📚 Mastering Trigonometry: Sine Rule, Cosine Rule and Area Formula | 掌握三角学：正弦定理、余弦定理与面积公式
Trigonometry is a core IGCSE Mathematics topic that extends Pythagoras' theorem to non-right-angled triangles. In your exam you will often need to find missing sides, missing angles or the area of a triangle using sine, cosine and area formulas. This revision guide explains the sine rule, cosine rule and area formula in a clear, step-by-step way.
三角学是 IGCSE 数学的核心主题，它将勾股定理推广到非直角三角形。考试中你经常需要利用正弦、余弦和面积公式求缺失的边、角或三角形面积。本复习指南将清晰、逐步地讲解正弦定理、余弦定理和面积公式。
1. Trigonometry Basics and Notation | 三角学基础与符号
In any triangle ABC, standard notation uses a for the side opposite angle A, b for the side opposite angle B, and c for the side opposite angle C. Angles are usually measured in degrees, and side lengths are in units such as cm or m.
在任意三角形 ABC 中，标准记法用 a 表示角 A 的对边，b 表示角 B 的对边，c 表示角 C 的对边。角度通常以度为单位，边长单位为 cm 或 m 等。
The sine rule and cosine rule work for all triangles, not only right-angled triangles. This is why they are called 'non-right-angled triangle methods' in many IGCSE questions.
正弦定理和余弦定理适用于所有三角形，而不仅仅是直角三角形。这就是为什么在许多 IGCSE 题目中它们被称为“非直角三角形方法”。
Before choosing a method, always label the triangle clearly. Put angles at the vertices with capital letters, and place each lower-case side directly opposite the matching angle.
选择方法之前，务必清晰地标注三角形。用大写字母标出顶点处的角，并把每个小写边放在对应角的直接对边位置。
2. The Sine Rule | 正弦定理
The sine rule connects the sides and angles of any triangle. It is most useful when you know either two angles and one side, or two sides and a non-included angle.
正弦定理将任意三角形的边和角联系起来。当你已知两角一边，或两边及一个非夹角时，它最有用。
a / sin A = b / sin B = c / sin C
sin B / b = sin C / c. Choose the form that makes the unknown quantity easier to isolate.
你也可以写出倒数形式：sin A / a = sin B / b = sin C / c。选择更容易分离未知量的形式。
Always label the triangle correctly first: each side must be matched with its opposite angle. If the labelling is wrong, the sine rule will produce incorrect answers.
首先必须正确标注三角形：每条边必须与其对角对应。如果标注错误，正弦定理会产生错误答案。
3. Using the Sine Rule to Find a Missing Side | 用正弦定理求缺失边
other side by setting up a proportion. For example, suppose A = 40°, B = 60° and a = 8 cm.
如果已知两角和一边，可以通过比例关系求另一边。例如，假设 A = 40°，B = 60°，a = 8 cm。
= a / sin A, substitute the known values, and solve for b. This gives b = a sin B / sin A.
利用 b / sin B = a / sin A，代入已知值并解出 b。得到 b = a sin B / sin A。
b = a × sin B / sin A
example, b = 8 × sin 60° / sin 40° ≈ 10.78 cm, provided your calculator is in degree mode.
在此例中，b = 8 × sin 60° / sin 40° ≈ 10.78 cm，前提是你的计算器处于度数模式。
The sine rule is especially efficient for finding a side when you have an angle-side opposite pair already known.
当你已经知道一组对角和对边时，正弦定理尤其适合求边。
4. Using the Sine Rule to Find a Missing Angle | 用正弦定理求缺失角
can find a missing angle. For instance, if a = 10 cm, b = 14 cm and A = 35°, find angle B.
当已知两边和一个非夹角时，正弦定理可以求缺失角。例如，如果 a = 10 cm，b = 14 cm，A = 35°，求角 B。
B / b = sin A / a, so sin B = b sin A / a. Substitute: sin B = 14 × sin 35° / 10 ≈ 0.802.
利用 sin B / b = sin A / a，所以 sin B = b sin A / a。代入：sin B = 14 × sin 35° / 10 ≈ 0.802。
sin B = b × sin A / a
er, be careful: the ambiguous case may give another possible angle, 180° − 53.3° = 126.7°.
然后 B = sin⁻¹(0.802) ≈ 53.3°。但要小心：不定情形可能给出另一个可能角，180° − 53.3° = 126.7°。
The ambiguous case occurs because sine is positive in both the first and second quadrants. IGCSE questions sometimes ask you to state both possible values or to explain why one value is impossible in a given diagram.
不定情形出现的原因是正弦在第一、第二象限
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