📚 The Cosine Rule | 余弦定理
The cosine rule is a fundamental theorem in trigonometry that extends the Pythagorean theorem to any triangle. It allows you to calculate an unknown side or angle when you are given sufficient information, particularly in non-right-angled triangles. For Edexcel A-Level Mathematics, mastering the cosine rule is essential for solving triangles, proving identities, and tackling vector problems.
余弦定理是三角学中的一个基本定理，它将勾股定理推广到任意三角形。当已知足够的信息时，特别是在非直角三角形中，你可以用它来计算未知的边或角。对于 Edexcel A-Level 数学来说，掌握余弦定理是解三角形、证明恒等式和解决向量问题的基础。
1. The Cosine Rule Formula | 余弦定理公式
For any triangle with sides a, b, c and opposite angles A, B, C respectively, the cosine rule states:
对于任意三角形，其边长 a、b、c 分别对应角 A、B、C，余弦定理可表述为：
a² = b² + c² − 2bc cos A
This relates side a to the other two sides and the included angle A, which is the angle between sides b and c. Notice that the side on the left is squared, and the term 2bc cos A always uses the two sides that form the angle.
该式将边 a 与另外两边及其夹角 A 联系起来，其中 A 是边 b 与边 c 之间的夹角。注意等式左边的边取平方，而 2bc cos A 这一项使用的正是构成该角的两条边。
There are two equivalent forms for the other sides:
其他两边也有两个等价形式：
b² = a² + c² − 2ac cos B
c² = a² + b² − 2ab cos C
When you need to find an angle, the formula can be rearranged to isolate the cosine term:
当需要求角度时，可将公式变形，分离出余弦项：
cos A = (b² + c² − a²) ÷ (2bc)
b² + c² − a²) / (2bc). The same rearrangement works for cos B and cos C, and these forms are extremely useful in exam questions.
写成繁分数形式即 cos A = (b² + c² − a²) / (2bc)。对 cos B 和 cos C 可以做同样的变形，这些形式在考试题目中非常实用。
2. Triangle Labelling Conventions | 三角形标记约定
Before applying the cosine rule, you must label the triangle consistently. The side opposite angle A is always labelled a, opposite B is b, and opposite C is c. The angle between sides b and c is A, between a and c is B, and between a and b is C.
在应用余弦定理之前，必须一致地标记三角形。角 A 的对边总是标为 a，角 B 的对边标为 b，角 C 的对边标为 c。边 b 和 c 之间的夹角是 A，边 a 和 c 之间的夹角是 B，边 a 和 b 之间的夹角是 C。
In many Edexcel problems, the triangle is already labelled for you, so the first step is simply to check which side is opposite which angle. If the labels do not match the convention, rename the vertices yourself before substituting any numbers.
在许多 Edexcel 题目中，三角形的标记已经给出，因此第一步只需检查哪条边对应哪个角。如果标记不符合约定，请在代入任何数值之前自行重新命名顶点。
| Angle 角 | Opposite side 对边 | Included sides 夹边 |
| A | a | b and c |
| B | b | a and c |
| C | c | a and b |
A common exam slip is to pair a side with an angle that is not opposite to it. Always sketch the triangle, write the letters in capitals for angles and lower case for sides, and only then substitute.
常见的考试失误是把一条边与并非其对角的角配对。请务必先画出三角形，用大写字母表示角、小写字母表示边，然后再代入数值。
3. Where the Cosine Rule Comes From: A Proof | 余弦定理的来源：证明
Understanding the proof helps you remember the formula and gives you a reliable method for "show that" the origin, side AB along the x-axis, and let AB = c and AC = b with angle A between them.
理解证明过程有助于记住公式，也能为"证明题"提供可靠方法。把三角形放在坐标系中，让顶点 A 位于原点，边 AB 沿 x 轴，设 AB = c、AC = b，两者夹角为 A。
Then the coordinates of the vertices are A(0, 0), B(c, 0) and C(b cos A, b sin A). The length a is the distance BC, so by the distance formula:
于是三个顶点的坐标为 A(0, 0)、B(c, 0)、C(b cos A, b sin A)。长度 a 就是 BC 的距离，由两点间距离公式得：
a² = (b cos A − c)² + (b sin A)²
sin²A = 1, this simplifies to a² = b² + c² − 2bc cos A, which is exactly the cosine rule.
展开括号得 a² = b² cos²A − 2bc cos A + c² + b² sin²A。由于 cos²A + sin²A = 1，化简后即得 a² = b² + c² − 2bc cos A，这正是余弦定理。
An alternative proof drops a perpendicular from C to AB and applies Pythagoras twice. Both proofs are accepted in Edexcel examinations, and the coordinate proof is usually quicker to reproduce under time pressure.
另一种证法是从 C 向 AB 作垂线，两次使用勾股定理。两种证明在 Edexcel 考试中都被接受，而在时间紧张时，坐标法通常更快写出。
4. When to Use the Cosine Rule | 何时使用余弦定理
The cosine rule is used in exactly two situations. The first is when you know two sides and the included angle (SAS) and you want the third side. The second is when you know all three sides (SSS) and you want an angle.
余弦定理只在两种情形下使用。第一，已知两边及其夹角（SAS），要求第三边。第二，已知三边（SSS），要求某个角。
- SAS: two sides and the angle between them → find the opposite side. 已知两边及夹角 → 求对边。
- SSS: all three sides → find any angle. 已知三边 → 求任意角。
- Sine rule territory: two angles plus a side, or two sides plus a non-included angle. 正弦定理适用：两角加一边，或两边加非夹角。
If the information you are given describes the sine rule, do not force the cosine rule into the problem: you will either create extra work or make an algebraic error.
如果题目给出的信息属于正弦定理的适用范围，就不要硬套余弦定理：否则要么增加计算量，要么造成代数错误。
| Given 已知 | Use 使用 | Find 求 |
| Two sides + included angle 两边加夹角 | Cosine rule 余弦定理 | Third side 第三边 |
| Three sides 三边 | Cosine rule 余弦定理 | Any angle 任意角 |
| Two sides + non-included angle 两边加非夹角 | Sine rule 正弦定理 | Angle or side 角或边 |
| Two angles + a side 两角加一边 | Sine rule 正弦定理 | Remaining sides 其余边 |
5. Finding a Missing Side: Worked Example | 求未知边：例题
= 7 cm, c = 9 cm and the included angle A = 52°. Substitute into a² = b² + c² − 2bc cos A:
设三角形中 b = 7 cm，c = 9 cm，夹角 A = 52°。代入 a² = b² + c² − 2bc cos A：
a² = 7² + 9² − 2 × 7 × 9 × cos 52°
Now a² = 49 + 81 − 126 × 0.6157 = 130 − 77.58 = 52.42, so a = √52.42 ≈ 7.24 cm to three significant figures.
于是 a² = 49 + 81 − 126 × 0.6157 = 130 − 77.58 = 52.42，故 a = √52.42 ≈ 7.24 cm（保留三位有效数字）。
Keep the full calculator value in the display before square-rooting, and only round at the very end. Rounding the cosine term too early is one of the most frequent sources of lost marks in A-Level papers.
在开平方之前，请保留计算器显示的完整数值，只在最后一步取近似值。过早对余弦项取近似，是 A-Level 试卷中最常见的失分原因之一。
6. Finding a Missing Angle: Worked Example | 求未知角：例题
= 6 cm and c = 7 cm. To find angle A, the angle opposite side a, use the rearranged form:
某三角形三边为 a = 8 cm，b = 6 cm，c = 7 cm。要求边 a 的对角 A，使用变形公式：
cos A = (b² + c² − a²) ÷ (2bc) = (36 + 49 − 64) ÷ (84) = 21 ÷ 84 = 0.25
Therefore A = cos⁻¹(0.25) ≈ 75.5°. To find B, apply the same form with the roles rotated:
因此 A = cos⁻¹(0.25) ≈ 75.5°。要求 B，可用同样公式并轮换角色：
cos B = (a² + c² − b²) ÷ (2ac) = (64 + 49 − 36) ÷ (112) = 0.6875
Because cos is one-to-one on the interval 0° to 180°, the inverse cosine gives exactly one valid angle with no ambiguous case.
所以 B ≈ 46.6°，再由三角形内角和得 C ≈ 180° − 75.5° − 46.6° = 57.9°。由于 cos 在 0° 到 180° 区间内是一一对应的，反余弦只会给出唯一有效角，不存在"多解"问题。
7. The Cosine Rule and Pythagoras | 余弦定理与勾股定理
agorean theorem. The cosine rule is therefore a strict generalisation: Pythagoras is the special case where the included angle is a right angle.
若 A = 90°，则 cos A = 0，公式退化为 a² = b² + c²，即勾股定理。因此余弦定理是一个严格的推广：勾股定理正是夹角为直角时的特例。
The sign of cos A also tells you the shape of the triangle, which is a favourite exam question. If A is obtuse then cos A is negative, so the − 2bc cos A term becomes positive and a² is larger than b² + c².
cos A 的正负还能告诉你三角形的形状，这是考试常考的一点。若 A 为钝角，则 cos A 为负，− 2bc cos A 这一项变为正，于是 a² 大于 b² + c²。
- A = 90°: cos A = 0 and a² = b² + c² (right-angled). 直角：a² = b² + c²。
- A < 90°: cos A > 0 and a² < b² + c² (acute). 锐角：a² < b² + c²。
- A > 90°: cos A < 0 and a² > b² + c² (obtuse). 钝角：a² > b² + c²。
Remember that in an obtuse triangle the obtuse angle must be opposite the longest side, so always test the largest side against the other two when classifying a triangle.
要注意，钝角三角形中钝角必定对着最长边，因此在判断三角形形状时，应始终用最长边与另外两边进行比较。
8. Exact Values and Non-Calculator Questions | 特殊角与不可使用计算器的题目
Edexcel sometimes sets the cosine rule in a non-calculator paper. In that case the included angle will almost certainly be 60°, 90°, 120° or 45°, because those angles have exact cosine values that you are expected to know.
Edexcel 有时会在不允许使用计算器的试卷中考查余弦定理。此时夹角几乎必然是 60°、90°、120° 或 45°，因为这些角的余弦值有精确值，属于必须掌握的内容。
cos 60° = ½, cos 90° = 0, cos 120° = −½, cos 45° = √2 ÷ 2
. If instead A = 120°, then a² = 25 + 64 − 80 × (−½) = 89 + 40 = 129, so a = √129 ≈ 11.36.
例如 b = 5、c = 8、A = 60° 时，a² = 25 + 64 − 2 × 5 × 8 × ½ = 89 − 40 = 49，故 a = 7（精确值）。若 A = 120°，则 a² = 25 + 64 − 80 × (−½) = 89 + 40 = 129，故 a = √129 ≈ 11.36。
The two answers above illustrate the effect of an obtuse angle clearly: the same two sides produce a longer third side when the included angle increases from 60° to 120°.
上面两个结果清楚展示了钝角的影响：同样的两条边，当夹角从 60° 增大到 120° 时，第三边变长了。
9. Bearings, Navigation and Real-World Problems | 方位角、导航与实际应用
Cosine rule questions often appear in disguise. A typical setting is a ship or an aeroplane travelling two legs of a journey, where you must find the direct distance from the starting point to the finishing point.
余弦定理的题目常常以"伪装"的形式出现。典型情境是轮船或飞机分两段航行，需要求起点到终点的直线距离。
oint R. The turn at Q is 115° − 40° = 75°, so the interior angle PQR is 180° − 75° = 105°.
假设一艘船沿方位角 040° 航行 12 km 到达点 Q，再沿方位角 115° 航行 9 km 到达点 R。在 Q 处的转向角为 115° − 40° = 75°，因此三角形内角 PQR = 180° − 75° = 105°。
PR² = 12² + 9² − 2 × 12 × 9 × cos 105° = 225 + 55.9 = 280.9
diagram with north lines, mark the interior angle at the turning point, and check that your angle is between the two known sides before substituting.
因此 PR ≈ 16.8 km。请务必画出带北向线的清晰示意图，标出转弯处的内角，并在代入前确认该角确实是两条已知边之间的夹角。
10. Vectors and the Cosine Rule | 向量与余弦定理
The cosine rule is really a statement about vectors. If two vectors b and c have magnitudes |b| and |c|
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