📚 Sine Rule, Cosine Rule and Area of a Triangle | 正弦定理、余弦定理与三角形面积
In IGCSE Mathematics, many problems involve triangles that do not contain a right angle. The sine rule and cosine rule allow you to find unknown sides and angles in any triangle, and the sine area formula links two sides with the included angle. This revision guide explains the formulas, shows when to use each rule, and gives worked examples with bearings.
在 IGCSE 数学中，许多问题涉及不含直角的三角形。正弦定理和余弦定理可以帮助你求任意三角形中的未知边和角，正弦面积公式则将两边及其夹角联系起来。本复习指南解释这些公式、说明何时使用各定理，并给出方位角应用示例。
1. The Problem with Non-Right-Angled Triangles | 非直角三角形的难题
Pythagoras' acent only work in right-angled triangles. However, many real-world shapes and exam questions involve acute or obtuse triangles, such as triangular fields, roof structures and navigation routes.
勾股定理以及基本三角比 sin θ = 对边 ÷ 斜边、cos θ = 邻边 ÷ 斜边、tan θ = 对边 ÷ 邻边只适用于直角三角形。然而，许多现实图形和考试题涉及锐角三角形或钝角三角形，例如三角形田地、屋顶结构和导航路线。
To solve these triangles, you need two powerful tools: the sine rule and the cosine rule. They are especially useful in bearings problems, triangles in 3D, and mensuration questions where perpendicular height is not given directly.
要解这些三角形，你需要两个强有力的工具：正弦定理和余弦定理。它们在方位角问题、三维三角形以及未直接给出垂直高度的测量题中尤其有用。
2. Standard Notation for Triangles | 三角形的标准记号
By convention, the angles of a triangle are labelled with capital letters A, B and C. The side opposite each angle is labelled with the matching lowercase letter: side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.
按照惯例，三角形的角用大写字母 A、B、C 表示，每个角所对的边用相应的小写字母表示：边 a 对着角 A，边 b 对着角 B，边 c 对着角 C。
a ↔ A, b ↔ B, c ↔ C
This notation is essential because the sine and cosine rules are written in terms of opposite side-angle pairs. Before substituting into either formula, draw and label the triangle clearly so that you do not confuse which side is opposite which angle.
这种记号非常重要，因为正弦定理和余弦定理都是按“边与对角”的对应关系书写的。在代入任一公式之前，先清晰地画出并标记三角形，以免混淆哪条边对哪个角。
3. The Sine Rule: Formula and Purpose | 正弦定理：公式与用途
The sine rule connects the sides of any triangle with the sines of their opposite angles. It is usually written in the form:
正弦定理将任意三角形的边与其对角的正弦联系起来。它通常写成：
a ÷ sin A = b ÷ sin B = c ÷ sin C
fficient when you know two angles and one side, or two sides and a non-included angle. It is not suitable when you only know three sides or two sides with the included angle.
也可以反过来写成 sin A ÷ a = sin B ÷ b = sin C ÷ c。正弦定理最适用于已知两角一边，或已知两边及一个非夹角的情形。它不适合已知三边或已知两边及其夹角的情形。
4. Using the Sine Rule to Find a Missing Side | 用正弦定理求未知边
B = 60° and side a = 7 cm. To find side b, write the two relevant parts of the sine rule:
假设你知道角 A = 40°、角 B = 60°，边 a = 7 cm。要求边 b，写出正弦定理中相关的两项：
a ÷ sin A = b ÷ sin B
Substitute the known values:
代入已知值：
7 ÷ sin 40° = b ÷ sin 60°
cm. You only need two terms, so always pick the pair that involves the unknown and one complete known opposite side-angle pair.
整理得 b = 7 × sin 60° ÷ sin 40°。用计算器可得 b ≈ 9.43 cm。你只需要两项，因此始终选择包含未知量和一组已知边-角对应关系的两项。
5. Using the Sine Rule to Find a Missing Angle | 用正弦定理求未知角
g angle. For example, given a = 8, b = 10 and A = 50°, use the form b ÷ sin B = a ÷ sin A.
如果你知道两边和一个非夹角，就可以求未知角。例如，已知 a = 8、b = 10、A = 50°，使用 b ÷ sin B = a ÷ sin A。
÷ sin B = 8 ÷ sin 50°, so sin B = 10 × sin 50° ÷ 8 ≈ 0.958. Then B = sin⁻¹(0.958) ≈ 73.4°.
代入得 10 ÷ sin B = 8 ÷ sin 50°，所以 sin B = 10 × sin 50° ÷ 8 ≈ 0.958。然后 B = sin⁻¹(0.958) ≈ 73.4°。
106.6° is also a possible value. This is called the ambiguous case of the sine rule, and it occurs when the known angle is acute and the side opposite the known angle is shorter than the other given side.
要注意：正弦函数首先给出锐角。如果角 B 可能是钝角，需要检验 180° − 73.4° = 106.6° 是否也是一个可能值。这就是正弦定理的“歧义情形”，它发生在已知角为锐角且已知角的对边短于另一已知边时。
6. The Cosine Rule: Formula and Purpose | 余弦定理：公式与用途
The cosine rule is a generalisation of Pythagoras' theorem for non-right-angled triangles. It has two main forms. To find a missing side:
余弦定理是勾股定理在非直角三角形中的推广。它有两种主要形式。求边时：
a² = b² + c² − 2bc cos A
To find a missing angle, rearrange the same identity:
求角时，将同一等式变形：
cos A = (b² + c² − a²) ÷ (2bc)
The cosine rule is used when you know three sides, or two sides and the included angle. It is especially reliable because it can always determine a side or angle without the ambiguity that sometimes occurs in the sine rule.
余弦定理适用于已知三边，或已知两边及其夹角的情形。它尤其可靠，因为它总能确定边或角，而不会出现正弦定理中有时会出现的歧义。
7. Using the Cosine Rule to Find a Missing Side | 用余弦定理求未知边
cm, c = 8 cm and angle A = 45°. To find side a, substitute into a² = b² + c² − 2bc cos A.
假设 b = 6 cm、c = 8 cm、角 A = 45°。要求边 a，代入 a² = b² + c² − 2bc cos A。
a² = 6² + 8² − 2 × 6 ×
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