📚 IGCSE Trigonometry: Sine Rule, Cosine Rule and Triangle Area | IGCSE 三角学：正弦定理、余弦定理与三角形面积
In IGCSE Mathematics, questions often move beyond right-angled triangles and ask you to solve non-right-angled triangles. The sine rule, cosine rule, and the sine area formula are essential tools for these problems. This revision guide explains when and how to use each formula, with worked examples and exam tips.
在 IGCSE 数学中，题目经常超越直角三角形，要求你解非直角三角形。正弦定理、余弦定理和正弦面积公式是解决这些问题的核心工具。本复习指南将解释每个公式的使用时机和方法，并提供例题与考试技巧。
1. Why Trigonometry Beyond Right Angles | 为什么需要超出直角的三角学
Right-angled trigonometry using SOH CAH TOA is limited to triangles with a 90° angle. However, many real-world and exam problems involve acute or obtuse triangles where no right angle is given. For these cases, the sine rule and cosine rule allow you to find unknown sides or angles from partial information.
使用 SOH CAH TOA 的直角三角形三角学仅限于含有 90° 角的三角形。然而，许多实际问题和考试题涉及锐角三角形或钝角三角形，其中没有给出直角。对于这些情况，正弦定理和余弦定理允许你从部分已知信息中求出未知的边或角。
- Right-angled trig: needs one angle = 90°.
- Sine and cosine rules: work for any triangle.
直角三角形三角学需要一个 90° 角；正弦定理和余弦定理适用于任何三角形。
2. Labelling Triangles Consistently | 一致地标记三角形
Before using either rule, label the sides and angles consistently. Convention: angles are capital letters A, B, C, and sides opposite them are lowercase a, b, c. Usually side a is opposite angle A, side b opposite angle B, and side c opposite angle C.
在使用任一公式之前，请一致地标记边和角。惯例是：角用大写字母 A、B、C 表示，它们所对的边用小写字母 a、b、c 表示。通常边 a 对角 A，边 b 对角 B，边 c 对角 C。
| Angle | 角 | Opposite side | 对边 |
|---|---|
| A | a |
| B | b |
| C | c |
3. The Sine Rule: Formula and When to Use It | 正弦定理：公式与使用条件
The sine rule links each side of a triangle to the sine of its opposite angle. It is usually written as:
正弦定理将三角形的每条边与其对角的正弦联系起来。它通常写作：
a / sin A = b / sin B = c / sin C
e is most useful when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA), though the SSA case may be ambiguous.
你也可以使用倒数形式：sin A / a = sin B / b = sin C / c。正弦定理最适用于已知两个角和一个边（角角边 AAS 或角边角 ASA），或已知两边和一个非夹角（边边角 SSA）的情况，不过 SSA 可能出现歧义。
- Use for: AAS, ASA, SSA (with care).
- Do not use when you only know three sides or two sides and the included angle.
适用于 AAS、ASA 和 SSA（需小心）；不要在三边已知或仅知两边及夹角时使用。
4. The Cosine Rule: Finding Sides and Angles | 余弦定理：求边与求角
The cosine rule is a generalisation of Pythagoras' theorem that works for any triangle. To find a side when you know the other two sides and the included angle, use:
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