📚 Trigonometric Ratios | 三角函数比
Trigonometric ratios are the foundation of A-level trigonometry. In the Edexcel specification, you are expected to use sine, cosine and tangent confidently in right-angled triangles, on the unit circle, in exact-value problems and in solving equations. This revision guide covers the key definitions, identities, graphs and exam techniques you need.
三角比是 A-level 三角学的基础。在 Edexcel 考纲中，你需要熟练运用正弦、余弦和正切，解决直角三角形、单位圆、精确值以及三角方程等问题。本复习指南涵盖关键定义、恒等式、图像和考试技巧。
1. Defining Sine, Cosine and Tangent | 正弦、余弦与正切的定义
For a right-angled triangle, the three primary trigonometric ratios are defined relative to a given acute angle θ. They compare the lengths of two sides of the triangle.
对于直角三角形，三个基本三角比是相对于给定锐角 θ 定义的，它们比较三角形两条边的长度。
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Here, the hypotenuse is the longest side, opposite the right angle. The opposite side is the side facing angle θ, and the adjacent side is the side next to θ that is not the hypotenuse.
这里，斜边是最长边，对着直角；对边是角 θ 正对的边；邻边是与 θ 相邻但不是斜边的边。
These ratios depend only on the angle θ, not on the size of the triangle, because similar triangles have proportional side lengths.
这些比值只取决于角 θ，而与三角形的大小无关，因为相似三角形的边长成比例。
2. SOHCAHTOA and Right-Angled Triangles | SOHCAHTOA 与直角三角形
The mnemonic SOHCAHTOA helps you remember which sides are used in each ratio.
助记口诀 SOHCAHTOA 可以帮助你记住每个比值使用的边。
- SOH: Sin = Opposite / Hypotenuse — 正弦 = 对边 / 斜边
- CAH: Cos = Adjacent / Hypotenuse — 余弦 = 邻
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