1. Let's state the problem: We want to understand and apply the cosine rule and sine rule to solve triangles. 2. The cosine rule states: $$c^2 = a^2 + b^2 - 2ab\cos(C)$$ where $a$, $b$, and $c$ are sides of a triangle and $C$ is the angle opposite side $c$. 3. The sine rule states: $$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$$ where $A$, $B$, and $C$ are angles opposite sides $a$, $b$, and $c$ respectively. 4. Important rules: - Use the cosine rule when you know two sides and the included angle or all three sides. - Use the sine rule when you know one side and its opposite angle and want to find another side or angle. 5. Example: Given sides $a=7$, $b=10$, and angle $C=60^\circ$, find side $c$ using the cosine rule. 6. Apply the cosine rule: $$c^2 = 7^2 + 10^2 - 2 \times 7 \times 10 \times \cos(60^\circ)$$ 7. Calculate: $$c^2 = 49 + 100 - 140 \times 0.5 = 149 - 70 = 79$$ 8. Find $c$: $$c = \sqrt{79} \approx 8.89$$ 9. Now, to find angle $A$ using the sine rule: $$\frac{a}{\sin(A)} = \frac{c}{\sin(C)} \Rightarrow \sin(A) = \frac{a \sin(C)}{c} = \frac{7 \times \sin(60^\circ)}{8.89}$$ 10. Calculate: $$\sin(A) = \frac{7 \times 0.866}{8.89} \approx 0.681$$ 11. Find angle $A$: $$A = \sin^{-1}(0.681) \approx 43.0^\circ$$ 12. Summary: Using the cosine rule, we found side $c \approx 8.89$. Using the sine rule, we found angle $A \approx 43.0^\circ$. These rules help solve triangles when certain sides and angles are known.