📚 The Sine and Cosine Rules | 正弦定理和余弦定理
In IGCSE Mathematics, solving triangles that are not right-angled is a core skill. The sine rule and the cosine rule extend trigonometry beyond right-angled triangles, allowing you to find unknown sides and angles in any triangle. This revision guide explains both rules, their applications, and common pitfalls, with worked examples to help you master the topic.
在 IGCSE 数学中，解非直角三角形是一项核心技能。正弦定理和余弦定理将三角函数从直角三角形扩展到任意三角形，使你能够求出任意三角形中的未知边和未知角。本复习指南将解释这两个定理、它们的应用和常见陷阱，并通过例题帮助你掌握这一考点。
1. Labelling a Triangle | 三角形的标记方法
Before using either rule, you must understand the standard labelling convention. In any triangle ABC, the side opposite angle A is labelled a, the side opposite angle B is labelled b, and the side opposite angle C is labelled c. This consistency allows the formulas to be applied directly.
在使用这两个定理之前，你必须理解标准的标记约定。在任意三角形 ABC 中，角 A 所对的边记为 a，角 B 所对的边记为 b，角 C 所对的边记为 c。这种一致的标记方式使我们能直接套用公式。
a / sin A = b / sin B = c / sin C
a² = b² + c² − 2bc × cos A
These two formulas are the foundation for this entire topic. Make sure you can write them from memory and understand what each symbol represents.
这两个公式是整个专题的基础。请确保你能默写它们，并理解每个符号所代表的含义。
2. The Sine Rule | 正弦定理
The sine rule is used when you know a pair of matching side and angle values, along with one more side or angle. It is particularly useful in non-right-angled triangles when you need to find an unknown side or an unknown angle.
正弦定理适用于已知一组对应边和角，以及另一条边或另一个角的情况。它在非直角三角形中特别有用，可用于求未知边或未知角。
a / sin A = b / sin B = c / sin C
To find a side, you take two equal fractions and solve for the unknown numerator. To find an angle, you invert the fractions and solve for sin A.
求边时，取两个相等的分式，解出未知的分子；求角时，将分式取倒数，然后解出 sin A。
3. Applying the Sine Rule to Find a Side | 用正弦定理求边
Suppose you know angle A, side b, and angle B. You can find side a by writing:
假设你已知角 A、边 b 和角 B，则可以通过下面的式子求出边 a：
a / sin A = b / sin B
y both sides by sin A to isolate a. For example, if A = 40°, B = 70°, and b = 12 cm, then:
将等式两边同时乘以 sin A，即可分离出 a。例如，若 A = 40°，B = 70°，b = 12 cm，则：
a = 12 × sin 40° / sin 70° ≈ 8.21 cm
Always check that your calculator is in degree mode before evaluating trigonometric functions.
在计算三角函数之前，务必检查计算器是否处于角度制模式。
4. Applying the Sine Rule to Find an Angle | 用正弦定理求角
To find an unknown angle, use the inverted form of the sine rule:
求未知角时，使用正弦定理的倒数形式：
sin A / a = sin B / b
Consider a triangle with side a = 8 cm, side b = 6 cm, and angle B = 35°. Then:
考虑一个三角形，其中边 a = 8 cm，边 b = 6 cm，角 B = 35°，那么：
sin A = 8 × sin 35° / 6 ≈ 0.7648
A ≈ 49.9°
ers because sin θ = sin(180° − θ). Always decide which answer is geometrically reasonable.
用正弦定理求角时，可能存在两个解，因为 sin θ = sin(180° − θ)。务必根据几何图形判断哪个解是合理的。
5. The Ambiguous Case | 解不唯一的情况
The ambiguous case occurs when you are given two sides and a non-included angle. In this situation, the sine rule can produce two different triangles: one with an acute angle and one with an obtuse angle.
当已知两边和其中一边的对角时，会出现“解不唯一”的情况。此时正弦定理可能产生两个不同的三角形：一个含有锐角，另一个含有钝角。
If the given angle is obtuse, only the acute solution is possible.
如果已知角是钝角，则只可能取锐角解。
If the sum of the calculated angle and the given angle is less than 180°, both solutions may be valid.
如果计算出的角与已知角之和小于 180°，则两个解都可能成立。
Always use the triangle angle sum to eliminate invalid answers.
始终利用三角形内角和定理排除无效答案。
Sketching the triangle first is the most reliable way to decide which answer makes sense.
先画出三角形草图是判断哪个解合理的最可靠方法。
6. The Cosine Rule | 余弦定理
The cosine rule is used when you know two sides and the included angle, or when you know all three sides and need an angle. It is a generalisation of Pythagoras' theorem.
余弦定理适用于已知两边及其夹角，或已知三边求角的情况。它是勾股定理的推广形式。
a² = b² + c² − 2bc × cos A
works for any triangle, including right-angled triangles where it reduces to a² = b² + c².
其中 a 是角 A 所对的边，b 和 c 是另外两条边。该公式适用于任意三角形，包括直角三角形；在直角三角形中它简化为 a² = b² + c²。
7. Finding an Angle Using the Cosine Rule | 用余弦定理求角
To find an angle, rearrange the cosine rule to make cos A the subject:
要求角，将余弦定理变形，使 cos A 成为主体：
cos A = (b² + c² − a²) / (2bc)
For example, if a = 7 cm, b = 9 cm, and c = 12 cm, then:
例如，如果 a = 7 cm，b = 9 cm，c = 12 cm，那么：
cos A = (9² + 12² − 7²) / (2 × 9 × 12)
cos A = (81 + 144 − 49) / 216 = 176 / 216 ≈ 0.8148
A ≈ 35.4°
This version of the cosine rule never produces an ambiguous answer because cos A is unique for angles between 0° and 180°.
这个形式的余弦定理不会产生歧义解，因为在 0° 到 180° 之间，cos A 的取值是唯一的。
8. Area of a Triangle | 三角形面积公式
When you know two sides and the included angle, the area of a triangle can be found directly without needing to calculate the height.
已知两边及其夹角时，无需计算高即可直接求出三角形面积。
Area = ½ × a × b × sin C
essed using trigonometry. In an exam, you may need to identify which two sides and included angle you have.
这个公式等价于标准公式：面积 = ½ × 底 × 高，只不过用三角函数表示。考试中你需要判断哪两条边和它们的夹角是已知的。
9. Sine Rule or Cosine Rule? | 何时用正弦定理，何时用余弦定理？
The table below summarises the decision process when solving a non-right-angled triangle.
下表总结了在解非直角三角形时的选择过程。
| Given information | 已知条件 | Suggested rule | 建议定理 |
| Two angles and one side | 两角一边 | Sine rule | 正弦定理 |
| Two sides and a non-included angle | 两边及其中一边对角 | Sine rule, check ambiguous case | 正弦定理，注意双解 |
| Two sides and the included angle | 两边及其夹角 | Cosine rule then sine rule | 先用余弦定理，再用正弦定理 |
| Three sides | 三边 | Cosine rule for angles | 用余弦定理求角 |
Write down what you know and what you need. If you have a matching side-angle pair, the sine rule is often quicker. If you have three sides or two sides with the included angle, use the cosine rule.
先写下已知条件和所求量。如果有一组对应的边和角，正弦定理通常更快；如果有三边或两边加夹角，则使用余弦定理。
10. Real-world Application: Bearings | 实际应用：方位角
Trigonometry is frequently tested through bearing problems. Bearings are angles measured clockwise from north, expressed as three-digit numbers. A typical question gives two positions relative to a point and asks for the distance between them.
三角学经常通过方位角问题进行考查。方位角是从正北方向顺时针测量的角度，用三位数表示。典型题目会给出两个位置相对于某点的方位，并求出它们之间的距离。
Consider a ship from point A to point B: the distance AB is 20 km and the bearing is 060°. From point B to point C: the distance BC is 15 km and the bearing is 120°. To find AC, you can use the cosine rule because you know two sides and the included angle at B, which is 60°.
考虑一艘船从 A 点驶向 B 点：距离 AB = 20 km，方位角为 060°；从 B 点驶向 C 点：距离 BC = 15 km，方位角为 120°。要求 AC，因为已知两边和夹角 B = 60°，可用余弦定理。
AC² = 20² + 15² − 2 × 20 × 15 × cos 60°
AC² = 400 + 225 − 300 = 325
AC ≈ 18.03 km
Always draw a clear diagram with north lines marked before attempting the calculation.
计算前务必画出清晰的图形，并标出北向参考线。
11. Exam-style Worked Example | 考试题型完整例题
In triangle ABC, AB = 8 cm, BC = 11 cm, and angle ABC = 38°. Find AC and angle BAC.
在三角形 ABC 中，AB = 8 cm，BC = 11 cm，角 ABC = 38°。求 AC 和角 BAC。
Use the cosine rule first because we have two sides and the included angle.
由于已知两边及其夹角，先用余弦定理。
AC² = 8² + 11² − 2 × 8 × 11 × cos 38°
AC² = 64 + 121 − 176 × 0.7880 ≈ 46.32
AC ≈ 6.81 cm
Now use the sine rule to find angle BAC. Let angle BAC be A.
然后用正弦定理求角 BAC。设角 BAC 为 A。
sin A / 11 = sin 38° / 6.81
sin A ≈ 11 × 0.6157 / 6.81 ≈ 0.9948
A ≈ 84.1°
The angle 95.9° is rejected because it would make the sum of angles in the triangle exceed 180°.
角度 95.9° 需舍去，因为加上其他角后会使三角形内角和超过 180°。
12. Summary and Exam Tips | 总结与考试建议
To solve any triangle problem, first identify what information is given. Use the sine rule when you have a side-angle pair, and the cosine rule when you have two sides with an included angle or three sides. Always round your final answer to three significant figures or one decimal place as the question requires.
解任何三角形问题时，先判断已知条件。有对应边角对时用正弦定理；已知两边夹角或三边时用余弦定理。最终答案应按题目要求保留三位有效数字或精确到一位小数。
Draw a labelled diagram for every problem.
每道题都画一个标注清楚的图形。
Check whether the ambiguous case applies before giving a single answer.
在给出唯一答案前检查是否存在双解情况。
Make sure your calculator is in degree mode.
确保计算器处于角度制。
Quote the formula in your working to earn method marks.
在作答中写出公式，以获得过程分。
Mastering the sine and cosine rules unlocks the entire geometry and trigonometry section of the IGCSE syllabus. Practice past paper questions until you can choose the correct rule instantly and apply it with confidence.
掌握正弦定理和余弦定理后，IGCSE 考纲中的几何与三角函数部分就能迎刃而解。请反复练习历年真题，直到你能迅速选择正确的定理并自信地完成计算。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图