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

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

When you move beyond right-angled trigonometry, many IGCSE questions ask you to find missing sides or angles in any triangle. The sine rule, cosine rule and area formula give you a complete toolkit for non-right-angled triangles.

当你从直角三角形三角学进入更广泛的内容时，许多 IGCSE 题目要求你求任意三角形中的未知边或角。正弦定理、余弦定理和面积公式为你提供了处理非直角三角形的完整工具。

1. Why we need the sine and cosine rules | 为什么需要正弦定理和余弦定理

In a right-angled triangle, Pythagoras’ theorem and the basic ratios sin, cos and tan are enough. However, most triangles in real life and in IGCSE exam questions do not have a right angle.

在直角三角形中，勾股定理和基本的 sin、cos、tan 比已经足够。然而，现实生活和 IGCSE 考题中的大多数三角形并不含直角。

The sine rule and cosine rule work for any triangle, so they are essential for problems involving non-right-angled shapes such as bearings, navigation, surveying and 3D geometry.

正弦定理和余弦定理适用于任何三角形，因此对于方位角、航海、测量和三维几何等涉及非直角图形的问题，它们是必不可少的。

Questions on this topic often appear in both the core and extended IGCSE papers, especially when a triangle is given with side lengths and angles in non-standard positions.

这一主题的题目经常出现在 IGCSE 核心和扩展试卷中，尤其是当三角形以非标准位置给出边长和角度时。

2. Labelling triangles and standard notation | 三角形标记与标准符号

In IGCSE trigonometry, vertices are usually labelled A, B and C. The side opposite vertex A is called a, the side opposite B is b, and the side opposite C is c.

在 IGCSE 三角学中，顶点通常标记为 A、B 和 C。顶点 A 的对边称为 a，顶点 B 的对边称为 b，顶点 C 的对边称为 c。

Angles are written as capital letters and sides as lower-case letters. This convention makes the sine rule and cosine rule much easier to remember and apply.

角用大写字母表示，边用小写字母表示。这一约定使正弦定理和余弦定理更容易记忆和应用。

Always draw a clear diagram and place each side opposite its matching angle. A correct diagram is one of the simplest ways to avoid careless mistakes later on.

一定要画出清晰的图形，并把每条边放在其对应角的对边位置。正确的图形是避免后续粗心错误最简单的方法之一。

3. The sine rule | 正弦定理

For any triangle ABC, the ratio of a side to the sine of its opposite angle is constant:

对于任意三角形 ABC，边与其对角正弦的比值是常数：

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