📚 Solving Triangle Problems | 解三角形问题
In Edexcel A-Level Mathematics, solving triangle problems means using the sine rule, cosine rule and area formula to find unknown sides and angles in non-right-angled triangles. These techniques appear in pure exam questions, often mixed with bearings, angles of elevation and 3D geometry.
在 Edexcel A-Level 数学中，解三角形问题是指运用正弦定理、余弦定理和面积公式，求非直角三角形中未知的边和角。这类技巧经常出现在纯数学考题中，并常与方位角、仰角和三维几何结合考查。
1. Toolkit for Any Triangle | 任意三角形的工具
For a triangle ABC with side lengths a, b, c opposite angles A, B, C, the three essential results are:
对于三角形 ABC，边长 a、b、c 分别对应角 A、B、C，三个核心公式如下：
a / sin A = b / sin B = c / sin C
a² = b² + c² − 2bc cos A
cos A = (b² + c² − a²) / (2bc)
Area = ½ ab sin C = ½ bc sin A = ½ ac sin B
with the sine rule, the reciprocal form sin A / a = sin B / b = sin C / c is often easier.
正弦定理将边与其对角的正弦联系起来。余弦定理将三条边和一个角联系起来。面积公式使用两条边及其夹角。用正弦定理求角时，倒数形式 sin A / a = sin B / b = sin C / c 通常更方便。
2. Choosing the Right Rule | 选择正确的规则
Use the sine rule when you know two angles and one side, or two sides and a non-included angle. Use the cosine rule when you know three sides, or two sides and the included angle. Always label the triangle first and write down exactly what is known.
当已知两角和一边，或已知两边及一个非夹角时，使用正弦定理。当已知三边，或已知两边及其夹角时，使用余弦定理。解题时应先标注三角形，并明确写出已知条件。
| Known information | Rule to use first |
|---|---|
| AAS or ASA: two angles and one side | Sine rule |
| SSA: two sides and a non-included angle | Sine rule, check the ambiguous case |
| SAS: two sides and the included angle | Cosine rule |
| SSS: three sides | Cosine rule |
Choosing the correct rule at the start prevents long algebraic detours and reduces rounding errors.
一开始就选择正确规则，可以避免冗长的代数绕路并减少舍入误差。
3. The Sine Rule in Action | 正弦定理的应用
Example: In triangle ABC, A = 40°, B = 65° and a = 8 cm. Find side b.
例题：在三角形 ABC 中，A = 40°，B = 65°，a = 8 cm。求边 b。
Using the sine rule:
使用正弦定理：
b / sin 65° = 8 / sin 40°
So:
因此：
b = 8 sin 65° / sin 40° ≈ 11.28 cm
4° or B ≈ 147.6°, but the second value is impossible here because A + B would exceed 180°.
正弦定理也可用于求角。例如，若 a = 10，b = 7，A = 50°，则 sin B = 7 sin 50° / 10 ≈ 0.536。由此可得 B ≈ 32.4° 或 B ≈ 147.6°，但第二个值在此不成立，因为 A + B 会超过 180°。
4. The Cosine Rule in Action | 余弦定理的应用
Example 1: Given b = 5, c = 7 and A = 60°, find side a.
例题 1：已知 b = 5，c = 7，A = 60°，求边 a。
a² = 5² + 7² − 2 × 5 × 7 × cos 60°
a² = 25 + 49 − 70 × 0.5 = 39
a = √39 ≈ 6.24
Example 2: Given a = 5, b = 6 and c = 7, find angle A.
例题 2：已知 a = 5，b = 6，c = 7，求角 A。
cos A = (6² + 7² − 5²) / (2 × 6 × 7)
cos A = (36 + 49 − 25) / 84 = 60 / 84 = 5 / 7
A ≈ 44.4°
The cosine rule is essential for SAS and SSS cases because the sine rule cannot start with those combinations.
余弦定理对 SAS 和 SSS 情形至关重要，因为正弦定理无法从这些条件组合直接开始求解。
5. Area of a Triangle | 三角形面积
The area of a non-right-angled triangle can be found using two sides and the included angle:
非直角三角形的面积可以用两条边及其夹角求得：
Area = ½ ab sin C
Example: If b = 8 cm, c = 6 cm and A = 30°, then:
例题：若 b = 8 cm，c = 6 cm，A = 30°，则：
Area = ½ × 8 × 6 × sin 30° = 12 cm²
If the three sides are known but no angle is given, use the cosine rule first to find one angle, then apply the area formula.
若已知三边但未给出任何角，可先用余弦定理求出一个角，再应用面积公式。
6. The Ambiguous Case (SSA) | 不确定情形（SSA）
When two sides and a non-included angle are known, the triangle may have two possible solutions, one solution, or no solution. For given a, b and angle A, where a is opposite angle A, compare a with b sin A.
当已知两边和一个非夹角时，三角形可能存在两个解、一个解或无解。对于给定的 a、b 和角 A，其中 a 是角 A 的对边，需要将 a 与 b sin A 作比较。
- If A is obtuse: one triangle exists if a > b; no triangle exists if a ≤ b.
- 如果 A 为钝角：若 a > b，则有一个三角形；若 a ≤ b，则无三角形。
- If A is acute and a < b sin A: no triangle exists.
- 如果 A 为锐角且 a < b sin A：不存在三角形。
- If A is acute and a = b sin A: exactly one right-angled triangle exists.
- 如果 A 为锐角且 a = b sin A：存在唯一一个直角三角形。
- If A is acute and b sin A < a < b: two triangles exist, one with B acute and one with B obtuse.
- 如果 A 为锐角且 b sin A < a < b：存在两个三角形，一个 B 为锐角，另一个 B 为钝角。
- If A is acute and a ≥ b: exactly one triangle exists.
- 如果 A 为锐角且 a ≥ b：存在唯一一个三角形。
riangles. Using the sine rule, sin B = 10 sin 30° / 6 ≈ 0.833, so B ≈ 56.4° or B ≈ 123.6°.
例题：已知 a = 6，b = 10，A = 30°。因为 b sin A = 10 sin 30° = 5，且 5 < 6 < 10，所以存在两个可能的三角形。使用正弦定理，sin B = 10 sin 30° / 6 ≈ 0.833，因此 B ≈ 56.4° 或 B ≈ 123.6°。
7. Bearings and Navigation Problems | 方位角与航海问题
A bearing is measured clockwise from north and is always written as a three-figure angle. For example, east is 090°, south is 180° and north-west is 315°.
方位角从正北方向顺时针测量，并始终写成三位数角。例如，正东是 090°，正南是 180°，西北是 315°。
Example: A walker travels 6 km on a bearing of 030°, then 8 km on a bearing of 120°. Find the distance and bearing from the starting point.
例题：一名步行者先沿方位角 030° 行走 6 km，再沿方位角 120° 行走 8 km。求他离起点的距离和方位角。
The angle between the two paths is 120° − 030° = 90°, so the distance from the start is:
两段路径之间的夹角为 120° − 030° = 90°，因此离起点的距离为：
d = √6² + 8² = √100 = 10 km
ing θ satisfies tan θ = east / north, so θ ≈ 83.1°. Therefore the final bearing is 083.1°.
求方位角时，分解位移：东向分量 = 6 sin 30° + 8 sin 120° ≈ 9.928 km；北向分量 = 6 cos 30° + 8 cos 120° ≈ 1.196 km。方位角 θ 满足 tan θ = 东向分量 / 北向分量，因此 θ ≈ 83.1°。最终方位角为 083.1°。
8. Angles of Elevation and Depression | 仰角与俯角
The angle of elevation is measured upwards from the horizontal, and the angle of depression is measured downwards from the horizontal. Both are commonly used in practical triangle problems.
仰角是从水平线向上测量的角，俯角是从水平线向下测量的角。两者都常用于实际三角形问题。
Example: From a point A, the angle of elevation of the top of a tower is 20°. After walking 50 m directly towards the tower to point B, the angle of elevation is 30°. Find the height h of the tower.
例题：从点 A 测得塔顶的仰角为 20°。朝塔的方向直行 50 m 到点 B 后，仰角变为 30°。求塔高 h。
Let x be the distance from B to the base of the tower. Then:
设 B 到塔底的距离为 x，则：
