📚 The Cosine Rule and Its Applications | 余弦定理及应用
The cosine rule is one of the most powerful tools in trigonometry. It generalises the Pythagorean theorem to all triangles, providing a direct way to relate side lengths to angles. In this article we explore its statement, proof, and practical applications, with worked examples tailored for A-level students.
余弦定理是三角学中最强大的工具之一。它将勾股定理推广到任意三角形，直接建立边长与角之间的联系。本文将探讨其公式、推导、实际应用，并结合适合 A-level 学生的例题进行讲解。
1. Introducing the Cosine Rule | 余弦定理的引入
The cosine rule, also called the law of cosines, is used to solve triangles that do not contain a right angle. It connects the length of one side of a triangle to the other two sides and the cosine of the angle opposite the first side.
余弦定理，也称为余弦定律，用于解不含直角的三角形。它将三角形一条边的长度与另外两条边以及该边所对角（即这条边对面的角）的余弦联系起来。
It is especially useful when we know two sides and the included angle, or when we know all three sides and need to find an angle.
当我们已知两边及其夹角，或已知三边需要求角时，它特别有用。
2. The Formula | 公式
For any triangle with sides a, b, c opposite angles A, B, C respectively, the cosine rule states:
对于任意三角形，设边 a、b、c 分别对应角 A、B、C，余弦定理表示为：
a² = b² + c² − 2bc cos A
Similarly: b² = c² + a² − 2ca cos B and c² = a² + b² − 2ab cos C.
同理：b² = c² + a² − 2ca cos B，c² = a² + b² − 2ab cos C。
3. Proving the Cosine Rule | 证明余弦定理
We prove the rule for side a. Consider triangle ABC with sides as usual.
我们对边 a 证明余弦定理。考虑三角形 ABC，边长按通常方式标记。
Place A at the origin, B on the positive x-axis at (c, 0), and C at (b cos A, b sin A).
将点 A 置于原点，B 放在 x 轴正半轴上的 (c, 0)，C 的坐标为 (b cos A, b sin A)。
Then the distance BC is a, so:
于是边 BC 的长度为 a，因此：
a² = (b cos A − c)² + (b sin A)²
Expanding the right-hand side:
将右边展开：
a² = b² cos² A − 2bc cos A + c² + b² sin² A
Using cos² A + sin² A = 1:
利用 cos² A + sin² A = 1：
a² = b² + c² − 2bc cos A
Similar arguments give the other two versions.
同理可得到另外两个形式。
4. Finding a Side When Two Sides and the Included Angle Are Known | 已知两边及夹角求第三边
If two sides and the angle between them are known, the third side can be found directly using the cosine rule.
若已知两边及其夹角，可直接用余弦定理求出第三边。
Example: b = 5, c = 7 and A = 60°. Then:
例如：b = 5，c = 7，A = 60°。则：
a² = 5² + 7² − 2 × 5 × 7 × cos 60° = 25 + 49 − 35 = 39
Therefore a = √39 ≈ 6.245.
因此 a = √39 ≈ 6.245。
Notice that the side we find is opposite the given angle; this avoids ambiguous cases that appear with the sine rule.
注意，我们求出的边是给定角所对的边；这避免了正弦定理可能产生的两解歧义情况。
5. Finding an Angle When Three Sides Are Known | 已知三边求角
Rearranging the cosine rule allows us to compute any angle from three known sides.
将余弦定理变形后，可以在已知三边的情况下计算任意一个角。
From a² = b² + c² − 2bc cos A, we get:
由 a² = b² + c² − 2bc cos A，可得：
cos A = (b² + c² − a²) / (2bc)
Similarly:
类似地：
cos B = (c² + a² − b²) / (2ca)，cos C = (a² + b² − c²) / (2ab)
Example: a = 7, b = 8, c = 9. The largest angle is C opposite c = 9:
例：a = 7，b = 8，c = 9。最大角为 c = 9 所对的角 C：
cos C = (7² + 8² − 9²) / (2 × 7 × 8) = (49 + 64 − 81) / 112 = 32 / 112 = 2/7
So C ≈ 73.4°.
因此 C ≈ 73.4°。
6. Determining the Shape of a Triangle | 判断三角形形状
The cosine rule can detect whether a triangle is acute, right, or obtuse without calculating the angle.
余弦定理无需实际计算角度，就能判断三角形是锐角三角形、直角三角形还是钝角三角形。
Compare a² with b² + c² for the largest side a:
对于最大边 a，比较 a² 与 b² + c²：
If a² = b² + c², then A = 90°, so the triangle is right-angled.
若 a² = b² + c²，则 A = 90°，因此三角形为直角三角形。
If a² > b² + c², then A > 90°, so the triangle is obtuse.
若 a² > b² + c²，则 A > 90°，因此三角形为钝角三角形。
If a² < b² + c², then A < 90°, and if this holds for all sides, the triangle is acute.
若 a² < b² + c²，则 A < 90°；若所有边均满足此条件，则三角形为锐角三角形。
Because A is opposite the largest side, checking only the largest side is enough for oblique triangles.
因为 A 是最大边所对的角，对于斜三角形，只需检查最大边即可。
7. Applying the Cosine Rule in Real-World Contexts | 余弦定理的实际应用
In real situations such as navigation, surveying and architecture, distances cannot always be measured directly. A triangle can often be constructed using two known distances and the angle between them.
在航海、测量和建筑等实际情境中，距离往往无法直接测量。通常可以利用两个已知距离及其夹角构造三角形。
For example, two ships leave a port at different bearings. The distance between them after a certain time can be obtained from the cosine rule.
例如，两艘船从同一港口沿不同方位航行，经过一段时间后，它们之间的距离可由余弦定理求出。
Another common problem is finding the width of a river by measuring a baseline and an angle from each endpoint.
另一个常见问题是通过测量基线和两端点处的角度来求河宽。
8. The Vector Form of the Cosine Rule | 余弦定理的向量形式
The cosine rule also appears in vector algebra. For two vectors u and v, the length of their difference is given by:
余弦定理同样出现在向量代数中。对于两个向量 u 和 v，它们的差向量的长度满足：
|u − v|² = |u|² + |v|² −
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