📚 The Cosine Rule | 余弦定理
The cosine rule, also known as the law of cosines, is one of the most powerful tools in trigonometry and triangle geometry. It extends the famous Pythagorean theorem to any triangle – not only right-angled ones – by introducing the cosine of an angle into the relation between the three sides. In the IB Mathematics curriculum, whether you are following Analysis & Approaches or Applications & Interpretation, mastering the cosine rule is essential for solving non-right-angled triangles in contexts ranging from pure geometry to navigation, construction, and vector analysis.
余弦定理，也称为余弦定律，是三角学和三角形几何中最强有力的工具之一。它将著名的毕达哥拉斯定理推广到任意三角形——而不仅仅是直角三角形——通过在三条边的关系中引入角的余弦。在 IB 数学课程中，无论你修读的是分析与方法还是应用与解释，掌握余弦定理都是解决非直角三角形问题的关键，这些情境涵盖从纯几何到导航、建筑以及向量分析等多个领域。
1. Introduction to the Cosine Rule | 余弦定理简介
The cosine rule states that in any triangle with sides a, b, c and opposite angles A, B, C respectively, the square of one side equals the sum of the squares of the other two sides minus twice the product of those two sides multiplied by the cosine of the included angle.
余弦定理指出，在任意三角形中，设边长分别为 a、b、c，对角分别为 A、B、C，则一边的平方等于另外两边的平方和减去这两边乘积的两倍再乘以它们夹角的余弦。
In other words, it provides a direct algebraic link between the three sides and one angle, allowing us to determine an unknown side or an unknown angle without having to split the triangle into right-angled components.
换句话说，它在三条边与一个角之间建立了直接的代数联系，使我们无需将三角形分割成直角三角形就能求出未知的边或角。
2. Deriving the Cosine Rule | 余弦定理的推导
nate geometry. Place triangle ABC with vertex A at the origin, side c along the positive x‑ in² A. Using the identity cos² A + sin² A = 1 simplifies this to a² = b² + c² − 2bc cos A.
一种经典推导方法使用坐标几何。将三角形 ABC 的顶点 A 置于原点，边 c 沿 x 轴正方向放置，因此 B 点坐标为 (c, 0)。顶点 C 可定位于 (b cos A, b sin A)。那么边 a 就是线段 BC 的长度。应用距离公式可得：a² = (b cos A − c)² + (b sin A − 0)² = b² cos² A − 2bc cos A + c² + b² sin² A。利用恒等式 cos² A + sin² A = 1，便简化为 a² = b² + c² − 2bc cos A。
An alternative derivation drops an altitude from one vertex to split the triangle into two right‑angled triangles. By applying Pythagoras' theorem to both smaller triangles and eliminating the altitude, the cosine rule emerges naturally. This shows that the law of cosines is a generalisation of the Pythagorean relationship.
另一种推导方法是从一个顶点作高，将三角形分割成两个直角三角形。对这两个小三角形分别应用毕达哥拉斯定理，并消去高，余弦定理就自然显现出来。这表明余弦定律是毕达哥拉斯关系的推广。
3. The Cosine Rule Formula and Its Variations | 余弦定理公式及其变体
The standard form of the cosine rule is usually written with respect to side a:
余弦定理的标准形式通常关于边 a 写为：
a² = b² + c² − 2bc · cos A
By symmetry, we also have the corresponding expressions for sides b and c:
由对称性，我们同样有关于边 b 和 c 的相应表达式：
b² = a² + c² − 2ac · cos B
c² = a² + b² − 2ab · cos C
When solving for an angle, it is convenient to rearrange the formula:
当需要求解角度时，将公式重新排列更为方便：
cos A = (b² + c² − a²) / (2bc)
and similarly for cos B and cos C. These rearrangements are the key to unlocking an unknown angle when all three side lengths are known.
对于 cos B 和 cos C 也有类似的式子。当三条边长均已知时，这些变形便是求出未知角度的关键。
4. When to Use the Cosine Rule | 何时使用余弦定理
In triangle problems, you should reach for the cosine rule in two typical cases: (1) when you are given two sides and the included angle (SAS) and need the third side; (2) when you are given all three sides (SSS) and need to find any angle. In both situations, the sine rule alone is insufficient or leads to ambiguity.
在三角形求解问题中，有两种典型情形应使用余弦定理：(1) 已知两边及其夹角（SAS）需求第三边；(2) 已知三边（SSS）需求任意一个角度。在这两种情形下，仅用正弦定理是不够的或会产生多解性问题。
The cosine rule is a deterministic tool: in the SSS case it always yields a unique angle (between 0° and 180°), because the cosine function is one‑to‑one over this interval. This eliminates the ambiguous case that sometimes appears with the sine rule.
余弦定理是一个确定性的工具：在 SSS 情形下，它总是给出一个唯一的角度（在 0° 到 180° 之间），因为余弦函数在这个区间是一一对应的。这消除了正弦定理有时会出现的多解情况。
5. Finding a Side – The SAS Case | 求边——已知两边及其夹角
d A = 60°. Then a² = 8² + 6² − 2×8×6×cos 60° = 64 + 36 − 96×0.5 = 100 − 48 = 52. Hence a = √52 ≈ 7.21 cm (to 3 s.f.).
假设已知边 b 和 c 及其夹角 A。要求边 a，只需代入 a² = b² + c² − 2bc cos A 再开平方根即可。例如，已知 b = 8 cm，c = 6 cm，A = 60°。那么 a² = 8² + 6² − 2×8×6×cos 60° = 64 + 36 − 96×0.5 = 100 − 48 = 52。因此 a = √52 ≈ 7.21 cm（保留三位有效数字）。
It is crucial to use the correct order of operations and ensure your calculator is in degree mode when working with degree measures. The cosine rule works equally well with angles in radians if the problem is set in radian measure.
使用正确的运算顺序至关重要，并且当角度以度为单位时，要确保计算器处于度模式。如果题目中使用弧度制，余弦定理同样适用。
6. Finding an Angle – The SSS Case | 求角——已知三边
− 7²) / (2×5×6) = (25 + 36 − 49) ⁄ 60 = 12 ⁄ 60 = 0.2. Therefore A = arccos(0.2) ≈ 78.5°.
当三条边的长度均已知时，可将余弦定理重新排列以求得任意角度。例如，要求边 a 所对的角 A，可使用 cos A = (b² + c² − a²) / (2bc)。设三角形的边长为 a = 7，b = 5，c = 6。那么 cos A = (5² + 6² − 7²) / (2×5×6) = (25 + 36 − 49) / 60 = 12 / 60 = 0.2。因此 A = arccos(0.2) ≈ 78.5°。
Always check that the fraction lies between −1 and 1 before applying the inverse cosine. If it does not, a triangle with the given sides cannot exist – this is a useful check for triangle inequality violations.
在使用反余弦之前，务必检查分数是否落在 −1 到 1 之间。如果不在，则给定的边长无法构成三角形——这是检验三角形不等式是否违背的有效方法。
7. Choosing Between the Sine and Cosine Rules | 正弦定理与余弦定理的选择
A common question in IB exams asks students to decide which rule to apply. The general guideline is: if you have a side and the angle opposite it plus another side or angle, the sine rule is often quicker. However, if you are given two sides and the included angle (SAS) or three sides (SSS), the cosine rule is the correct starting point. Mixing them up can lead to circular calculations or the ambiguous case.
IB 考试中常见的一类问题是让学生决定适用哪条定理。一般准则是：如果你已知一条边及其对角，再加上另一条边或另一个角，正弦定理通常更快捷。然而，若已知两边及其夹角（SAS）或三边（SSS），余弦定理才是正确的出发点。将两者混淆可能导致循环计算或出现多解情形。
Sometimes both rules are used in sequence – for example, the cosine rule is first applied to find a missing side, then the sine rule finds a second angle, and the third angle is obtained from the 180° sum. This hybrid approach is very common in multi‑step triangle solving.
有时需要按顺序使用两种定理——例如，先用余弦定理求出缺失的边，再用正弦定理求出第二个角，第三个角则由三角形内角和为 180° 求得。这种混合方法在分步求解三角形中非常普遍。
8. Applications in Navigation and Bearings | 在导航与方位角中的应用
In navigation problems, bearings are measured clockwise from north. The cosine rule is ideal for calculating the distance between two points after following a route with two legs and a turn. For instance, a plane flies 100 km on a bearing of 030°, then changes course to 120° and flies 150 km. The angle between the two legs is the difference in bearings, and the straight‑line distance back to the start is found by the cosine rule.
在导航问题中，方位角是从正北方向顺时针测量的。余弦定理非常适合计算飞机经过两段带有转弯的航线后两点间的距离。例如，一架飞机先沿方位 030° 飞行 100 km，然后转向方位 120° 飞行 150 km。两段航程间的夹角即方位角之差，返回起点的直线距离就可用余弦定理求出。
This type of real‑life context not only tests the cosine rule but also asks students to interpret a diagram, convert bearings into interior angles, and apply rounding correctly to the final answer with appropriate units.
这类实际情境不仅考察余弦定理，还要求学生在图中解释信息、将方位角转换为三角形的内角，并在最终答案中正确地进行舍入并带上合适的单位。
9. Vector Proof and Connection to the Dot Product | 向量证明与点积的联系
s u and v, and the third side is u − v (or v − u). The squared length of the third side is |u − v|² = |u|² + |v|² − 2 u · v. By definition, the dot product u · v = |u||v| luded angle of the triangle. Substituting gives the cosine rule: a² = b² + c² − 2bc cos A.
在向量方法中，三角形的两边用向量 u 和 v 表示，第三边则为 u − v（或 v − u）。第三边长度的平方等于 |u − v|² = |u|² + |v|² − 2 u · v。根据定义，点积 u · v = |u||v| cos θ，其中 θ 是两向量之间的夹角，也正是三角形的夹角。代入后即得余弦定理：a² = b² + c² − 2bc cos A。
This connection is particularly important for HL students who study vectors in depth. It shows that the cosine rule is a consequence of the geometric properties of the dot product, reinforcing the unity of algebra, geometry, and trigonometry.
这一联系对于深入学习向量的 HL（高级）学生尤为重要。它表明余弦定理是点积几何性质的一个结论，更加强了代数、几何与三角学之间的统一性。
10. Common Mistakes and How to Avoid Them | 常见错误与如何避免
Students frequently mislabel sides and angles, especially when the triangle is rotated. Always check that the angle you use in the cosine rule is the one opposite the side whose square appears on the left-hand side of the formula.
学生经常将边和角的标注搞混，特别是当三角形被旋转时。务必检查余弦定理中使用的角是否与公式左侧出现平方的那条边相对。
Another common error is forgetting to take the square root when solving for a side, or misapplying the rearranged formula for an angle. A small slip like using 2ab instead of 2bc can cost marks. Writing out the full formula before substituting values helps prevent such mistakes.
另一个常见错误是在求边长时忘记开平方根，或在求角度时将重新排列的公式用错。像把 2bc 误写成 2ab 这样的小差错可能导致扣分。在代入数值之前先写出完整的公式有助于防止这类错误。
Calculator mistakes are also frequent: leaving the device in radian mode for a degree problem, or misusing the inverse cosine function. Always sense‑check your answer – an angle in an SSS triangle should never exceed 180°, and a side length must be positive.
计算器设置错误也屡见不鲜：做度量题目时仍留在弧度模式，或者误用反余弦功能。务必对答案进行合理性检查——SSS 三角形中的角不应超过 180°，边长则必须为正数。
11. Worked Examples | 例题解析
Example 1 (SAS): In triangle ABC, side b = 12 cm, side c = 9 cm, and angle A = 48°. Find side a.
例题 1（SAS）： 在三角形 ABC 中，边 b = 12 cm，边 c = 9 cm，角 A = 48°，求边 a。
a² = 12² + 9² − 2×12×9×cos 48° = 144 + 81 − 216×0.6691 ≈ 225 − 144.5 = 80.5. Therefore a = √80.5 ≈ 8.97 cm (3 s.f.).
代入公式 a² = b² + c² − 2bc cos A 得：a² = 12² + 9² − 2×12×9×cos 48° = 144 + 81 − 216×0.6691 ≈ 225 − 144.5 = 80.5。因此 a = √80.5 ≈ 8.97 cm（保留三位有效数字）。
Example 2 (SSS): Triangle XYZ has sides x = 5, y = 6, z = 7. Find the measure of the largest angle.
例题 2（SSS）： 三角形 XYZ 的三边为 x = 5，y = 6，z = 7，求最大角。
The remaining angles can be found with the sine rule or a second cosine rule application.
最大角对最长边，即角 Z 对边 z = 7。使用 cos Z = (x² + y² − z²) / (2xy) = (25 + 36 − 49) / 60 = 12/60 = 0.2，得 Z = arccos(0.2) ≈ 78.5°。剩余各角可用正弦定理或再次使用余弦定理求得。
These structured steps – identify the type of given information, choose the appropriate form of the cosine rule, substitute carefully, and check the reasonableness – are your blueprint for success in IB problems.
这些结构化的步骤——识别已知信息的类型、选择合适的余弦定理形式、仔细代入并进行合理性检查——是你在 IB 题目中取得成功的蓝图。
12. Summary and Key Takeaways | 总结与要点
b² + c² − 2bc cos A is a cornerstone of IB Mathematics trigonometry, indispensable for non‑right‑ alculation of a side (SAS) or an angle (SSS) and eliminates the ambiguity sometimes encountered with the sine rule. Its derivation ties together geometry, algebra, and vectors, and its applications stretch from theoretical triangle solving to real‑world navigation and engineering.
余弦定理 a² = b² + c² − 2bc cos A 是 IB 数学三角学的基石，对非直角三角形而言不可或缺。它能直接计算边长（SAS）或角度（SSS），并消除正弦定理有时会遇到的多解性。其推导将几何、代数和向量结合在一起，其应用则从理论上的三角形求解延伸到现实世界的导航和工程问题。
To master the topic, practise identifying the given configuration, selecting the correct formula variation, using your calculator correctly, and always verifying that your answers make physical sense. With the cosine rule in your toolkit, no triangle will remain a mystery.
要掌握这一专题，需多练习识别已知条件类型、选择正确的公式变体、正确使用计算器，并始终验证答案的物理合理性。有了余弦定理这一工具，任何三角形都将不再神秘。
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