The Sine and Cosine Rules | 正弦定理与余弦定理

📚 The Sine and Cosine Rules | 正弦定理与余弦定理

Trigonometry is one of the most powerful tools in the IGCSE Mathematics syllabus. While most students first learn sine, cosine and tangent in right-angled triangles, many exam questions in Paper 4 involve non-right-angled triangles. In these cases, the sine rule and the cosine rule become essential. This article explains both rules, when to use each one, and how to apply them confidently in your exams.

三角学是 IGCSE 数学课程中最强大的工具之一。大多数学生最初是在直角三角形中学习正弦、余弦和正切的，但 Paper 4 中的许多考题涉及非直角三角形。在这种情况下，正弦定理和余弦定理就变得至关重要。本文将讲解这两个定理、何时使用它们，以及如何在考试中自信地应用。

1. The Sine Rule | 正弦定理

The sine rule relates the sides of a triangle to the sines of their opposite angles. For any triangle ABC, where side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C, the rule is stated as follows:

正弦定理将三角形的边与其对角的正弦值联系起来。对于任意三角形 ABC，其中边 a 是角 A 的对边，边 b 是角 B 的对边，边 c 是角 C 的对边，该定理表述如下：

a / sin A = b / sin B = c / sin C

This form is used when you know a side and its opposite angle, plus one other side or angle. It can be rearranged to sin A / a = sin B / b = sin C / c, which is often more convenient when finding an unknown angle.

当你已知一条边及其对角，以及另一条边或角时，使用这一形式。它也可以改写为 sin A / a = sin B / b = sin C / c，在求未知角时通常更为方便。

Use the sine rule when you know two angles and one side (AAS or ASA).
Use the sine rule when you know two sides and a non-included angle (SSA).

当已知两个角和一个边（AAS 或 ASA）时，使用正弦定理。
当已知两条边和一个非夹角（SSA）时，使用正弦定理。

A key point to remember: when finding an angle using the sine rule, the inverse sine function may give two possible answers in the range 0° to 180° — one acute and one obtuse. You must decide which is correct based on the diagram or the context of the problem.

一个关键点需要记住：使用正弦定理求角时，反正弦函数在 0° 到 180° 范围内可能给出两个答案——一个锐角和一个钝角。你必须根据图形或题目背景判断哪个是正确的。

2. The Cosine Rule | 余弦定理

The cosine rule is used when the sine rule cannot be applied directly. It is particularly useful for finding a side when two sides and the included angle are known, or for finding an angle when all three sides are known.

当正弦定理无法直接应用时，使用余弦定理。它特别适用于已知两边及其夹角求第三边，或已知三边求角的情况。

a² = b² + c² − 2bc cos A

Here, angle A is the angle between sides b and c, and side a is opposite angle A. To find an angle, the formula is rearranged:

这里，角 A 是边 b 和边 c 之间的夹角，边 a 是角 A 的对边。要求角时，公式可变形为：

cos A = (b² + c² − a²) / 2bc

Notice the pattern: the side on the left of the equation is always the side opposite the angle you are working with. This symmetry makes the rule easier to memorise and apply.

请注意这个规律：等式左边的边始终是你所求角所对的边。这种对称性使该定理更容易记忆和应用。

Use the cosine rule for finding a side when you know two sides and the included angle (SAS).
Use the cosine rule for finding an angle when you know all three sides (SSS).

当已知两边及其夹角（SAS）求边时，使用余弦定理。
当已知三边（SSS）求角时，使用余弦定理。

3. Choosing Between the Sine and Cosine Rules | 如何选择正弦定理或余弦定理

Many students struggle with deciding which rule to apply. A simple decision tree can help. First, identify what information you have and what you are asked to find.

许多学生在决定应用哪个定理时感到困惑。一个简单的判断流程可以帮助你。首先，明确你已知的信息以及要求的内容。

Given information	Unknown	Rule to use
Two angles, one side (AAS / ASA)	Another side	Sine rule
Two sides, one opposite angle (SSA)	Another angle	Sine rule
Two sides and included angle (SAS)	Third side	Cosine rule
Three sides (SSS)	Any angle	Cosine rule

If you have a right-angled triangle, do not use either rule — basic SOH-CAH-TOA and Pythagoras are simpler and faster.

如果是直角三角形，不要使用上述任一规则——基本的 SOH-CAH-TOA 和勾股定理更简单、更快捷。

4. Area of a Triangle Using Sine | 利用正弦求三角形面积

For any triangle, the area can be found using the formula involving two sides and the included angle:

对于任意三角形，面积可以通过两边及其夹角的公式求得：

Area = ½ ab sin C

In this formula, a and b are two sides of the triangle, and C is the angle between them. Notice that the capital letter C here does not necessarily refer to a specific vertex; it simply represents the included angle between the two chosen sides.

在此公式中，a 和 b 是三角形的两条边，C 是它们之间的夹角。注意，这里的大写字母 C 不一定指特定的顶点；它只是表示所选两条边之间的夹角。

This formula is extremely useful because it works for any triangle, not just right-angled ones. It is also a common exam question item, often combined with the sine or cosine rule in multi-part problems.

该公式非常实用，因为它适用于任何三角形，而不仅仅是直角三角形。它也是常见的考试题目，通常与正弦定理或余弦定理结合在多步问题中。

5. Worked Example — Finding an Unknown Side | 例题：求未知边长

Problem. In triangle ABC, angle A = 35°, angle B = 72°, and side a = 8 cm. Find side b.

题目。在三角形 ABC 中，角 A = 35°，角 B = 72°，边 a = 8 cm。求边 b。

Solution. Since we know two angles and one opposite side, we use the sine rule:

解答。由于已知两个角和一个对边，我们使用正弦定理：

a / sin A = b / sin B

Substitute the known values:

代入已知数值：

8 / sin 35° = b / sin 72°

Multiply both sides by sin 72°:

两边乘以 sin 72°：

b = 8 × sin 72° / sin 35°

Using a calculator: b = 8 × 0.9511 / 0.5736 ≈ 13.26 cm. Always give your final answer to a sensible degree of accuracy, typically 1 decimal place or 3 significant figures.

使用计算器计算：b = 8 × 0.9511 / 0.5736 ≈ 13.26 cm。最终答案应保留合理的精度，通常为 1 位小数或 3 位有效数字。

6. Worked Example — Finding an Unknown Angle | 例题：求未知角度

Problem. In triangle ABC, side a = 7 cm, side b = 9 cm, and side c = 11 cm. Find angle A.

题目。在三角形 ABC 中，边 a = 7 cm，边 b = 9 cm，边 c = 11 cm。求角 A。

Solution. Since all three sides are known, we use the cosine rule rearranged for an angle:

解答。因为已知三边，我们使用余弦定理的求角形式：

cos A = (b² + c² − a²) / 2bc

Substitute the known values:

代入已知数值：

cos A = (9² + 11² − 7²) / (2 × 9 × 11)

cos A = (81 + 121 − 49) / 198 = 153 / 198 ≈ 0.7727

Therefore A = cos⁻¹(0.7727) ≈ 39.4°. Note that the cosine rule for an angle always produces a unique answer between 0° and 180°, so there is no ambiguity here.

因此 A = cos⁻¹(0.7727) ≈ 39.4°。注意，余弦定理求角在 0° 到 180° 之间总是产生唯一答案，因此这里没有歧义。

7. Solving Bearings Problems | 解方位角问题

Bearings are one of the most common real-world applications of the sine and cosine rules. A bearing is an angle measured clockwise from north, always written with three digits, such as 045° or 120°. Bearing problems often involve navigating between points and calculating distances — exactly the kind of context in which the sine and cosine rules shine.

方位角是正弦定理和余弦定理最常见的现实应用之一。方位角是从正北方向顺时针测量的角度，始终用三位数字表示，如 045° 或 120°。方位角问题通常涉及在点之间导航和计算距离——这正是正弦定理和余弦定理发挥作用的场景。

Worked example. A ship sails from port P on a bearing of 060° for 15 km to point Q, then changes course and sails on a bearing of 150° for 20 km to point R. Find the distance PR.

例题。一艘船从港口 P 沿方位角 060° 航行 15 km 到达点 Q，然后改变航向沿方位角 150° 航行 20 km 到达点 R。求距离 PR。

Solution. First, draw a diagram showing triangle PQR with all given information. The angle at Q is the angle between the two paths, which is 150° − 60° = 90°. Wait — that seems too simple; indeed here we can use Pythagoras: PR = √(15² + 20²) = 25 km. However, if the angle at Q were not 90°, we would use the cosine rule:

解答。首先，画出三角形 PQR 的示意图，标出所有已知信息。Q 处的角是两条路径之间的夹角，即 150° − 60° = 90°。等等——这似乎太简单了；确实，这里可以直接使用勾股定理：PR = √(15² + 20²) = 25 km。然而，如果 Q 处的角不是 90°，我们就需要使用余弦定理：

PR² = PQ² + QR² − 2(PQ)(QR) cos Q

In a general bearing problem, carefully subtract the bearings to find the included angle at the intermediate point, then apply the appropriate rule.

在一般的方位角问题中，仔细计算方位角之差以求出中间点的夹角，然后应用相应的定理。

8. Common Mistakes and Pitfalls | 常见错误与易错点

Even strong students lose marks on trigonometric problems due to avoidable errors. Here are the most frequent mistakes found in IGCSE exam scripts.

即使是优秀的学生也会在三角题目中因可避免的错误而失分。以下是 IGCSE 考试卷中最常见的错误。

Using degrees instead of radians — always check your calculator mode! IGCSE at this level almost always uses degrees unless stated otherwise.
Applying the sine rule when the known angle is not opposite any known side.
Forgetting that sin⁻¹ can yield a second valid angle when solving SSA problems.
Using the wrong pair of sides in the cosine rule — the side on the left must be opposite the angle being used.
Rounding values too early in a multi-step problem, causing a large final error.

使用弧度而不是角度——始终检查计算器模式！IGCSE 在此级别几乎总是使用角度，除非另有说明。
当已知角不是任何已知边的对角时，错误地使用了正弦定理。
忘记在 SSA 问题中 sin⁻¹ 可能产生第二个有效角度。
在余弦定理中用错了边——左边的边必须与所用角相对。
在多步计算中过早取近似值，导致最终误差较大。

To avoid these pitfalls, always sketch the triangle, label all known sides and angles, and check that your answer is reasonable in the context of the diagram.

为了避免这些陷阱，始终先画出三角形草图，标出所有已知的边和角，并检查你的答案在图形背景下是否合理。

9. Practice Questions | 练习题目

Try these problems to test your understanding. Work through each one without looking at the answers first.

尝试以下问题来检验你的理解。先自己作答，再查看答案。

Question 1. In triangle ABC, a = 6 cm, b = 8 cm, and angle C = 50°. Find side c.

题目 1。在三角形 ABC 中，a = 6 cm，b = 8 cm，角 C = 50°。求边 c。

Question 2. In triangle XYZ, angle X = 40°, angle Y = 65°, and side x = 12 cm. Find side y.

题目 2。在三角形 XYZ 中，角 X = 40°，角 Y = 65°，边 x = 12 cm。求边 y。

Question 3. A triangle has sides of length 5 cm, 6 cm, and 7 cm. Use the cosine rule to find the smallest angle.

题目 3。一个三角形的三边长度为 5 cm、6 cm 和 7 cm。使用余弦定理求最小的角。

Answers: 1) c ≈ 6.15 cm 2) y ≈ 17.4 cm 3) ≈ 44.4°

For question 3, remember that the smallest angle is opposite the smallest side. Thus, the angle opposite the 5 cm side is the one to calculate.

对于题目 3，记住最小的角对应最小的边。因此，应计算 5 cm 边所对的角。

10. Summary | 总结

The sine rule and cosine rule are indispensable tools for solving problems involving non-right-angled triangles. The sine rule handles situations with two angles or two sides with an opposite angle, while the cosine rule covers two sides with an included angle and all-three-sides scenarios. The area formula ½ ab sin C extends your toolkit further.

正弦定理和余弦定理是解决非直角三角形问题不可或缺的工具。正弦定理适用于已知两个角或两条边及其一个对角的情形，而余弦定理适用于两边夹一角或已知三边的情形。面积公式 ½ ab sin C 进一步扩展了你的解题工具。

To succeed in IGCSE Mathematics Paper 4, practise drawing accurate diagrams, choosing the correct rule systematically, and communicating your working clearly. With consistent revision and plenty of practice problems, these rules will become second nature, and you will approach every trigonometry question with confidence.

要在 IGCSE 数学 Paper 4 中取得成功，请练习绘制准确的图形、系统地选择正确的定理，并清晰地表达你的解题过程。通过持续的复习和大量的练习，这些定理将成为你的本能反应，你将充满信心地应对每一道三角学题目。

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The Sine and Cosine Rules | 正弦定理与余弦定理