📚 Sine Rule, Cosine Rule and Triangle Area | 正弦定理、余弦定理与三角形面积
In IGCSE Mathematics, right-angled triangles are solved using Pythagoras' many exam questions require you to work with triangles that do not contain a right angle.
在 IGCSE 数学中，直角三角形使用毕达哥拉斯定理和基本三角比 sin θ = 对边/斜边、cos θ = 邻边/斜边、tan θ = 对边/邻边 来求解。但是，许多考试题目要求你处理不含直角的三角形。
This article revises the sine rule, the cosine rule, and the sine area formula. You will learn how to label triangles correctly, choose the right rule, calculate missing sides and angles, and avoid common calculator and substitution errors.
本文复习正弦定理、余弦定理和正弦面积公式。你将学会如何正确标注三角形、选择正确的定理、计算缺失的边和角，并避免常见的计算器和代入错误。
1. Why We Need Sine and Cosine Rules | 为什么需要正弦和余弦定理
nuse and cos θ = adjacent/hypotenuse do not apply directly because there is no hypotenuse.
基本三角比是通过直角定义的。在没有直角的三角形中，sin θ = 对边/斜边 和 cos θ = 邻边/斜边 的定义不能直接应用，因为没有斜边。
The sine rule and cosine rule extend trigonometry to all triangles. They are essential tools for solving non-right-angled triangles in the IGCSE syllabus, especially when given information is not enough for basic right-angled triangle methods.
正弦定理和余弦定理将三角学扩展到所有三角形。它们是 IGCSE 教学大纲中求解非直角三角形的重要工具，尤其是在已知信息不足以使用基本直角三角形方法时。
2. Labelling Convention | 标注约定
In IGCSE problems, the vertices of a triangle are usually labelled A, B, and C. The side opposite vertex A is labelled a, the side opposite B is labelled b, and the side opposite C is labelled c.
在 IGCSE 题目中，三角形的顶点通常标记为 A、B 和 C。顶点 A 的对边标记为 a，顶点 B 的对边标记为 b，顶点 C 的对边标记为 c。
Angles are written as capital letters A, B, C, and sides are written as lowercase letters a, b, c. Always pair each side with the angle directly opposite it before substituting into any formula.
角度写为大写字母 A、B、C，边写为小写字母 a、b、c。在代入任何公式之前，始终将每条边与其正对的角配对。
A clear labelling diagram makes it much easier to avoid substitution mistakes. If a question uses different letters such as P, Q, R, simply follow the same opposite-pair rule: side p is opposite angle P.
清晰的标注图可以大大避免代入错误。如果题目使用不同的字母，如 P、Q、R，只需遵循相同的对边对角规则：边 p 正对角 P。
3. The Sine Rule | 正弦定理
The sine rule states that the ratio of a side length to the sine of its opposite angle is constant for all three sides of a triangle.
正弦定理指出，三角形任意一边的长度与其对角的正弦之比是一个常数。
a / sin A = b / sin B = c / sin C
It can be used to find a missing side when you know two angles and one side. It can also be used to find a missing angle when you know two sides and a non-included angle.
它可以在已知两角和一边时求缺失的边。它也可以在已知两边和一个非夹角时求缺失的角。
For finding an angle, it is often easier to use the inverted form:
求角时，通常使用倒数形式更方便：
sin A / a = sin B / b = sin C / c
The rule can be derived from the area formula or by drawing an altitude, but the derivation is not required in most IGCSE exams. What matters is selecting the correct opposite pair and solving accurately.
该定理可以从面积公式推导，也可以通过作高来推导，但大多数 IGCSE 考试不要求推导过程。重要的是选择正确的对边对角组合并准确求解。
4. Using the Sine Rule to Find a Side | 使用正弦定理求边
Use the sine rule to find a side when you know two angles and one side. You need one complete opposite pair plus one extra angle or side.
当你已知两角和一边时，使用正弦定理求边。你需要一组完整的对边对角加上另一个角或边。
Example: in triangle ABC, A = 40°, B = 60°, and a = 7 cm. Find b.
示例：在三角形 ABC 中，A = 40°、B = 60°、a = 7 cm，求 b。
Write the relevant part of the sine rule and solve for b:
写出正弦定理的相关部分并求解 b：
b / sin 60° = 7 / sin 40°
b = (7 × sin 60°) / sin 40° ≈ 9.43 cm
Check your answer by comparing the angles: since B is larger than A, side b should be longer than side a. This confirms that b > a is correct.
通过比较角度来检查答案：因为 B 大于 A，
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图