Understanding and Applying the Cosine Rule | IB数学：余弦定理的理解与应用

📚 Understanding and Applying the Cosine Rule | IB数学：余弦定理的理解与应用

The cosine rule, also known as the law of cosines, is one of the most powerful tools in trigonometry. While the sine rule works beautifully when we know a pair of an angle and its opposite side, the cosine rule steps in when we have three sides or two sides with the included angle. In IB Mathematics, the cosine rule appears in both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses, at both Standard Level and Higher Level. Mastering this rule is essential for solving non-right-angled triangles, vector problems, and 3D geometry.

余弦定理，又称余弦法则，是三角学中最强大的工具之一。正弦定理在我们知道一个角与其对边这一配对时非常有效，但当我们已知三条边，或已知两条边及其夹角时，就需要用到余弦定理。在IB数学中，余弦定理出现在数学分析与方法（AA）以及应用与解释（AI）两门课程中，既涉及标准级别（SL），也涉及高级别（HL）。掌握这一定理是求解非直角三角形、向量问题以及三维几何的关键。

1. What Is the Cosine Rule? | 什么是余弦定理？

Consider a triangle ABC with side lengths a, b and c, where side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The cosine rule states that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice their product multiplied by the cosine of the included angle.

考虑一个三角形ABC，其中边长a、b、c分别对应角A、B、C的对边。余弦定理指出：三角形任意一边的平方，等于另外两边平方之和，减去这两边乘积的两倍乘以它们夹角的余弦。

The three equivalent forms of the cosine rule are:

余弦定理的三个等价形式如下：

a² = b² + c² − 2bc·cos A

b² = a² + c² − 2ac·cos B

c² = a² + b² − 2ab·cos C

Notice that the side on the left is always opposite the angle inside the formula. This symmetry makes the rule easy to remember: for any side, use the two other sides and the angle between them.

注意，等式左边的边永远是公式中那个角的对边。这种对称性使定理易于记忆：对于任意一条边，使用另外两条边以及它们之间的夹角即可。

When we need to find an angle, we can rearrange the rule. For example, solving for angle A gives:

当我们需要求角时，可以将公式变形。例如，解出角A得到：

cos A = (b² + c² − a²) ÷ (2bc)

2. Derivation of the Cosine Rule | 余弦定理的推导

To understand why the cosine rule works, we can derive it using coordinates. Place triangle ABC so that vertex C is at the origin and side CB lies along the positive x-axis. Let C = (0,0), B = (a,0), and A = (b·cos C, b·sin C). This is possible because side CA has length b and makes angle C with side CB.

为了理解余弦定理为何成立，我们可以用坐标法进行推导。将三角形ABC放置，使顶点C位于原点，边CB落在x轴正方向上。设C = (0,0)，B = (a,0)，A = (b·cos C, b·sin C)。这是可行的，因为边CA的长度为b，且与边CB的夹角为C。

Now side c is the distance between points A and B. Using the distance formula:

此时边c是点A与点B之间的距离。利用距离公式：

c² = (b·cos C − a)² + (b·sin C − 0)²

Expanding the brackets gives:

展开括号得到：

c² = b²·cos² C − 2ab·cos C + a² + b²·sin² C

Since sin² C + cos² C = 1, the b² terms combine:

由于sin² C + cos² C = 1，b²的项可以合并：

c² = a² + b² − 2ab·cos C

This elegant derivation shows that the cosine rule is simply the distance formula applied inside a triangle. The same logic works regardless of which vertex is placed at the origin.

这个简洁的推导表明，余弦定理本质上就是距离公式在三角形中的运用。无论将哪个顶点放在原点，逻辑都完全一致。

3. When to Use the Cosine Rule | 余弦定理的适用场景

In IB trigonometry problems, the cosine rule is used in two classic situations. Recognising these can save valuable exam time.

在IB三角学问题中，余弦定理用于两种经典情形。识别这些情形可以节省宝贵的考试时间。

Side-Side-Side (SSS): all three sides are known and an angle needs to be found.
Side-Angle-Side (SAS): two sides and the included angle are known and the third side needs to be found.

边边边（SSS）情形：已知三条边，需要求某个角。
边角边（SAS）情形：已知两条边及其夹角，需要求第三条边。

In contrast, the sine rule is used when we know two angles and one side (AAS or ASA), or when we know two sides and a non-included angle (SSA, the ambiguous case). If you have a pair consisting of an angle and its opposite side, the sine rule is usually the better choice.

相比之下，正弦定理用于已知两角一边（AAS或ASA），或已知两边及其中一边的对角（SSA，即歧义情况）。如果你有一个角与其对边的配对，正弦定理通常是更好的选择。

A quick diagnostic question: if the triangle is non-right-angled and the information given does not include an angle-opposite-side pair, reach for the cosine rule.

一个快速的判断方法：如果三角形不是直角三角形，并且题目给出的信息中没有“角—对边”的配对，那么就使用余弦定理。

4. Finding a Side: The SAS Case | 已知两边及夹角求第三边

Suppose we know two sides and the angle between them. To find the third side, we substitute directly into the cosine rule. Let us look at a complete worked example.

假设已知两条边以及它们之间的夹角。要求第三条边，可以直接代入余弦定理。我们来看一个完整的例题。

Example: In triangle ABC, side a = 5 cm, side b = 7 cm, and angle C = 60°. Find side c.

例题：在三角形ABC中，边a = 5 cm，边b = 7 cm，角C = 60°。求边c。

We use the form of the cosine rule that has c² on the left because angle C is the included angle:

我们使用左边为c²的余弦定理形式，因为角C是两个已知边的夹角：

c² = a² + b² − 2ab·cos C

Substitute the known values:

代入已知值：

c² = 5² + 7² − 2 × 5 × 7 × cos 60°

Since cos 60° = 0.5, this becomes:

因为cos 60° = 0.5，所以：

c² = 25 + 49 − 70 × 0.5 = 74 − 35 = 39

Therefore c = √39 ≈ 6.24 cm. Remember to take the positive square root because a length cannot be negative.

因此c = √39 ≈ 6.24 cm。记得取正的平方根，因为长度不可能是负数。

5. Finding an Angle: The SSS Case | 已知三边求角

When all three sides are known but no angle is given, we rearrange the cosine rule to make cos of the angle the subject of the formula. This is the SSS case.

当三条边都已知但没有给出任何角时，我们将余弦定理变形，使角度的余弦成为公式的主体。这就是SSS情形。

Example: In triangle ABC, a = 7 cm, b = 8 cm, c = 9 cm. Find angle A, which is opposite side a.

例题：在三角形ABC中，a = 7 cm，b = 8 cm，c = 9 cm。求角A，即边a所对的角。

Start from the first form of the cosine rule:

从余弦定理的第一种形式开始：

a² = b² + c² − 2bc·cos A

Rearrange it to isolate cos A:

将其变形以分离cos A：

cos A = (b² + c² − a²) ÷ (2bc)

Substitute the values:

代入数值：

cos A = (8² + 9² − 7²) ÷ (2 × 8 × 9) = (64 + 81 − 49) ÷ 144 = 96 ÷ 144 = 2/3

Thus A = arccos(2/3) ≈ 48.2°. Notice that the answer is the angle, not the cosine value; always use arccos or cos⁻¹ on your calculator.

因此A = arccos(2/3) ≈ 48.2°。注意，最终要求的是角度而不是余弦值；在计算器上一定要使用arccos或cos⁻¹功能。

6. Relationship with the Pythagorean Theorem | 余弦定理与勾股定理的关系

The cosine rule is actually a generalisation of the Pythagorean theorem. If the included angle C is 90°, then cos C = 0. Substituting this into the rule gives:

余弦定理实际上是勾股定理的一般化推广。如果夹角C为90°，则cos C = 0。将其代入定理中可得：

c² = a² + b² − 2ab × 0 = a² + b²

This is exactly the Pythagorean theorem. In other words, the cosine rule extends Pythagoras from right-angled triangles to all triangles. The extra term −2ab·cos C measures how the triangle “deviates” from being a right triangle.

这正是勾股定理。换言之，余弦定理把勾股定理从直角三角形推广到了任意三角形。多出来的项−2ab·cos C度量了三角形相对于直角三角形的“偏离程度”。

If angle C is acute, cos C is positive, so c² is smaller than a² + b². If angle C is obtuse, cos C is negative, so c² is larger than a² + b². This qualitative check is useful for verifying whether your final answer makes sense.

如果角C为锐角，cos C为正，则c²小于a² + b²。如果角C为钝角，cos C为负，则c²大于a² + b²。这种定性判断可以用来检验最终答案是否合理。

7. Applications in Vector Geometry | 余弦定理在向量几何中的应用

In IB Mathematics HL, the cosine rule connects beautifully with the dot product of vectors. For two vectors u and v with magnitudes |u| and |v|, placed tail to tail with angle θ between them, the distance between their heads is |u − v|. Squaring this distance gives:

在IB数学HL中，余弦定理与向量的点积有着美妙的联系。对于两个向量u和v，其大小分别为|u|和|v|，将它们尾对尾放置，夹角为θ，则它们终点之间的距离为|u − v|。对该距离平方，得到：

|u − v|² = |u|² + |v|² − 2|u||v|·cos θ

But we also know from the dot product that |u − v|² = u·u − 2u·v + v·v = |u|² + |v|² − 2u·v. Comparing the two expressions shows that:

但我们也知道，根据点积的定义，|u − v|² = u·u − 2u·v + v·v = |u|² + |v|² − 2u·v。比较两个表达式可得：

u·v = |u||v|·cos θ

Therefore the cosine rule and the dot product formula are two different languages describing the same geometric relationships. Understanding this duality helps you move smoothly between trigonometry and vector methods in Paper 2 and Paper 3.

因此，余弦定理与点积公式是描述同一几何关系的两种不同语言。理解这种双重性有助于你在试卷二和试卷三中自如地在三角学与向量方法之间切换。

In practice, when a triangle is formed by two vectors, you can use either the cosine rule or the dot product to find the angle between the vectors. Both approaches must give the same answer.

在实际问题中，当一个三角形由两个向量构成时，你可以使用余弦定理或点积来求向量之间的夹角。两种方法得出的答案必然一致。

8. Applications in 3D Geometry | 余弦定理在三维几何中的应用

Three-dimensional geometry problems in IB often require finding angles between line segments inside a cuboid or pyramid. The cosine rule is ideal because cross-sections often form non-right-angled triangles.

IB的三维几何问题常常需要求长方体或棱锥内部线段之间的角度。余弦定理非常适用，因为这些截面通常形成非直角三角形。

Example: A cuboid has base edges 6 m and 8 m, and height 10 m. Find the angle between the diagonal of the base and the space diagonal of the cuboid that starts from the same vertex.

例题：一个长方体的底面边长为6 m和8 m，高为10 m。求底面对角线与从同一顶点出发的空间对角线之间的夹角。

The base diagonal length is √(6² + 8²) = 10 m. The space diagonal length is √(6² + 8² + 10²) = √200 = 10√2 m. The third side of the relevant triangle is the vertical edge of length 10 m, which connects the endpoints of the two diagonals.

底面对角线长度为√(6² + 8²) = 10 m。空间对角线长度为√(6² + 8² + 10²) = √200 = 10√2 m。相关三角形的第三条边是长度为10 m的竖直棱，它连接着两条对角线的端点。

Let β be the angle between the base diagonal and the space diagonal. Applying the cosine rule:

设β为底面对角线与空间对角线之间的夹角。应用余弦定理：

cos β = (10² + (10√2)² − 10²) ÷ (2 × 10 × 10√2) = 200 ÷ (200√2) = 1/√2

Therefore β = 45°. This triangle is actually a right isosceles triangle in disguise, but the cosine rule handles it directly without any special insight.

因此β = 45°。这个三角形实际上是一个隐藏的等腰直角三角形，但余弦定理无需任何特殊观察即可直接处理。

9. Real-World Applications | 余弦定理的实际应用

The cosine rule is not just an abstract exercise; it is widely used in navigation, surveying, engineering and even computer graphics. Whenever a triangle is not right-angled, the cosine rule is often the fastest route to the answer.

余弦定理不仅仅是抽象的练习，它广泛应用于导航、测量、工程，甚至计算机图形学中。只要三角形不是直角三角形，余弦定理通常就是最快的求解途径。

Navigation and bearings: a ship travels 40 km on bearing 030° and then 60 km on bearing 120°. The cosine rule can find the direct distance back to the starting point.
Surveying: to measure the width of a river, surveyors measure two distances on the bank and the angle between them, then use the cosine rule to compute the required distance.
Engineering structures: cable-stayed bridges and roof trusses contain many non-right-angled triangles; engineers use the cosine rule to determine member lengths and forces.
Computer graphics: calculating the angle between a light source and a 3D surface normal, often via the equivalent dot product form.

导航与方位角：一艘船先以方位角030°航行40 km，再以方位角120°航行60 km。余弦定理可以求出直接返回起点的距离。
测量学：为了测量河的宽度，测量员在岸上测量两段距离以及它们之间的夹角，然后用余弦定理计算所需的距离。
工程结构：斜拉桥和屋架中包含大量非直角三角形；工程师用余弦定理确定杆件长度和受力。
计算机图形学：计算光源与三维表面法线之间的夹角，通常使用等价的点积形式。

In IB Applications and Interpretation (AI), such contextual problems are particularly important. Paper 1 and Paper 2 often embed the cosine rule in real-world scenarios, testing not only computation but also the selection of the correct formula.

在IB应用与解释（AI）课程中，这类情境问题尤为重要。试卷一和试卷二常常把余弦定理嵌入真实情境中，不仅考查计算能力，还考查能否选择正确的公式。

10. Common Mistakes and Exam Tips | 常见错误与备考建议

Even strong students lose marks on the cosine rule because of small errors. Below are the most common pitfalls and practical strategies to avoid them.

即使是优秀的学生，也常因一些小错误在余弦定理上丢分。下面是最常见的陷阱以及避免它们的实用策略。

Using the wrong formula: check whether the given information is SAS (find a side) or SSS (find an angle). Using the sine rule without a known pair is a classic error.
Forgetting the square root: the cosine rule gives c², not c. Always take the positive square root when finding a side length.
Calculator mode: if angles are given in degrees, ensure the calculator is in degree mode; if given in radians, use radian mode. Mixing these leads to wildly incorrect answers.
Rearranging incorrectly: for an angle, the numerator is the sum of squares of the two adjacent sides minus the square of the opposite side, divided by twice the product of the adjacent sides.
Not checking final units: a side length should be stated in the appropriate unit, and an angle should be given in degrees or radians as required by the question.
Ignoring rounding instructions: IB often asks for answers to three significant figures or to the nearest degree. Follow the instruction exactly.

用错公式：检查已知信息是SAS（求边）还是SSS（求角）。没有已知对边配对却使用正弦定理是典型错误。
忘取平方根：余弦定理求出的是c²而非c。求边长时一定要取正的平方根。
计算器模式：如果角度以度给出，确保计算器处于度模式；如果以弧度给出，使用弧度模式。混用会导致答案严重错误。
变形错误：求角时，分子是两条邻边平方之和减去对边平方，再除以两条邻边乘积的两倍。
不检查最终单位：边长应给出相应单位，角度则应按题目要求以度或弧度表示。
忽略四舍五入说明：IB通常要求保留三位有效数字或精确到最近整数度。严格按照要求执行。

A solid step-by-step exam routine is essential. First, draw a carefully labelled diagram. Second, write down the exact form of the cosine rule you plan to use. Third, substitute numbers with units. Fourth, solve step by step, showing all working. Finally, check that your answer is plausible: a side must be shorter than the sum of the other two sides, and an angle must be between 0° and 180°.

一个稳健的分步考试流程至关重要。首先，画出标好标签的示意图。其次，写下你计划使用的余弦定理的准确形式。再次，带单位代入数值。接着，逐步求解，展示所有过程。最后，检查答案是否合理：边长必须小于另外两边之和，角度必须在0°到180°之间。

With regular practice, the cosine rule will become an automatic reflex. Every time you see a non-right-angled triangle with three pieces of information, ask yourself which rule applies — and let the cosine rule handle the cases where the sine rule has nothing to pair up with.

通过规律练习，余弦定理会成为你的条件反射。每次看到一个非直角三角形并已知三个量时，问自己该用

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Understanding and Applying the Cosine Rule | IB数学：余弦定理的理解与应用