{14} = \frac{5.8289}{14} \approx 0.4164$$ 6. **Find $\beta$:** $$\beta = \arcsin(0.4164) \approx 24.62^\circ$$ 7. **Find $\gamma$ using triangle angle sum:** $$\gamma = 180^\circ - \alpha - \beta = 180^\circ - 32^\circ - 24.62^\circ = 123.38^\circ$$ **Final answers:** $$\beta \approx 24.62^\circ, \quad \gamma \approx 123.38^\circ$$