📚 The Cosine Rule | 余弦定理
The cosine rule is a fundamental result in A-Level Mathematics that connects the three sides of any triangle with the cosine of one of its angles. It extends Pythagoras' theorem to non-right-angled triangles and is an essential tool for solving triangles when the sine rule cannot be applied directly.
余弦定理是 A-Level 数学中的一个基本结论，它将任意三角形的三条边与其中一个角的余弦联系起来。它把勾股定理推广到非直角三角形，是正弦定理无法直接使用时解三角形的重要工具。
For triangle ABC, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The main form of the cosine rule is shown below.
在三角形 ABC 中，边 a 对角 A，边 b 对角 B，边 c 对角 C。余弦定理的主要形式如下所示。
a² = b² + c² - 2bc cos A
1. What the Cosine Rule Does | 余弦定理的作用
The cosine rule relates the lengths of all three sides of a triangle to the cosine of one angle. It is used when you know either two sides and the included angle (SAS) to find the third side, or all three sides (SSS) to find an angle.
余弦定理将三角形三条边的长度与一个角的余弦联系起来。当你已知两边及其夹角（SAS）求第三边，或已知三边（SSS）求一个角时，就可以使用余弦定理。
Unlike the sine rule, the cosine rule does not require a known opposite pair of side and angle. It works for all triangles, including acute, obtuse, and right-angled triangles.
与正弦定理不同，余弦定理不需要已知一组对边对角。它适用于所有三角形，包括锐角三角形、钝角三角形和直角三角形。
2. Standard Formula and Labelling | 标准公式与边角标记
In Edexcel exam questions, triangles are usually labelled so that side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. This standard convention must be followed carefully before applying the formula.
在 Edexcel 考试题中，三角形通常标记为边 a 对角 A，边 b 对角 B，边 c 对角 C。在应用公式之前，必须仔细遵循这一标准约定。
The cosine rule can be written in three equivalent forms, one for each angle:
余弦定理可以写成三种等价形式，分别对应三个角：
a² = b² + c² - 2bc cos A
b² = c² + a² - 2ca cos B
c² = a² + b² - 2ab cos C
Always identify which side and opposite angle you are working with before substituting values. A common error is to mix up sides such as using b and c with angle A incorrectly.
在代入数值之前，一定要明确你正在处理的是哪条边和哪个对角。一个常见错误是把边与角弄混，例如在角 A 的公式中错误地使用了 b 和 c。
3. Rearranged Forms for Finding Angles | 求角的变形公式
When all three sides are known, the cosine rule is rearranged to find an angle. The three angle forms are:
当已知三条边时，将余弦定理变形即可求出角度。三个求角公式为：
cos A = (b² + c² - a²) / (2bc)
cos B = (c² + a² - b²) / (2ca)
cos C = (a² + b² - c²) / (2ab)
(value). Make sure your calculator is in degree mode unless the question asks for radians.
要求角 A，先计算 cos A 的值，然后使用反余弦函数：A = cos⁻¹(值)。除非题目要求使用弧度，否则确保计算器处于角度模式。
4. Derivation Using Coordinates | 坐标法推导
One clean way to derive the cosine rule is to place the triangle on a coordinate grid. Put vertex A at the origin (0,0), vertex B at (c,0), and vertex C at (b cos A, b sin A).
推导余弦定理的一种简洁方法是把三角形放在坐标系中。将顶点 A 放在原点 (0,0)，顶点 B 放在 (c,0)，顶点 C 放在 (b cos A, b sin A)。
The distance BC is side a, so using the distance formula gives:
线段 BC 的长度就是边 a，因此使用距离公式可得：
a² = (c - b cos A)² + (0 - b sin A)²
Expanding and simplifying:
展开并化简：
a² = c² - 2bc cos A + b² cos² A + b² sin² A
Since sin² A + cos² A = 1, the expression becomes:
因为 sin² A + cos² A = 1，所以表达式变为：
a² = b² + c² - 2bc cos A
This derivation also works when angle A is obtuse if directed coordinates are used, but the final formula remains the same.
如果使用有向坐标，当角 A 为钝角时该推导同样成立，最终公式保持不变。
5. Derivation by Splitting the Triangle | 分割三角形推导
cular be D, and let AD = b cos A, CD = b sin A, and DB = c - b cos A, assuming A is acute.
另一种常见证法是从顶点 C 向边 AB 作垂线。设垂足为 D，并设 AD = b cos A，CD = b sin A，DB = c - b cos A，这里假设角 A 为锐角。
Applying Pythagoras' theorem to the right-angled triangle CDB gives:
对直角三角形 CDB 应用勾股定理：
a² = (b sin A)² + (c - b cos A)²
Expanding:
展开得：
a² = b² sin² A + c² - 2bc cos A + b² cos² A
Using sin² A + cos² A = 1 gives the cosine rule:
利用 sin² A + cos² A = 1 得到余弦定理：
a² = b² + c² - 2bc cos A
For an obtuse angle at A, the perpendicular falls outside the triangle, but the same algebra can be adapted using signed lengths.
如果角 A 为钝角，垂线落在三角形外部，但使用有向长度可以同样进行代数推导。
6. When to Use the Cosine Rule | 何时使用余弦定理
The cosine rule is the best choice for two standard situations. The table below summarises when to use the cosine rule and when the sine rule is more suitable.
余弦定理是两种标准情形下的最佳选择。下表总结了何时使用余弦定理以及何时正弦定理更合适。
| Given information | Method | 中文 |
|---|---|---|
| Two sides and the included angle (SAS) | Cosine rule to find the third side | 两边及其夹角：余弦定理求第三边 |
| Three sides (SSS) | Cosine rule to find an angle | 三边：余弦定理求角 |
| Two angles and one side (AAS or ASA) | Sine rule | 两角一边：正弦定理 |
| Two sides and a non-included angle (SSA) | Sine rule with care, or cosine rule quadratic | 两边及非夹角：正弦定理注意多解，或余弦定理二次方程 |
Remember that for SAS, the angle must be the included angle between the two known sides. If the angle is not included, the sine rule or a quadratic approach may be needed.
请记住，对于 SAS 情形，已知角必须是两条已知边之间的夹角。如果角不是夹角，可能需要使用正弦定理或二次方程方法。
7. Worked Example: Finding a Side (SAS) | 例题：已知两边及其夹角求边
b = 8 cm, c = 5 cm, and included angle A = 60°. Find side a to three significant figures.
一个三角形中，边 b = 8 cm，边 c = 5 cm，夹角 A = 60°。求边 a，保留三位有效数字。
Use the cosine rule:
使用余弦定理：
a² = b² + c² - 2bc cos A
Substitute the values:
代入数值：
a² = 8² + 5² - 2 × 8 × 5 × cos 60°
Since cos 60° = 0.5, this becomes:
因为 cos 60° = 0.5，所以：
a² = 64 + 25 - 80 × 0.5
a² = 89 - 40 = 49
Therefore a = √49 = 7 cm. The third side is exactly 7 cm.
因此 a = √49 = 7 cm。第三边为 7 cm。
8. Worked Example: Finding an Angle (SSS) | 例题：已知三边求角
A triangle has sides a = 7 cm, b = 8 cm, and c = 5 cm. Find angle A to one decimal place.
一个三角形中，边 a = 7 cm，边 b = 8 cm，边 c = 5 cm。求角 A，保留一位小数。
Use the rearranged cosine rule for angle A:
使用求角 A 的余弦定理变形公式：
cos A = (b² + c² - a²) / (2bc)
Substitute the values:
代入数值：
cos A = (8² + 5² - 7²) / (2 × 8 × 5)
cos A = (64 + 25 - 49) / 80
cos A = 40 / 80 = 0.5
Then A = cos⁻¹(0.5) = 60°. Angle A is exactly 60°.
因此 A = cos⁻¹(0.5) = 60°。角 A 为 60°。
ms, if the question asks for the angle in radians, you would give A = π/3 radians instead.
在 Edexcel 考试中，如果题目要求以弧度制表示角度，则应写 A = π/3 弧度。
9. Connection to Pythagoras and Triangle Inequalities | 与勾股定理及三角形不等式的关系
s 90°, then cos A = 0, so the cosine rule becomes a² = b² + c². This is exactly Pythagoras' theorem, showing that the cosine rule is a generalisation of Pythagoras.
如果角 A 为 90°，则 cos A = 0，于是余弦定理变为 a² = b² + c²
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