1. The problem is to solve a triangle using the sine rule and cosine rule without using the semiperimeter. 2. The sine rule states that for any triangle with sides $a$, $b$, $c$ and opposite angles $A$, $B$, $C$ respectively: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ 3. The cosine rule states: $$c^2 = a^2 + b^2 - 2ab \cos C$$ 4. To solve a triangle, you need at least one side and two angles or two sides and one angle. 5. Use the cosine rule to find the unknown side if you have two sides and the included angle. 6. Use the sine rule to find unknown angles or sides once you have enough information. 7. Example: Given sides $a=7$, $b=10$, and angle $C=60^\circ$, find side $c$. 8. Apply cosine rule: $$c^2 = 7^2 + 10^2 - 2 \times 7 \times 10 \times \cos 60^\circ = 49 + 100 - 140 \times 0.5 = 149 - 70 = 79$$ 9. So, $$c = \sqrt{79} \approx 8.89$$ 10. Use sine rule to find angle $A$: $$\frac{a}{\sin A} = \frac{c}{\sin C} \Rightarrow \sin A = \frac{a \sin C}{c} = \frac{7 \times \sin 60^\circ}{8.89} = \frac{7 \times 0.866}{8.89} \approx 0.681$$ 11. Therefore, $$A = \sin^{-1}(0.681) \approx 43^\circ$$ 12. Finally, find angle $B$: $$B = 180^\circ - A - C = 180^\circ - 43^\circ - 60^\circ = 77^\circ$$ This completes the triangle solution using sine and cosine rules without semiperimeter.