Sine Rule | 正弦定理

📚 Sine Rule | 正弦定理

The sine rule is one of the most powerful tools in trigonometry for solving non-right-angled triangles. If you are studying Edexcel IGCSE Mathematics, mastering the sine rule is essential for the exam. This article will walk you through the formula, its applications, and the pitfalls to avoid.

正弦定理是三角学中求解非直角三角形最强有力的工具之一。如果你正在学习 Edexcel IGCSE 数学，掌握正弦定理对你的考试至关重要。本文将带你全面了解正弦定理的公式、应用以及需要避免的陷阱。

1. What is the Sine Rule? | 什么是正弦定理

The sine rule relates the lengths of the sides of a triangle to the sines of its angles. For any triangle ABC, with sides a, b, and c opposite angles A, B, and C respectively, the sine rule states:

正弦定理将三角形的边长与其对应角的正弦值联系起来。对于任意三角形 ABC，其中边 a、b、c 分别对应角 A、B、C，正弦定理表述为：

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

This single equation is the key to solving many triangle problems. Note that the rule works for any triangle, not just right-angled ones.

这个方程是解决许多三角形问题的关键。请注意，正弦定理适用于任何三角形，而不仅仅是直角三角形。

2. The Two Forms of the Rule | 正弦定理的两种形式

The sine rule can be written in two useful forms. When finding a missing side, use the form:

正弦定理可以写成两种有用的形式。当求未知边时，使用以下形式：

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

When finding a missing angle, it is often easier to use the reciprocal form:

当求未知角时，通常使用倒数形式更为简便：

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

Both forms are equivalent; the choice depends on whether you are solving for a side or an angle. In the Edexcel IGCSE exam, you may be asked to state the formula from memory, so make sure you can write both forms confidently.

这两种形式是等价的，选择哪种取决于你要求的是边还是角。在 Edexcel IGCSE 考试中，你可能会被要求凭记忆写出公式，因此请确保你能自信地写出这两种形式。

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

The sine rule is used when you are given:

正弦定理适用于以下已知条件：

Two angles and one side (AAS or ASA). You can find the third angle easily using the angle sum of a triangle (180°), then find the other sides.

已知两角一边（AAS 或 ASA）。利用三角形内角和为 180°，你可以轻松求出第三个角，然后求出其他边。
Two sides and a non-included angle (SSA). This is the ambiguous case, which we will discuss in detail later.

已知两边及其中一边的对角（SSA）。这就是模糊情况，我们稍后将详细讨论。

If you are given two sides and the included angle (SAS), or all three sides (SSS), you should use the cosine rule instead. Choosing the correct rule is a key exam skill that will be tested throughout your IGCSE paper.

如果已知两边及其夹角（SAS），或已知三边（SSS），则应使用余弦定理。选择正确的定理是一项关键的考试技能，在整个 IGCSE 试卷中都会考查。

4. Finding Missing Sides | 求未知边

Let us work through an example. In triangle ABC, angle A = 40°, angle B = 65°, and side b = 10 cm. Find side a.

让我们通过一个例子来学习。在三角形 ABC 中，角 A = 40°，角 B = 65°，边 b = 10 cm。求边 a。

Using the sine rule with the two known angles and one known side:

使用正弦定理，代入两个已知角和一条已知边：

a / sin 40° = 10 / sin 65°

Rearrange to find a:

移项求解 a：

a = 10 × sin 40° / sin 65°

Using a calculator: sin 40° ≈ 0.6428, and sin 65° ≈ 0.9063. So a ≈ 10 × 0.6428 / 0.9063 ≈ 7.09 cm. Always round to an appropriate degree of accuracy (usually 1 decimal place or 3 significant figures in IGCSE questions).

使用计算器：sin 40° ≈ 0.6428，sin 65° ≈ 0.9063。因此 a ≈ 10 × 0.6428 / 0.9063 ≈ 7.09 cm。请始终将结果保留到适当的精确度（在 IGCSE 题目中通常是 1 位小数或 3 位有效数字）。

5. Finding Missing Angles | 求未知角

Finding a missing angle uses the reciprocal form of the sine rule. Suppose in triangle ABC, side a = 8 cm, side b = 12 cm, and angle B = 50°. Find angle A.

求未知角使用正弦定理的倒数形式。假设在三角形 ABC 中，边 a = 8 cm，边 b = 12 cm，角 B = 50°。求角 A。

Using the reciprocal form:

使用倒数形式：

sin A / 8 = sin 50° / 12

Multiply both sides by 8:

两边同时乘以 8：

sin A = 8 × sin

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Sine Rule | 正弦定理