Edexcel IGCSE Mathematics: The Cosine Rule and Its Applications | Edexcel IGCSE数学：余弦定理及其应用

📚 Edexcel IGCSE Mathematics: The Cosine Rule and Its Applications | Edexcel IGCSE数学：余弦定理及其应用

The cosine rule is a fundamental tool in trigonometry for solving non-right-angled triangles. In the Edexcel IGCSE syllabus, it appears both in Paper 1 and Paper 2, often combined with the sine rule, area formulas, and practical geometry problems. This article provides a complete, exam-focused guide to the cosine rule, its derivation, applications, and common pitfalls.

余弦定理是解非直角三角形时不可或缺的三角学工具。在 Edexcel IGCSE 考纲中，它在 Paper 1 和 Paper 2 中都会出现，常与正弦定理、面积公式以及实际几何问题结合考查。本文为同学们提供一份完整、紧扣考点的余弦定理指南，包括公式、推导、应用和常见错误。

1. What Is the Cosine Rule? | 余弦定理简介

The cosine rule connects the lengths of three sides of a triangle to the cosine of one of its angles. It generalises Pythagoras’ theorem to any triangle, not just right-angled ones. When you know two sides and the included angle, or three sides, the cosine rule gives you the missing measurement directly.

余弦定理将三角形的三条边与其中一个角的余弦值联系起来。它把勾股定理推广到任意三角形，而不仅仅适用于直角三角形。当你知道两条边及其夹角，或知道三条边时，余弦定理可以直接求出未知量。

For a triangle with sides \(a\), \(b\), \(c\) and angles \(A\), \(B\), \(C\) opposite those sides respectively, the cosine rule states:

对于边长分别为 \(a\)、\(b\)、\(c\)，且对应角分别为 \(A\)、\(B\)、\(C\) 的三角形，余弦定理表述为：

a² = b² + c² − 2bc cos A

Equivalently, you can rearrange it to find an angle:

等价地，可以重新排列来求角：

cos A = (b² + c² − a²) / 2bc

Here, angle \(A\) is always opposite side \(a\). This notation is consistent throughout the Edexcel IGCSE exams.

这里，角 \(A\) 始终是边 \(a\) 的对角。这种符号约定在 Edexcel IGCSE 考试中保持一致。

2. Formula and Notation | 公式与符号

In the Edexcel formula sheet, the cosine rule is given as:

在 Edexcel 公式表中，余弦定理以如下形式给出：

a² = b² + c² − 2bc cos A

You may also see it written in cyclic forms:

你还会看到它的循环形式：

b² = a² + c² − 2ac cos B
c² = a² + b² − 2ab cos C

Notice the pattern: the side on the left is opposite the angle on the right. When you are finding a side, use the version that places the unknown side on the left. When you are finding an angle, use the rearranged formula.

注意规律：左边的边对应右边的角。求边长时，把未知边放在左边；求角度时，使用重排后的公式。

Always label your triangle carefully. Write down the values of \(a\), \(b\), \(c\), \(A\), \(B\), \(C\) before substituting. This prevents sign errors and keeps your working clear.

务必仔细标注三角形。代入前先写下 \(a\)、\(b\)、\(c\)、\(A\)、\(B\)、\(C\) 的值，这样可以避免符号错误，让过程更清晰。

3. Understanding the Formula | 理解公式

The cosine rule is a generalisation of Pythagoras’ theorem. If angle \(A = 90°\), then \(\cos 90° = 0\), and the formula becomes \(a² = b² + c²\), which is exactly Pythagoras’ theorem.

余弦定理是勾股定理的推广。如果角 \(A = 90°\)，则 \(\cos 90° = 0\)，公式变为 \(a² = b² + c²\)，正是勾股定理。

Why does the extra term \(-2bc \cos A\) appear? When \(A\) is acute, \(\cos A > 0\), so the side \(a\) is shorter than the simple sum \(b² + c²\). When \(A\) is obtuse, \(\cos A < 0\), so the term tends to be positive, making \(a²\) larger. This matches the geometric fact that an obtuse angle opens up a longer opposite side.

为什么会出现额外的 \(-2bc \cos A\) 项？当 \(A\) 为锐角时，\(\cos A > 0\)，所以边 \(a\) 比简单的 \(b² + c²\) 要小。当 \(A\) 为钝角时，\(\cos A < 0\)，因此这一项变为正数，使 \(a²\) 更大。这符合几何事实：钝角对应更长的对边。

You do not need to prove the cosine rule in the exam, but understanding its connection to Pythagoras helps you remember it and spot when to apply it.

考试中不需要证明余弦定理，但理解它与勾股定理的联系有助于记忆，并能帮你判断何时使用。

4. When to Use the Cosine Rule | 何时使用余弦定理

The cosine rule is used in two main situations:

余弦定理主要有两种应用场景：

Two sides and the included angle (SAS): You are given sides \(b\) and \(c\) and the included angle \(A\). Use \(a² = b² + c² − 2bc \cos A\) to find the opposite side \(a\).
Three sides (SSS): You are given all three sides. Use \(\cos A = (b² + c² − a²) / 2bc\) to find any angle.

两边及其夹角（SAS）：已知边 \(b\)、\(c\) 和夹角 \(A\)，使用 \(a² = b² + c² − 2bc \cos A\) 求对边 \(a\)。
三边（SSS）：已知三条边，使用 \(\cos A = (b² + c² − a²) / 2bc\) 求任意角。

If you have two angles and one side (AAS or ASA), the sine rule is usually more direct. If you have two sides and a non-included angle (SSA), the situation is ambiguous and the sine rule may give two possible answers; the cosine rule can also be used to check.

如果已知两角一边（AAS 或 ASA），通常用正弦定理更直接。如果已知两边和其中一边的对角（SSA），情况存在歧义，正弦定理可能给出两个答案；此时也可以用余弦定理来检验。

5. Finding a Side Length | 求边长

Example 1: In triangle \(ABC\), \(AB = 8 \text{ cm}\), \(AC = 6 \text{ cm}\), and angle \(A = 60°\). Find \(BC\).

例 1：在三角形 \(ABC\) 中，\(AB = 8 \text{ cm}\)，\(AC = 6 \text{ cm}\)，角 \(A = 60°\)。求 \(BC\)。

Here \(BC\) is opposite angle \(A\), so let \(a = BC\), \(b = AC = 6\), \(c = AB = 8\), \(A = 60°\). Substitute into the formula:

这里 \(BC\) 是角 \(A\) 的对边，所以令 \(a = BC\)，\(b = AC = 6\)，\(c = AB = 8\)，\(A = 60°\)。代入公式：

a² = 6² + 8² − 2 × 6 × 8 × cos 60°

Since \(\cos 60° = 0.5\):

因为 \(\cos 60° = 0.5\)：

a² = 36 + 64 − 48 = 52

Therefore \(a = \sqrt{52} ≈ 7.21 \text{ cm}\). Always include units and give your answer to a sensible degree of accuracy, usually 3 significant figures in IGCSE.

因此 \(a = \sqrt{52} ≈ 7.21 \text{ cm}\)。记得写单位，并保留到合理的精度，IGCSE 通常保留 3 位有效数字。

6. Finding an Angle | 求角

Example 2: A triangle has sides \(a = 7\), \(b = 9\), \(c = 11\). Find angle \(A\) opposite side \(a\).

例 2：三角形三边为 \(a = 7\)、\(b = 9\)、\(c = 11\)。求边 \(a\) 的对角 \(A\)。

Use the rearranged formula:

使用重排后的公式：

cos A = (b² + c² − a²) / 2bc = (9² + 11² − 7²) / (2 × 9 × 11)

Calculate the numerator and denominator:

计算分子和分母：

cos A = (81 + 121 − 49) / 198 = 153 / 198 ≈ 0.7727

Then \(A = \cos^{-1}(0.7727) ≈ 39.4°\). Always use the inverse cosine function on your calculator, not a division by cosine.

所以 \(A = \cos^{-1}(0.7727) ≈ 39.4°\)。计算器上要用反余弦函数，而不是除以余弦。

Remember that the cosine rule gives a unique angle between \(0°\) and \(180°\). If the result is negative, the angle is obtuse.

注意余弦定理给出的角度在 \(0°\) 到 \(180°\) 之间是唯一的。如果结果为负数，则该角为钝角。

7. Real-World Applications | 实际应用

The cosine rule appears in many real contexts: navigation, surveying, vector problems, and coordinate geometry.

余弦定理在导航、测量、向量问题和坐标几何等许多实际场景中都有应用。

Navigation example: A ship sails 12 km north, then 9 km on a bearing of \(120°\). How far is it from the starting point?

导航例子：一艘船先向北航行 12 km，然后沿方位角 \(120°\) 航行 9 km。它离起点多远？

The angle between the two paths is \(180° − 120° = 60°\) (since the first path is due north). So the two known sides are 12 and 9, with included angle \(60°\). Let \(x\) be the distance from the start:

两条路径之间的夹角为 \(180° − 120° = 60°\)（因为第一条路径指向正北）。所以两条已知边为 12 和 9，夹角为 \(60°\)。设 \(x\) 为离起点的距离：

x² = 12² + 9² − 2 × 12 × 9 × cos 60° = 144 + 81 − 108 = 117

Thus \(x ≈ 10.8 \text{ km}\).

因此 \(x ≈ 10.8 \text{ km}\)。

Coordinate geometry example: Given two points \(P(1,2)\) and \(Q(4,6)\), you can find the distance using Pythagoras directly, but if you also have a third point with a known angle, the cosine rule helps find an unknown distance.

坐标几何例子：给定两点 \(P(1,2)\) 和 \(Q(4,6)\)，可以直接用勾股定理求距离，但如果还有第三点且已知某个夹角，余弦定理可以帮助求未知距离。

8. Comparing with the Sine Rule | 与正弦定理比较

Both the sine rule and cosine rule solve non-right-angled triangles. The diagram below summarises when to use each.

正弦定理和余弦定理都用于解非直角三角形。下表总结了它们的适用情况。

Given information	Use
Two angles and one side (AAS/ASA)	Sine rule
Two sides and a non-included angle (SSA)	Sine rule first (may be ambiguous)
Two sides and the included angle (SAS)	Cosine rule
Three sides (SSS)	Cosine rule

已知条件	使用
两角一边（AAS/ASA）	正弦定理
两边及其中一边的对角（SSA）	正弦定理优先（可能有歧义）
两边及夹角（SAS）	余弦定理
三边（SSS）	余弦定理

Note that the sine rule can also be turned around, but the cosine rule is especially useful when no angle is known but all three sides are given.

注意正弦定理也可以变式使用，但在未知任何角、只知道三边时，余弦定理特别有用。

9. Common Mistakes | 常见错误

Using the wrong formula form: If you are finding an angle, do not use the side version directly. Rearrange first or use the \(\cos A\) version.
Mislabelling sides: Side \(a\) must be opposite angle \(A\). Forgetting this leads to wrong substitutions.
Calculator in degree mode: IGCSE exams expect degree mode unless otherwise stated. Check your calculator has a small “D” or “DEG” indicator.
Incorrectly handling the \(-2bc\) term: Do not forget the factor 2 in front of \(bc\). A common careless error is writing \(a² = b² + c² − bc \cos A\).
Not taking the square root: When solving for a side, you must take the positive square root. Sometimes students square the sides but forget the final step.

用错公式形式：求角度时不要直接使用边长版本，应先变形或直接使用 \(\cos A\) 的公式。
标错边与角：边 \(a\) 必须与角 \(A\) 相对。标错会导致代入错误。
计算器未调到角度模式：IGCSE 考试默认使用角度制，除非另有说明。检查计算器显示 “D” 或 “DEG”。
忽略 \(-2bc\) 中的 2：忘记 \(bc\) 前的系数 2 是常见失误，例如写成 \(a² = b² + c² − bc \cos A\)。
忘记开方：求边长时最后要取正平方根。有些同学只计算了平方却忘记最后一步。

10. Step-by-Step Problem Solving | 分步解题

Follow these steps for any cosine rule question:

解余弦定理问题时，按照以下步骤：

Sketch the triangle and label the given sides and angles.
Choose the correct form: side formula if finding a side, \(\cos^{-1}\) if finding an angle.
Write the formula down in symbols, then substitute numbers.
Calculate carefully, using bracket buttons on the calculator for the numerator.
If finding a side, square-root at the end; if finding an angle, use \(\cos^{-1}\).
Round to 3 significant figures or the required accuracy. Include units for sides.

画出三角形草图，标出已知的边和角。
选择正确的形式：求边用边长公式，求角用 \(\cos^{-1}\)。
先写出公式符号，再代入数字。
仔细计算，在计算器上用括号包住分子。
求边时最后开方；求角时使用 \(\cos^{-1}\)。
四舍五入到 3 位有效数字或题目要求的精度。边长要写单位。

Worked example: Find the largest angle in a triangle with sides 6 cm, 7 cm, and 8 cm.

完整示例：求边长分别为 6 cm、7 cm 和 8 cm 的三角形中的最大角。

The largest angle is opposite the largest side, 8 cm. Let \(a = 8\), \(b = 6\), \(c = 7\). Then angle \(A\) is the largest angle:

最大边是 8 cm，因此其对角最大。令 \(a = 8\)，\(b = 6\)，\(c = 7\)，则角 \(A\) 为最大角：

cos A = (6² + 7² − 8²) / (2 × 6 × 7) = (36 + 49 − 64) / 84 = 21 / 84 = 0.25

So \(A = \cos^{-1}(0.25) ≈ 75.5°\).

所以 \(A = \cos^{-1}(0.25) ≈ 75.5°\)。

11. Practice Questions | 练习题

Try these exam-style questions on your own:

请独立尝试以下考试风格题目：

Question 1: In triangle \(ABC\), \(AB = 7.5 \text{ cm}\), \(BC = 9.2 \text{ cm}\), angle \(B = 48°\). Find \(AC\).

题 1：三角形 \(ABC\) 中，\(AB = 7.5 \text{ cm}\)，\(BC = 9.2 \text{ cm}\)，角 \(B = 48°\)。求 \(AC\)。

Question 2: A triangle has sides 10, 14, and 18 cm. Calculate the smallest angle.

题 2：三角形三边为 10、14 和 18 cm。求最小的角。

Question 3: Two planes leave an airport at the same time. Plane A flies 400 km at a bearing of \(080°\), plane B flies 350 km at a bearing of \(160°\). Find the distance between the two planes.

题 3：两架飞机同时离开机场。飞机 A 沿方位角 \(080°\) 飞行 400 km，飞机 B 沿方位角 \(160°\) 飞行 350 km。求两架飞机之间的距离。

Answers: 1) \(AC ≈ 6.92 \text{ cm}\) (use \(AC² = 7.5² + 9.2² − 2×7.5×9.2×\cos 48°\)); 2) Smallest side is 10, so \(\cos C = (14² + 18² − 10²)/(2×14×18)\), \(C ≈ 33.6°\); 3) Bearing difference \(80°\), distance \(c² = 400² + 350² − 2×400×350×\cos 80°\), \(c ≈ 483 \text{ km}\).

答案：1) \(AC ≈ 6.92 \text{ cm}\)（使用 \(AC² = 7.5² + 9.2² − 2×7.5×9.2×\cos 48°\)）；2) 最短边为 10，所以 \(\cos C = (14² + 18² − 10²)/(2×14×18)\)，\(C ≈ 33.6°\)；3) 方位角差 \(80°\)，距离 \(c² = 400² + 350² − 2×400×350×\cos 80°\)，\(c ≈ 483 \text{ km}\)。

12. Summary | 总结

The cosine rule is a reliable method for solving triangles when you have SAS or SSS. Memorise the two equivalent forms:

余弦定理是解决 SAS 或 SSS 三角形问题的可靠方法。请牢记两个等价形式：

a² = b² + c² − 2bc cos A and cos A = (b² + c² − a²) / 2bc

Always check your calculator mode, label your triangle correctly, and take the square root when finding a side. With practice, cosine rule questions become quick marks in the exam.

始终检查计算器模式，正确标注三角形，求边时记得开方。通过练习，余弦定理类问题会成为考试中的送分题。

For more revision notes and past-paper practice, visit aleveler.com.

更多复习笔记和真题练习，请访问 aleveler.com。

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Edexcel IGCSE Mathematics: The Cosine Rule and Its Applications | Edexcel IGCSE数学：余弦定理及其应用