1. **Stating the problem:** We are given the cosine rules for angles in a triangle: $$\cos A=\frac{b^2+c^2-a^2}{2bc}$$ and $$\cos C=\frac{a^2+b^2-c^2}{2ab}$$ We want to understand these formulas and their relation to the Pythagorean theorem and the general law of cosines. 2. **Formula explanation:** The cosine rule relates the lengths of sides of any triangle to the cosine of one of its angles. For angle $C=\theta$, the law of cosines states: $$c^2 = a^2 + b^2 - 2ab \cos \theta$$ This formula generalizes the Pythagorean theorem. 3. **Important property:** If $\theta = 90^\circ$, then $\cos 90^\circ = 0$. Substituting into the law of cosines: $$c^2 = a^2 + b^2 - 2ab \times 0 = a^2 + b^2$$ which is the Pythagorean theorem. 4. **Intermediate work:** Using the cosine rule for angle $A$: $$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$ and for angle $C$: $$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$$ These formulas allow calculation of angles if side lengths are known. 5. **Summary:** The cosine rule is a powerful tool for solving any triangle when you know two sides and the included angle or all three sides. It reduces to the Pythagorean theorem for right triangles. Final answer: $$c^2 = a^2 + b^2 - 2ab \cos \theta$$ and $$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$ $$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$$