Sine Rule, Cosine Rule and Area of a Triangle | 正弦定理、余弦定理与三角形面积

2026-08-21 18:09 · 本文部分内容由AI辅助创作

📚 Sine Rule, Cosine Rule and Area of a Triangle | 正弦定理、余弦定理与三角形面积

nderstanding when to use each formula is just as important as being able to substitute values accurately.

在 IGCSE 数学中，非直角三角形的题目通常需要用到正弦定理、余弦定理或面积公式 Area = 1/2 ab sin C。理解每个公式的使用时机与能够准确代入数值同样重要。

1. Standard Notation for Triangles | 三角形的标准记号

In any triangle ABC, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. This pairing is used in all the formulas below, so correct labelling is essential.

在任意三角形 ABC 中，边 a 对角 A，边 b 对角 B，边 c 对角 C。这个对应关系用于下面所有公式，因此正确标注非常重要。

Always sketch and label the triangle before solving a problem. If no diagram is given, draw one and place the known values on the correct sides and angles. Use the same units for all lengths.

解题前一定要先画图并标注三角形。如果没有给出图形，就自己画一个，把已知值标在正确的边和角上。所有长度要使用相同的单位。

2. The Sine Rule | 正弦定理

The sine rule connects side lengths to the sines of their opposite angles. For triangle ABC, it states:

正弦定理把边长与其对角的正弦联系起来。对于三角形 ABC，它表示为：

a / sin A = b / sin B = c / sin C

This form is used when you are trying to find a missing side. The reciprocal form is often more convenient when finding a missing angle:

这个形式用于求缺失的边长。当你求缺失的角时，倒数形式通常更方便：

sin A / a = sin B / b = sin C / c

in A = b / sin B. This gives one equation that can be rearranged to solve for the unknown.

你每次只需要使用其中的两部分，例如 a / sin A = b / sin B。这样就得到一个方程，可以变形求解未知量。

3. When to Use the Sine Rule | 何时使用正弦定理

Use the sine rule when you know either two angles and one side, or two sides and a non-included angle. In the first case, the known pattern is AAS or ASA, and you usually find a missing side.

当你已知两个角和一个边，或者两个边和一个非夹角时，使用正弦定理。第一种情况的已知模式是 AAS 或 ASA，通常用来求缺失的边。

In the second case, the known pattern is SSA, and you usually find a missing angle. However, the SSA case can produce two possible angles, so it is called the ambiguous case.

第二种情况的已知模式是 SSA，通常用来求缺失的角。但是，SSA 情况可能产生两个可能的角，因此称为多解情况。

d be about 53° or about 127°. You must check whether the obtuse option is valid by seeing if the angles still add up to less than 180°.

例如，如果 sin B = 0.8，则 B 可能约为 53° 或约 127°。你必须检查钝角选项是否有效，也就是看这些角度加起来是否仍然小于 180°。

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