Sine Rule and Solving Triangles | 正弦定理与三角形解法

📚 Sine Rule and Solving Triangles | 正弦定理与三角形解法

Triangles appear regularly in A-level Mathematics, and the sine rule is one of the most important tools for solving them. It connects the sides of a triangle to the sines of their opposite angles, allowing us to find missing lengths and missing angles when certain information is known.

在 A-level 数学中，三角形问题非常常见，而正弦定理是求解三角形最重要的工具之一。它将三角形的边与其对角的正弦值联系起来，使我们在已知部分条件时能够求出未知的边长和角度。

1. What Is the Sine Rule? | 什么是正弦定理？

For any triangle ABC, label the sides so that side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The sine rule states that each side divided by the sine of its opposite angle is equal to the same constant.

对于任意三角形 ABC，我们通常规定：边 a 对应角 A，边 b 对应角 B，边 c 对应角 C。正弦定理表示：每条边与其对角正弦值的比值都相等。

a / sin A = b / sin B = c / sin C = 2R

Here, R is the circumradius of the triangle, so 2R is the diameter of the circle that passes through all three vertices. In most examinations, you will use the first three ratios rather than the circumradius unless the question specifically asks about the circumcircle.

其中 R 是三角形的外接圆半径，2R 就是经过三个顶点的外接圆直径。在大多数考试中，你主要使用前三个比值，除非题目明确要求计算外接圆。

2. Notation and When to Use the Sine Rule | 符号约定与适用条件

The sine rule can be used in two common situations. First, when two angles and any side are known, which is often called AAS or ASA. Second, when two sides and a non-included angle are known, which is often called SSA. The second case needs extra care because it may produce zero, one, or two possible triangles.

正弦定理通常用于两类常见情形。第一类是已知两角和任意一边，也就是 AAS 或 ASA。第二类是已知两边和其中一边的对角，也就是 SSA。第二种情况需要特别小心，因为可能产生零个、一个或两个三角形。

Known information	What you can find
Two angles and any side (AAS / ASA)	The remaining angle and the remaining sides
Two sides and a non-included angle (SSA)	A possible angle, but check the ambiguous case

Always check that you are using a side and the angle directly opposite it. If you mix up corresponding pairs, the sine rule will give a wrong answer.

使用正弦定理时，必须确保所用边与角是对应的，即边所对的角就是方程中的那个角。如果把对应关系弄错，结果必然错误。

3. Derivation of the Sine Rule | 正弦定理的推导

Consider triangle ABC. Draw a perpendicular line from C to AB, and call the length of this altitude h. In the right triangle formed on the left, h = b sin A. In the right triangle formed on the right, h = a sin B.

以三角形 ABC 为例。从顶点 C 向 AB 作垂线，设垂线长度为 h。在左侧直角三角形中，有 h = b sin A；在右侧直角三角形中，有 h = a sin B。

Since both expressions describe the same altitude, we can write b sin A = a sin B. Rearranging gives a / sin A = b / sin B. Repeating the same argument from another vertex gives the full sine rule.

因为这两条表达式表示同一条垂线，所以 b sin A = a sin B。整理后得到 a / sin A = b / sin B。从其他顶点重复同样推理，就能得到完整的正弦定理。

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

4. Using the Sine Rule to Find an Unknown Side | 用正弦定理求未知边长

The sine rule is very convenient when you need to find a missing side and you already know its opposite angle plus one complete side-angle pair.

当需要求未知边，且已知这条边所对的角以及另一组完整的边角对应关系时，正弦定理非常方便。

Example: In triangle ABC, angle A = 35°, angle B = 65°, and side a = 8. Find side b.

例如：在三角形 ABC 中，角 A = 35°，角 B = 65°，边 a = 8。求边 b。

Using the sine rule, b / sin B = a / sin A, so b = 8 sin 65° / sin 35°.

根据正弦定理，b / sin B = a / sin A，因此 b = 8 sin 65° / sin 35°。

b = 8 × 0.9063 / 0.5736 ≈ 12.64

The missing side is approximately 12.64 units long.

因此所求边约为 12.64 个单位长度。

5. Using the Sine Rule to Find an Unknown Angle | 用正弦定理求未知角度

To find an angle, use the reciprocal form of the sine rule. For example, to find angle A, use sin A / a = sin B / b.

求角度时，可以使用正弦定理的反向形式。例如，要求角 A，可以使用 sin A / a = sin B / b。

Example: In triangle ABC, angle B = 50°, side a = 10, and side b = 12. Find angle A.

例如：在三角形 ABC 中，角 B = 50°，边 a = 10，边 b = 12。求角 A。

sin A = a sin B / b = 10 sin 50° / 12 ≈ 0.6385.

sin A = a sin B / b = 10 sin 50° / 12 ≈ 0.6385。

The acute solution is A ≈ 39.7°. The alternative solution is 180° – 39.7° = 140.3°, but 140.3° + 50° > 180°, so it is invalid. Therefore A ≈ 39.7°.

锐角解为 A ≈ 39.7°。另一个可能的解是 180° – 39.7° = 140.3°，但 140.3° + 50° > 180°，所以这个解不成立。因此 A ≈ 39.7°。

6. The Ambiguous Case | 三角形解的不确定情况

When you know two sides and a non-included angle, SSA, there may be more than one possible triangle. This is called the ambiguous case. For an acute angle A, with side a opposite A and side b adjacent to A, compare a with b sin A.

当已知两边及其中一边的对角（SSA）时，可能出现不止一个符合条件的三角形，这就是“三角形解的不确定情况”。若角 A 为锐角，a 是角 A 的对边，b 是角 A 的邻边，则需要比较 a 与 b sin A 的关系。

Condition	Number of triangles
a < b sin A	0
a = b sin A	1 right-angled triangle
b sin A < a < b	2 triangles
a ≥ b	1 triangle

The reason for the ambiguity is that sin B = sin(180° – B). If both B and 180° – B can be placed into the triangle without making the angles exceed 180°, then both configurations are mathematically valid.

出现这种不确定性的原因是 sin B = sin(180° – B)。如果 B 和 180° – B 都能放入同一个三角形且不使内角和超过 180°，那么两种情形在数学上都成立。

7. Area of a Triangle Using Sine | 用正弦公式求三角形面积

The area of a triangle can also be found using two sides and the included angle. The standard area formula is ½ × base × height, but when the perpendicular height is not given, we can use the sine of an included angle.

三角形的面积也可以通过两边及夹角求得。基本面积公式是 ½ × 底 × 高，但当垂直高度未给出时，我们可以使用夹角的正弦值。

Area = ½ ab sin C = ½ bc sin A = ½ ca sin B

For example, if b = 8, c = 6, and the included angle A = 30°, then the area is ½ × 8 × 6 × sin 30° = 12 square units.

例如，如果 b = 8，c = 6，夹角 A = 30°，则面积为 ½ × 8 × 6 × sin 30° = 12 平方单位。

8. Worked Example 1: Finding a Side and the Area | 例题一：求边与面积

In triangle ABC, angle A = 42°, angle B = 58°, and side c = 10 cm. Find angle C, side b, and the area of the triangle.

在三角形 ABC 中，角 A = 42°，角 B = 58°，边 c = 10 cm。求角 C、边 b 和三角形面积。

First, find angle C using the angle sum of a triangle.

首先，利用三角形内角和求角 C。

C = 180° – 42° – 58° = 80°

Now use the sine rule to find b. Since b is opposite angle B and c is opposite angle C, we write b / sin 58° = 10 / sin 80°.

然后用正弦定理求 b。因为 b 对的是角 B，c 对的是角 C，所以可写为 b / sin 58° = 10 / sin 80°。

b = 10 sin 58° / sin 80° ≈ 8.61 cm

The area uses two sides and the included angle A, so Area = ½ × b × c × sin A.

面积使用两边及其夹角 A，因此面积 = ½ × b × c × sin A。

Area = ½ × 8.61 × 10 × sin 42° ≈ 28.8 cm²

9. Worked Example 2: The Ambiguous Case in Action | 例题二：不确定情况的实际应用

Suppose angle A = 30°, side a = 4, and side b = 5. Find all possible values of angle B and side c.

已知角 A = 30°，边 a = 4，边 b = 5。求角 B 和边 c 的所有可能值。

Using the sine rule, sin B = b sin A / a = 5 × sin 30° / 4 = 0.625.

利用正弦定理，sin B = b sin A / a = 5 × sin 30° / 4 = 0.625。

The two possible values of B are B₁ ≈ 38.68° and B₂ ≈ 141.32°. Both are valid because A + B < 180° in both cases.

角 B 的两个可能值分别为 B₁ ≈ 38.68° 和 B₂ ≈ 141.32°。两者都成立，因为在这两种情况下都有 A + B < 180°。

For B₁ ≈ 38.68°, angle C₁ ≈ 180° – 30° – 38.68° = 111.32°.

当 B₁ ≈ 38.68° 时，角 C₁ ≈ 180° – 30° – 38.68° = 111.32°。

c₁ = a sin C₁ / sin A ≈ 4 × sin 111.32° / 0.5 ≈ 7.45

For B₂ ≈ 141.32°, angle C₂ ≈ 180° – 30° – 141.32° = 8.68°.

当 B₂ ≈ 141.32° 时，角 C₂ ≈ 180° – 30° – 141.32° = 8.68°。

c₂ = a sin C₂ / sin A ≈ 4 × sin 8.68° / 0.5 ≈ 1.21

Therefore this SSA information produces two different but valid triangles.

因此，这个 SSA 条件产生了两个不同但都成立的三角形。

10. The Sine Rule and the Circumradius | 正弦定理与外接圆半径

The extended sine rule states that a / sin A = b / sin B = c / sin C = 2R, where R is the radius of the circumcircle of the triangle.

正弦定理的扩展形式为 a / sin A = b / sin B = c / sin C = 2R，其中 R 是三角形外接圆的半径。

Example: If side a = 6 and angle A = 50°, then the circumradius is R = a / (2 sin A).

例如：如果边 a = 6，角 A = 50°，则外接圆半径 R = a / (2 sin A)。

R = 6 / (2 × sin 50°) ≈ 3.92

This relationship is useful in questions about the circumcircle, and it also explains why the sine rule ratios all have the same value.

这个关系在涉及外接圆的问题中很有用，也解释了为什么正弦定理中的各个比值相等。

11. Common Mistakes and Exam Tips | 常见错误与考试技巧

Always match each side with its opposite angle. The most common error is using a side with an angle that is not directly opposite it.

始终把边与它的对角对应。最常见的错误是使用了并不对应该角的边。
Check whether your calculator is in degree mode before calculating sin A or sin B.

计算 sin A 或 sin B 前，务必确认计算器处于“角度制”模式。
When solving sin B = k, remember that B could be acute or obtuse. If the obtuse option is geometrically possible, do not ignore it.

当解 sin B = k 时，记住 B 可能是锐角也可能是钝角。只要钝角在几何上成立，就不要忽略它。
Always check the angle sum: every triangle must satisfy A + B + C = 180°.

始终检查角度和：每个三角形都必须满足 A + B + C = 180°。
Draw a clear diagram before starting. A good diagram helps you choose the correct formula and reduces careless mistakes.

解题前先画一个清晰的图形。好的图形能帮助你选择正确公式，并减少粗心错误。

12. Summary | 总结

The sine rule is a central tool in triangle geometry. It is written as a / sin A = b / sin B = c / sin C, and it can be used to find missing sides and angles when you have the right information.

正弦定理是三角形几何中的核心工具。它的表达式为 a / sin A = b / sin B = c / sin C，当条件满足时，可以用来求未知的边和角。

Remember that the SSA case is ambiguous: it may give zero, one, or two triangles. When finding an angle, you must check both the acute and obtuse possibilities. You should also remember the area formula Area = ½ ab sin C and the extended form using the circumradius.

请记住，SSA 情形可能产生不确定解：可能得到零个、一个或两个三角形。求角度时，必须同时考虑锐角和钝角两种可能性。同时也要记住面积公式面积 = ½ ab sin C，以及含有外接圆半径的

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Sine Rule and Solving Triangles | 正弦定理与三角形解法