📚 A-Level Maths: The Cosine Rule and Its Typical Applications | A-Level 数学：余弦定理及其典型应用
The cosine rule is a fundamental tool in triangle geometry, essential for solving problems where the sine rule cannot be directly applied. It connects the lengths of sides of a triangle to the cosine of one of its angles, enabling us to find unknown sides, unknown angles, and even the area of a triangle in certain contexts.
余弦定理是三角形几何中的基本工具，对于解决正弦定理无法直接应用的问题至关重要。它将三角形的边长与某一内角的余弦值联系起来，使我们能够求解未知边长、未知角度，甚至在某些情形下计算三角形的面积。
1. Statement of the Cosine Rule | 余弦定理的表述
For any triangle ABC, with sides a, b, c opposite angles A, B, C respectively, the cosine rule states:
对于任意三角形 ABC，设边 a、b、c 分别对应角 A、B、C，则余弦定理表述为：
a² = b² + c² − 2bc·cos A
Equivalently, it can be rearranged to find an angle:
等价地，该公式可重排为求角的形式：
cos A = (b² + c² − a²) / (2bc)
The rule applies to any triangle, whether acute, obtuse, or right-angled. In the special case where A = 90°, cos A = 0, and the formula reduces to Pythagoras’ theorem: a² = b² + c².
这一定理适用于任意三角形，无论是锐角三角形、钝角三角形还是直角三角形。在特殊情况下，当 A = 90° 时，cos A = 0，公式退化为勾股定理：a² = b² + c²。
2. When to Use the Cosine Rule | 余弦定理的适用情形
The cosine rule is typically used in two main situations: (1) when two sides and the included angle (SAS) are known, and we wish to find the third side; (2) when all three sides (SSS) are known, and we wish to find an angle.
余弦定理通常在两种主要情形下使用：（1）已知两边及其夹角（边角边，SAS），要求第三边；（2）已知三边（边边边，SSS），要求某个角。
If two angles and a side are given (AAS or ASA), the sine rule is generally more efficient. If two sides and a non-included angle are given (SSA), the situation is ambiguous and requires careful analysis, often using both the sine and cosine rules.
如果已知两角及一边（角角边或角边角），通常使用正弦定理更为简便。如果已知两边及其中一边的对角（边边角），则会出现模棱两可的情形，需要仔细分析，通常同时使用正弦定理和余弦定理。
3. Derivation of the Cosine Rule | 余弦定理的推导
The cosine rule can be derived by placing triangle ABC on a coordinate plane. Let A be at the origin (0, 0), B at (c, 0), and C at (b·cos A, b·sin A). The distance between B and C is side a. Using the distance formula:
余弦定理可以通过将三角形 ABC 置于坐标平面中来推导。令 A 位于原点 (0, 0)，B 位于 (c, 0)，C 位于 (b·cos A, b·sin A)。B 与 C 之间的距离即为边 a。利用距离公式：
a² = (b·cos A − c)² + (b·sin A − 0)²
Expanding and simplifying:
展开并化简：
a² = b²cos²A − 2bc·cos A + c² + b²sin²A = b² + c² − 2bc·cos A
This derivation relies on the identity sin²A + cos²A = 1, which is true for all angles A between 0° and 180°.
这一推导依赖于恒等式 sin²A + cos²A = 1，该恒等式对所有介于 0° 和 180° 之间的角 A 均成立。
4. Finding an Unknown Side (SAS) | 已知两边及其夹角求第三边
Suppose we have a triangle with sides b = 7 cm, c = 5 cm, and the included angle A = 40°. To find side a, we apply the cosine rule directly:
假设一个三角形的边 b = 7 cm，c = 5 cm，夹角 A = 40°。要求边 a，我们直接应用余弦定理：
a² = 7² + 5² − 2 × 7 × 5 × cos 40°
Evaluating: a² = 49 + 25 − 70 × 0.7660 ≈ 74 − 53.62 = 20.38, so a ≈ √20.38 ≈ 4.51 cm. The unknown side is approximately 4.51 cm.
计算得：a² = 49 + 25 − 70 × 0.7660 ≈ 74 − 53.62 = 20.38，因此 a ≈ √20.38 ≈ 4.51 cm。未知边长约为 4.51 cm。
Always round your final answer to an appropriate degree of accuracy, typically 3 significant figures in A-Level examinations unless instructed otherwise.
务必根据题目要求对最终答案进行适当精度的舍入，通常在 A-Level 考试中保留 3 位有效数字，除非另有说明。
5. Finding an Unknown Angle (SSS) | 已知三边求角
When all three sides are known, we rearrange the cosine rule to solve for the cosine of the angle. For example, if a = 6 cm, b = 8 cm, c = 10 cm, and we wish to find angle A opposite side a:
当已知三边时，我们重排余弦定理以求解角的余弦值。例如，如果 a = 6 cm，b = 8 cm，c = 10 cm，要求边 a 所对的角 A：
cos A = (8² + 10² − 6²) / (2 × 8 × 10) = (64 + 100 − 36) / 160 = 128 / 160 = 0.8
Thus A = cos⁻¹(0.8) ≈ 36.87°. The inverse cosine function returns an angle between 0° and 180°, so there is no ambiguity when finding an angle in a triangle using this method.
因此 A = cos⁻¹(0.8) ≈ 36.87°。反余弦函数返回的角度范围在 0° 到 180° 之间，因此使用这种方法求三角形内角时不会产生歧义。
6. The Cosine Rule vs. The Sine Rule | 余弦定理与正弦定理的对比
Both rules are used to solve triangles, but they serve different purposes. The sine rule is ideal for AAS and ASA cases, while the cosine rule is better for SAS and SSS cases. Additionally, the sine rule can yield ambiguous results in SSA situations, whereas the cosine rule is always unambiguous.
这两个定理都用于解三角形，但用途不同。正弦定理适合角角边和角边角的情形，而余弦定理更适合边角边和边边边的情形。此外，正弦定理在边边角情形下可能产生两解，而余弦定理永远不会产生歧义。
When deciding which rule to use, first identify what information is given and what is required. If you have a pair consisting of an angle and its opposite side, consider the sine rule. If you have two sides and the included angle, or three sides only, use the cosine rule.
在决定使用哪个定理时，首先要明确已知条件和要求的目标。如果已知一组对边对角，考虑正弦定理；如果已知两边及其夹角，或仅知三边，则使用余弦定理。
7. Using the Cosine Rule to Find the Area of a Triangle | 利用余弦定理求三角形面积
Although the area of a triangle is often calculated using the formula ½ab·sin C, the cosine rule can help when we know all three sides. Heron’s formula, which uses the semi-perimeter s = (a + b + c)/2, states:
虽然三角形面积通常使用公式 ½ab·sin C 计算，但当我们已知三边时，余弦定理可以帮助我们。海伦公式利用半周长 s = (a + b + c)/2，其表述为：
Area = √[s(s − a)(s − b)(s − c)]
Alternatively, we can use the cosine rule to find the cosine of an angle, then the sine of that angle, and finally apply the standard area formula. This demonstrates the flexibility of the cosine rule in solving a wide range of geometric problems.
或者，我们可以用余弦定理求出某个角的余弦值，再求出该角的正弦值，最后应用标准面积公式。这体现了余弦定理在解决多种几何问题中的灵活性。
8. Application in Real-World Problems | 在现实问题中的应用
Cosine rule problems in A-Level often involve situations such as finding the distance between two ships after they have travelled in different directions, or calculating the length of a cable in a structural framework. These problems require translating a worded scenario into a triangle and then applying the rule.
A-Level 考试中的余弦定理题目通常涉及实际问题，例如求两艘船沿不同方向航行后的距离，或计算结构框架中缆绳的长度。这些问题需要将文字情境转化为三角形，然后应用定理。
For example, a ship sails 30 km on a bearing of 050°, then 45 km on a bearing of 130°. The angle between the two paths is 80°, so the distance from the start can be found using the cosine rule with sides 30 km and 45 km and included angle 80°.
例如，一艘船先按方位角 050° 航行 30 km，再按方位角 130° 航行 45 km。两条路径之间的夹角为 80°，因此可以用余弦定理，以 30 km 和 45 km 为两边，夹角为 80°，求起点到终点的距离。
9. Common Pitfalls and Exam Tips | 常见错误与考试技巧
One common mistake is applying the cosine rule incorrectly because the wrong angle is used. Remember that side a must be opposite angle A. Another error is forgetting to take the square root when finding a side, or failing to use the inverse cosine when finding an angle.
一个常见错误是因为用错了角而错误地应用余弦定理。记住，边 a 必须与角 A 相对。另一个错误是求边长时忘记开平方，或者求角时忘记使用反余弦。
In exams, always write down the formula before substituting values, and use the cosine rule with the angle in degrees unless the question specifies radians. Also, round only at the final step to avoid accumulating rounding errors.
在考试中，先写出公式再代入数值，并且除非题目指定弧度，否则余弦定理中的角使用度数。此外，只在最后一步进行舍入，以避免舍入误差的累积。
10. Worked Example: Mixed Triangle Problem | 综合例题：混合三角形问题
Consider triangle PQR, where PQ = 9 cm, PR = 6 cm, and angle P = 120°. Find the length QR and the angle R.
考虑三角形 PQR，其中 PQ = 9 cm，PR = 6 cm，角 P = 120°。求边 QR 的长度和角 R。
First, use the cosine rule to find QR (opposite angle P):
首先，用余弦定理求角 P 所对的边 QR：
QR² = 9² + 6² − 2 × 9 × 6 × cos 120° = 81 + 36 − 108 × (−0.5) = 117 + 54 = 171
So QR = √171 ≈ 13.08 cm. Then, use the cosine rule again to find angle R:
因此 QR = √171 ≈ 13.08 cm。然后，再次使用余弦定理求角 R：
cos R = (PR² + QR² − PQ²) / (2 × PR × QR) = (36 + 171 − 81) / (2 × 6 × 13.08) ≈ 126 / 156.96 ≈ 0.8028
Thus R ≈ cos⁻¹(0.8028) ≈ 36.6°. Checking: the sum of angles is 180°, so Q ≈ 23.4°, which is consistent.
因此 R ≈ cos⁻¹(0.8028) ≈ 36.6°。检验：三角形内角和为 180°，因此 Q ≈ 23.4°，计算结果一致。
11. Proof of the Cosine Rule Using Vectors | 使用向量的余弦定理证明
For students familiar with vectors, the cosine rule has an elegant vector proof. Let vectors u and v represent two sides of a triangle sharing a common vertex, with the third side given by vector u − v. The squared length of this vector is:
对于熟悉向量的学生，余弦定理有一个优雅的向量证明。设向量 u 和 v 表示从同一顶点出发的两条边，则第三边对应向量 u − v。该向量的长度平方为：
|u − v|² = (u − v)·(u − v) = |u|² + |v|² − 2u·v
Since u·v = |u||v|cos θ, where θ is the angle between u and v, this reduces exactly to the cosine rule. This viewpoint is useful in mechanics and further mathematics, where the cosine rule appears in the form of the law of cosines for resultant forces.
因为 u·v = |u||v|cos θ，其中 θ 是 u 与 v 之间的夹角，因此上式正好化为余弦定理。这种视角在力学和进阶数学中非常有用，此时余弦定律表现为合力的余弦定律形式。
12. Summary | 总结
The cosine rule is an indispensable part of A-Level mathematics. Mastering its statement, derivation, and application enables you to solve a wide range of triangle problems, from simple side and angle calculations to complex real-world contexts. Always check whether the sine rule or the cosine rule is more appropriate, and practise translating word problems into clear diagrams.
余弦定理是 A-Level 数学中不可或缺的一部分。掌握其表述、推导和应用，能使你解决各种三角形问题，从简单的边长和角度的计算到复杂的实际应用场景。始终判断正弦定理与余弦定理哪个更合适，并练习将文字问题转化为清晰的图形。
By understanding the underlying principles and potential pitfalls, you can approach exam questions with confidence and accuracy. The cosine rule is not merely a formula to memorise; it is a powerful mathematical connection that appears throughout geometry, trigonometry, and physics.
通过理解其背后的原理和潜在的易错点，你可以自信而准确地应对考试题目。余弦定理不仅仅是一个需要记忆的公式；它是贯穿几何、三角和物理学的强大数学联系。
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