IGCSE Maths: Right-Angled Trigonometry | IGCSE 数学：直角三角形三角学

📚 IGCSE Maths: Right-Angled Trigonometry | IGCSE 数学：直角三角形三角学

Right-angled trigonometry is one of the most important and frequently examined topics in IGCSE Mathematics. It gives you reliable tools to find unknown sides and angles in right-angled triangles, and it appears in problems ranging from simple diagrams to real-life contexts such as bearings, heights, and distances.

直角三角形三角学是 IGCSE 数学中最重要、最常考查的主题之一。它提供了在直角三角形中求未知边和角的可靠工具，并出现在从简单图形到方位角、高度和距离等实际情境的问题中。

This revision guide takes you through the key definitions, the three trigonometric ratios, typical question types, and common exam mistakes. Work through each section in order, and always write down the ratio equation before using your calculator.

本复习指南将带你梳理关键定义、三个三角比、常见题型以及常见考试错误。请按顺序学习每一节，并在使用计算器之前先写出比例方程。

1. Core Ideas: Hypotenuse, Opposite, Adjacent | 核心概念：斜边、对边、邻边

In any right-angled triangle, the longest side is called the hypotenuse. The opposite and adjacent sides are labelled relative to a chosen acute angle θ. The opposite side faces θ, while the adjacent side is the side joining θ to the right angle.

在任何直角三角形中，最长的一边称为斜边。对边和邻边是相对于所选锐角 θ 来标记的。对边正对着 θ，而邻边是连接 θ 与直角的那条边。

Useful labels are listed below:

常用标记如下：

• Hypotenuse: always opposite the right angle
• Opposite: opposite the angle you are using
• Adjacent: next to the angle and not the hypotenuse

• 斜边：始终与直角相对
• 对边：与你所使用的角相对
• 邻边：紧邻该角且不是斜边

A common mistake is to label the opposite and adjacent sides without first identifying the angle. The labels change when you switch from one acute angle to the other.

常见错误是在未先确定角度的情况下标记对边和邻边。当你从一个锐角切换到另一个锐角时，这些标记会发生变化。

In a right-angled triangle, the two acute angles are complementary, so if one acute angle is θ, the other is 90° − θ. The side that is opposite θ is adjacent to 90° − θ, and the side that is adjacent to θ is opposite 90° − θ.

在直角三角形中，两个锐角互为余角，因此如果一个锐角为 θ，另一个锐角为 90° − θ。与 θ 相对的边就是 90° − θ 的邻边，而与 θ 相邻的边则是 90° − θ 的对边。

2. The Three Trigonometric Ratios | 三个三角比

The three basic ratios are sine, cosine and tangent. They are often remembered by the abbreviation SOH CAH TOA.

三个基本三角比是正弦、余弦和正切。通常用缩写 SOH CAH TOA 来记忆。

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