A-Level Maths: The Cosine Rule and Its Applications | A-Level数学：余弦定理及其应用

📚 A-Level Maths: The Cosine Rule and Its Applications | A-Level数学：余弦定理及其应用

The cosine rule is one of the most powerful tools in A-Level trigonometry. It allows us to solve any triangle, whether right-angled or not, provided we know either two sides and the included angle, or all three sides. This article explains the rule, its derivation, and its practical applications in the Edexcel A-Level Maths syllabus.

余弦定理是A-Level三角学中最强大的工具之一。无论是否为直角三角形，只要我们知道两边及其夹角，或者三条边的长度，就可以利用余弦定理求解任意三角形。本文将详解该定理、推导过程以及其在Edexcel A-Level数学考纲中的实际应用。

1. What is the Cosine Rule | 什么是余弦定理

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分别对应于角A、B、C的对边，余弦定理表述为：

a² = b² + c² − 2bc·cos A

Equivalently, we may write the rule in two other cyclic forms:

等价地，我们还可以写出另外两种循环形式：

b² = a² + c² − 2ac·cos B

c² = a² + b² − 2ab·cos C

Notice the pattern: the side on the left-hand side is always the side opposite the angle used in the final term. This symmetry makes the rule easy to remember.

请注意规律：等式左侧的边始终是最终项中所用角所对的边。这种对称性使得余弦定理便于记忆。

There are two main situations where the cosine rule is used:

余弦定理主要适用于两种情形：

Two sides and the included angle are given (SAS) — use it to find the third side.
All three sides are given (SSS) — use it to find any angle.

已知两边及其夹角（SAS）—— 利用它求第三边。
已知三边（SSS）—— 利用它求任意角。

2. Deriving the Cosine Rule | 余弦定理的推导

To understand the cosine rule fully, it helps to see where it comes from. Consider triangle ABC, and drop a perpendicular from vertex B to side AC, meeting AC at point D. Let AD = x, so DC = b − x.

为了充分理解余弦定理，了解其来源很有帮助。考虑三角形ABC，从顶点B向边AC作垂线，垂足为D。设AD = x，则DC = b − x。

In right-angled triangle ABD, we have:

在直角三角形ABD中，我们有：

x = c·cos A

BD = c·sin A

In right-angled triangle BDC, applying Pythagoras’ theorem:

在直角三角形BDC中，应用勾股定理：

a² = BD² + DC² = (c·sin A)² + (b − c·cos A)²

Expanding and simplifying:

展开并化简：

a² = c²·sin²A + b² − 2bc·cos A + c²·cos²A

a² = b² + c²(sin²A + cos²A) − 2bc·cos A

Since sin²A + cos²A = 1, we arrive at the cosine rule. This derivation also shows why the rule works for obtuse angles: cos A is negative, and the perpendicular simply falls outside the triangle.

因为sin²A + cos²A = 1，我们便得到了余弦定理。这个推导也解释了为何该定理对钝角同样适用：cos A为负，垂足将落在三角形外部。

3. Finding the Third Side (SAS) | 已知两边及夹角求第三边

The most direct application of the cosine rule is to find the length of a side when two sides and the angle between them are known.

余弦定理最直接的应用是：已知两边及其夹角时，求第三边的长度。

Example: In triangle ABC, AB = 7 cm, AC = 5 cm, and angle A = 60°. Find BC.

例题：在三角形ABC中，AB = 7 cm，AC = 5 cm，角A = 60°。求BC。

Here b = 7, c = 5, and angle A = 60°. Applying the rule:

这里b = 7，c = 5，角A = 60°。代入公式：

a² = 7² + 5² − 2(7)(5)cos 60°

a² = 49 + 25 − 70(0.5) = 74 − 35 = 39

a = √39 ≈ 6.245 cm

Always give your final answer to an appropriate degree of accuracy — usually 3 significant figures in the exam unless the question states otherwise.

最终答案通常保留适当的精确度——考试中一般保留3位有效数字，除非题目另有要求。

4. Finding an 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 an angle:

当已知三边时，我们重新排列余弦定理来求某一角的余弦值：

cos A = (b² + c² − a²) / (2bc)

Example: A triangle has sides 6 cm, 8 cm and 10 cm. Find the largest angle.

例题：一个三角形的三边分别为6 cm、8 cm和10 cm。求最大角。

The largest angle is opposite the longest side, so a = 10, b = 8, c = 6.

最大角对应最长的边，因此设a = 10，b = 8，c = 6。

cos A = (8² + 6² − 10²) / (2 × 8 × 6)

cos A = (64 + 36 − 100) / 96 = 0 / 96 = 0

A = 90°

This confirms the well-known 6-8-10 right-angled triangle. The cosine rule works seamlessly even for right-angled triangles, degenerating into Pythagoras’ theorem when the angle is 90°.

这验证了著名的6-8-10直角三角形。余弦定理对于直角三角形同样适用，当角为90°时，定理退化为勾股定理。

When finding an angle, be careful:

求角时需要特别注意：

If cos A is positive, A is acute (0° < A < 90°).
If cos A is negative, A is obtuse (90° < A < 180°).
Use the inverse cosine function on your calculator: cos⁻¹.

若cos A为正，则A为锐角（0° < A < 90°）。
若cos A为负，则A为钝角（90° < A < 180°）。
使用计算器上的反余弦函数：cos⁻¹。

5. Cosine Rule or Sine Rule | 选用余弦定理还是正弦定理

A frequent question in exams is which rule to apply in a given situation. The decision chart is straightforward:

考试中的常见问题是：在给定情境下应该运用哪个定理。决策依据非常简单：

Given Information	Rule to Use	What to Find
Two sides and included angle (SAS)	Cosine rule	Third side
Three sides (SSS)	Cosine rule	Any angle
Two angles and one side (AAS or ASA)	Sine rule	Unknown side
Two sides and a non-included angle (SSA)	Sine rule	Missing angle (check ambiguous case)

已知条件	适用定理	所求量
两边及其夹角（SAS）	余弦定理	第三边
三边（SSS）	余弦定理	任意角
两角及一边（AAS或ASA）	正弦定理	未知边
两边及其中一边的对角（SSA）	正弦定理	未知角（注意模糊情形）

In short: use the cosine rule when the given information includes the angle between two known sides, or when you know all three sides. Use the sine rule in all other cases.

简言之：当已知两边及其夹角，或已知三边时使用余弦定理；其余情况使用正弦定理。

6. Real-World Applications | 实际应用

The cosine rule is widely used in real-life problems involving distances and directions. The Edexcel exam often frames these as bearings or navigation questions.

余弦定理广泛应用于涉及距离与方向的实际问题中。Edexcel考试常以方位角或航行问题为背景进行考查。

Example — Navigation: A ship sails 12 km on a bearing of 040°, then turns and sails 9 km on a bearing of 130°. Find the ship’s distance from its starting point.

例题——航行问题：一艘船沿方位角040°航行12 km，然后转向沿方位角130°航行9 km。求该船离出发点的距离。

The angle between the two paths is 130° − 40° = 90°, but more complex problems give non-right angles. Suppose instead the ship turns such that the angle between the two paths is 70°. Then the distance d from the start is:

两条路径之间的夹角为130° − 40° = 90°，但更复杂的题目会给出非直角。假设船转向后两条路径之间的夹角为70°，则起点到终点的距离d为：

d² = 12² + 9² − 2(12)(9)cos 70°

d² = 144 + 81 − 216(0.342) ≈ 225 − 73.87 = 151.13

d ≈ √151.13 ≈ 12.3 km

This approach works for any two vectors with a known angle between them — a key idea in resolving forces and velocities in mechanics as well.

这种方法适用于任意两个已知夹角的分量——这也是力学中分解力和速度的关键思想。

7. Combining with the Area Formula | 结合面积公式

For any triangle, the area is given by:

对于任意三角形，其面积公式为：

Area = ½·ab·sin C

where C is the angle between sides a and b. Once we find an angle using the cosine rule, we can immediately compute the area without dropping perpendiculars.

其中C是边a与b之间的夹角。一旦用余弦定理求出角，就可以立即计算面积，而无需作高。

Example: A triangle has sides a = 8, b = 11, and included angle C = 47°. Find the area.

例题：三角形的两边a = 8，b = 11，夹角C = 47°。求面积。

Area = ½(8)(11)sin 47° ≈ 44(0.731) ≈ 32.2 square units

In exam questions, you may need the cosine rule first to obtain a side, then the area formula. Read the question carefully to determine the required order of operations.

在考试中，你可能需要先用余弦定理求出一条边，再用面积公式。请仔细审题，确定运算的先后顺序。

8. Common Mistakes and Exam Tips | 常见错误与考试技巧

Students often lose marks on cosine rule questions due to avoidable errors. Here are the most frequent pitfalls:

学生在余弦定理题目中常因可避免的错误而失分。以下是最常见的陷阱：

Mislabeling sides: Side a must be opposite angle A. If you swap labels, your answer will be wrong.
Calculator mode: Ensure your calculator is in degree mode when angles are given in degrees.
Rounding too early: Keep full accuracy in intermediate steps; round only at the final answer.
Sign of cos: For obtuse angles, cos is negative. The term −2bc·cos A then adds to b² + c², making the side longer.
Forgetting brackets: When rearranging, remember that 2bc is a single denominator in the angle version.

标记错误：边a必须与角A相对。如果互换标记，答案将出错。
计算器模式：当角度以度为单位时，请确保计算器处于角度制模式。
过早四舍五入：中间步骤保留完整精度，仅在最终答案中四舍五入。
cos的符号：钝角的cos为负，此时−2bc·cos A将对b² + c²产生加成，使边更长。
遗漏括号：在变形公式时，记住求角版本中2bc是完整的分母。

A useful exam technique: when using the cosine rule to find an angle, always check whether your answer is geometrically sensible. The largest angle must be opposite the longest side.

一个实用的考试技巧：用余弦定理求角时，始终检查答案在几何上是否合理。最大的角必须对应最长的边。

9. Worked Exam Examples | 真题演练

Problem 1: In triangle PQR, PQ = 15 cm, PR = 10 cm, and angle QPR = 120°. Find the length QR.

题目1：在三角形PQR中，PQ = 15 cm，PR = 10 cm，角QPR = 120°。求QR的长度。

Let q = QR, p = 15, r = 10, and angle P = 120°. Using the cosine rule:

设q = QR，p = 15，r = 10，角P = 120°。应用余弦定理：

q² = 15² + 10² − 2(15)(10)cos 120°

q² = 225 + 100 − 300(−0.5)

q² = 325 + 150 = 475

q = √475 ≈ 21.8 cm

Notice how the negative cosine makes the side longer than it would be for a 60° angle — the triangle is stretched by the obtuse angle.

注意，负的余弦值使得边比60°角时更长——三角形因钝角而“拉伸”。

Problem 2: A triangle has sides 5 cm, 7 cm and 9 cm. Find the smallest angle.

题目2：三角形的三边为5 cm、7 cm和9 cm。求最小角。

The smallest angle is opposite the shortest side, so set a = 5, b = 7, c = 9.

最小角对应最短边，因此设a = 5，b = 7，c = 9。

cos A = (7² + 9² − 5²) / (2 × 7 × 9)

cos A = (49 + 81 − 25) / 126 = 105 / 126 ≈ 0.8333

A = cos⁻¹(0.8333) ≈ 33.6°

10. Summary | 小结

The cosine rule is a versatile formula that extends trigonometry beyond right-angled triangles. It is essential for solving SAS and SSS triangle problems, and it appears throughout the Edexcel A-Level Mathematics papers — from pure maths questions to mechanics and everyday contextual problems.

余弦定理是一个通用公式，将三角学从直角三角形拓展到任意三角形。它是解决SAS和SSS三角形问题的关键工具，贯穿Edexcel A-Level数学试卷——从纯数学题到力学以及日常情境问题。

Remember the core formula a² = b² + c² − 2bc·cos A, its rearranged form for angles, and the conditions for using it. With practice, choosing between the sine rule and the cosine rule becomes second nature, and you will approach triangle problems with confidence.

牢记核心公式a² = b² + c² − 2bc·cos A、用于求角的变形形式及其适用条件。通过练习，在正弦定理与余弦定理之间做出选择将变得驾轻就熟，你也能自信地应对各类三角形问题。

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A-Level Maths: The Cosine Rule and Its Applications | A-Level数学：余弦定理及其应用