📚 Sine Rule | 正弦定理
The sine rule is a fundamental relationship between the sides and angles of any triangle (not just right-angled triangles). It allows you to solve for unknown sides or angles when you have certain combinations of known values. In IGCSE Mathematics, the sine rule is an essential tool for non‑right‑angled triangle problems. Understanding how and when to apply it will help you tackle a wide range of geometry and trigonometry questions with confidence.
正弦定理是任意三角形（不仅仅是直角三角形）边与角之间的基本关系。它允许在已知某些值的组合时求解未知的边或角。在 IGCSE 数学中，正弦定理是处理非直角三角形问题的重要工具。理解如何以及何时应用它将帮助你自信地解决各种几何和三角问题。
1. Introduction to the Sine Rule | 正弦定理简介
The sine rule links each side of a triangle to the sine of its opposite angle. In any triangle ABC, with sides a opposite angle A, b opposite B, and c opposite C, the rule states that the ratio of a side length to the sine of its opposite angle is constant for all three sides.
正弦定理将三角形的每条边与其对角的正弦联系起来。在任意三角形 ABC 中，边 a 对应角 A，边 b 对应角 B，边 c 对应角 C，该定理指出边长与其对角正弦的比值对于三条边都是相同的常数。
2. The Formula | 公式
The standard form of the sine rule is:
正弦定理的标准形式为：
a / sin A = b / sin B = c / sin C
Equivalently, it can be written as:
等价地，也可写成：
sin A / a = sin B / b = sin C / c
Both forms are valid; the first is generally more convenient when finding a side, and the second when finding an angle. You will use whichever makes the algebra simpler for the unknown you are solving for.
两种形式都有效；求边时通常使用第一种更方便，求角时第二种更方便。你将根据所求未知数选择代数上更简单的形式。
3. When to Use the Sine Rule | 何时使用正弦定理
The sine rule is used for non‑right‑angled triangles in two main scenarios:
正弦定理在以下两种主要情形下用于非直角三角形：
When you know two angles and one side (AAS or ASA).
当你已知两个角和一个边（角角边或角边角）。
When you know two sides and a non‑included angle (SSA). Be careful – this case can produce two possible triangles, known as the ambiguous case.
当你已知两条边和一个非夹角（边边角）。要小心——这种情况可能产生两个可能的三角形，即所谓的模糊情形。
If you have two sides and the included angle (SAS) or three sides (SSS), the cosine rule is usually more appropriate.
如果你有两条边和夹角（边角边）或三条边（边边边），余弦定理通常更合适。
4. Using the Sine Rule to Find a Side | 使用正弦定理求边长
need two angles and one side, where the side you know is opposite one of the known angles.
求未知边时，使用 a / sin A = b / sin B 的形式。整理可得：a = (b × sin A) / sin B。你需要两个角和一个边，其中已知的边是其中一个已知角的对边。
in triangle ABC, if angle A = 40°, angle B = 60°, and side b = 8 cm, you can find side a:
例如，在三角形 ABC 中，如果角 A = 40°，角 B = 60°，边 b = 8 cm，那么可以求边 a：
a / sin 40° = 8 / sin 60°
a = (8 × sin 40°) / sin 60°
n a ≈ 5.94 cm (to 3 significant figures). Always ensure your calculator is in degree mode.
然后使用计算器计算正弦值，得到 a ≈ 5.94 cm（保留三位有效数字）。务必确保计算器处于角度模式。
5. Using the Sine Rule to Find an Angle | 使用正弦定理求角度
owever, remember that sin θ = k can have two solutions in the range 0° to 180°: θ and 180° – θ. This leads to the ambiguous case.
求角时，通常使用颠倒形式 sin A / a = sin B / b 更方便。整理得 sin A = (a × sin B) / b。然后使用反正弦函数 (sin⁻¹) 得到 A。但要记住，在 0° 到 180° 范围内，sin θ = k 可能有两个解：θ 和 180° – θ。这就导致了模糊情形。
For example, given a = 7 cm, b = 9 cm, and B = 50°, you can find angle A:
例如，已知 a = 7 cm，b = 9 cm，B = 50°，可以求角 A：
sin A / 7 = sin 50° / 9
sin A = (7 × sin 50°) / 9
7 × 0.7660) / 9 ≈ 0.5958. Then A = sin⁻¹(0.5958) ≈ 36.6°. The other possible angle is 180° – 36.6° = 143.4°, but this would need to be checked against the triangle's other angles to see if it is valid.
计算得 sin A ≈ (7 × 0.7660) / 9 ≈ 0.5958。然后 A = sin⁻¹(0.5958) ≈ 36.6°。另一个可能的角是 180° – 36.6° = 143.4°，但需要根据三角形其他角来检查是否有效。
6. The Ambiguous Case (SSA) | 模糊情形（边边角）
When you are given two sides and a non‑ ngth of a relative to b and the height (h = b × sin A) determines the number of solutions:
当给定两条边和一个非夹角（边边角）时，可能有两个可能的三角形、一个三角形或没有三角形。这是因为给定边 a、边 b 和角 A（a 是 A 的对边），a 的长度相对于 b 和高 (h = b × sin A) 决定了可能的三角形个数：
If a < h, no triangle is possible.
若 a < h，不可能构成三角形。
If a = h, a right‑angled triangle is formed (one solution).
若 a = h，构成直角三角形（一个解）。
If h < a < b, two distinct triangles may exist (the ambiguous case). One triangle has an acute angle, the other an obtuse angle.
若 h < a < b，可能存在两个不同的三角形（模糊情形）。一个三角形有锐角，另一个有钝角。
If a ≥ b, exactly one triangle is formed.
若 a ≥ b，恰好构成一个三角形。
In an exam, after calculating the acute angle using the sine rule, you must check whether the supplement (180° – angle) also fits the angle sum of 180° with the given angle. If the sum of the given angle and the supplementary angle is less than 180°, the second triangle is valid. Otherwise, only one solution exists.
在考试中，使用正弦定理计算出锐角后，必须检查补角（180° – 该角）加上已知角之和是否不超过 180°。如果已知角与补角之和小于 180°，则第二个三角形有效。否则，只有一个解。
7. Worked Example 1 – Finding a Missing Side | 示例1 – 求缺失边长
iangle PQR has angle P = 35°, angle Q = 72°, and side r = 15 cm (opposite R). Find side p.
三角形 PQR 中，角 P = 35°，角 Q = 72°，边 r = 15 cm（R 的对边）。求边 p。
First, find the missing angle R: R = 180° – 35° – 72° = 73°.
首先，求出缺失的角 R：R = 180° – 35° – 72° = 73°。
Apply the sine rule: p / sin 35° = 15 / sin 73°
应用正弦定理：p / sin 35° = 15 / sin 73°
Rearrange: p = (15 × sin 35°) / sin 73°
整理得：p = (15 × sin 35°) / sin 73°
Using sin 35° ≈ 0.5736, sin 73° ≈ 0.9563 → p ≈ (15 × 0.5736) / 0.9563 ≈
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