Sine Rule, Cosine Rule and Area of a Triangle | 正弦定理、余弦定理与三角形面积

📚 Sine Rule, Cosine Rule and Area of a Triangle | 正弦定理、余弦定理与三角形面积

Triangles appear throughout the IGCSE Mathematics syllabus, and not all of them are right-angled. To solve any triangle, you need three powerful tools: the sine rule, the cosine rule, and the area formula Area = ½ ab sin C. This revision guide explains when to use each rule, how to apply them step by step, and where students most often lose marks in the exam.

在 IGCSE 数学考纲中，三角形无处不在，而且并非所有三角形都是直角三角形。要解出任意三角形，你需要三大强大工具：正弦定理、余弦定理，以及面积公式 Area = ½ ab sin C。本复习指南将讲解何时使用每个定理、如何分步应用，以及考试中最常见的失分点。

1. Labelling a Triangle | 三角形标注

Every formula in this article uses the same convention. In triangle ABC, angle A is at vertex A, angle B at vertex B, and angle C at vertex C. The side opposite angle A is called side a, the side opposite angle B is side b, and the side opposite angle C is side c.

本文中的每一条公式都采用相同的标注约定。在三角形 ABC 中，角 A 位于顶点 A，角 B 位于顶点 B，角 C 位于顶点 C。与角 A 相对的边称为边 a，与角 B 相对的边称为边 b，与角 C 相对的边称为边 c。

Always sketch the triangle and write the letter labels on it before starting any calculation. Mixing up a side with its opposite angle is the single most common error in trigonometry questions on the IGCSE paper.

在开始任何计算之前，务必先画出三角形草图，并把字母标注上去。把边与其对角混为一谈，是 IGCSE 试卷中三角函数题目最常见的错误。

2. Choosing the Right Rule | 如何选择定理

The most important decision in a trigonometry problem is knowing which rule to use. The table below summarises the information you are given and the rule you should select.

解三角函数题最关键的一步，是判断该用哪个定理。下表总结了已知条件与应选定理的对应关系。

Notice that the sine rule works best when you already know a matching pair: a side and the angle opposite that side. The cosine rule works best when you know two sides and the angle between them, or three sides with no angles.

注意，正弦定理最适合在已知一组“边对角”配对时使用；而余弦定理则最适合在已知两边及其夹角，或已知三边但无任何角时使用。

3. The Sine Rule | 正弦定理

The sine rule connects each side to the sine of its opposite angle. For any triangle ABC, the rule states:

正弦定理将每条边与其对角的正弦联系起来。对任意三角形 ABC，定理表述为：

a ⁄ sin A = b ⁄ sin B = c ⁄ sin C

Use the sine rule when you know two angles and one side. To find a missing side, take two ratios that contain the known side and the unknown side, then rearrange. For example, to find side b:

当你已知两角一边时，使用正弦定理。求未知边时，选取含有已知边与未知边的两个比值，然后变形求解。例如求边 b：

b = a × sin B ⁄ sin A

To find a missing angle, use the inverted form of the sine rule. Take the known side and the opposite angle, together with the other known side, and solve for the sine of the unknown angle:

求未知角时，使用正弦定理的倒置形式。取已知边及其对角，再结合另一条已知边，解出未知角的正弦值：

sin B = b × sin A ⁄ a

After finding sin B, use the inverse sine function (sin⁻¹) to find angle B. One important warning: because sin θ = sin(180° − θ), there can be two possible angles between 0° and 180°. This is called the ambiguous case. If the given data is SSA, always check whether both angles produce a valid triangle.

求出 sin B 后，用反正弦函数（sin⁻¹）求角 B。一个重要提醒：因为 sin θ = sin(180° − θ)，在 0° 到 180° 之间可能存在两个符合条件的角，这称为“模棱两可情形”。当已知条件为 SSA（两边及一边的对角）时，务必检查两个角是否能构成有效三角形。

4. The Cosine Rule | 余弦定理

The cosine rule is a generalised version of Pythagoras’ theorem. It is useful when

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