1. **Problem:** Find the angle $x^\circ$ in a triangle with sides 8 cm, 7 cm, and 12 cm. 2. **Formula:** Use the Law of Cosines: $$c^2 = a^2 + b^2 - 2ab\cos(C)$$ where $C$ is the angle opposite side $c$. 3. **Identify sides:** Let $c=7$ cm (side opposite $x$), $a=8$ cm, $b=12$ cm. 4. **Apply Law of Cosines:** $$7^2 = 8^2 + 12^2 - 2 \times 8 \times 12 \times \cos(x)$$ $$49 = 64 + 144 - 192 \cos(x)$$ $$49 = 208 - 192 \cos(x)$$ 5. **Isolate $\cos(x)$:** $$192 \cos(x) = 208 - 49$$ $$192 \cos(x) = 159$$ 6. **Simplify:** $$\cos(x) = \frac{159}{192}$$ 7. **Calculate $x$:** $$x = \cos^{-1}\left(\frac{159}{192}\right)$$ 8. **Evaluate:** $$x \approx \cos^{-1}(0.8281) \approx 34.2^\circ$$ **Final answer:** $x \approx 34.2^\circ$