📚 The 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 (Extended), solving non-right-angled triangles is one of the most frequently tested skills. The sine rule, the cosine rule and the area formula Area = ½ab·sin C allow you to find unknown sides, unknown angles and the area of any triangle, whether it is acute, obtuse or scalene. This article explains exactly when to use each rule, how to apply it correctly, and what examiners look for in your working.
在 IGCSE 数学（扩展课程）中，解非直角三角形是最常考的技能之一。正弦定理、余弦定理以及面积公式 Area = ½ab·sin C 能帮助你求解任意三角形的未知边、未知角和面积，无论三角形是锐角、钝角还是不等边。本文将准确说明何时使用哪条规则、如何正确应用，以及考官在你的解题过程中看重什么。
1. Labelling the Triangle | 三角形标记
Before applying any rule, you must label the triangle consistently. The standard convention is to use capital letters A, B, C for the vertices, which also represent the angles at those vertices. Lower-case letters a, b, c are used for the sides opposite those vertices. So side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.
在应用任何公式之前，你必须以统一的方式标记三角形。标准惯例是用大写字母 A、B、C 表示顶点，同时也代表顶点处的角。小写字母 a、b、c 用于表示这些角的对边。因此，边 a 是角 A 的对边，边 b 是角 B 的对边，边 c 是角 C 的对边。
This notation matters because both the sine rule and the cosine rule are stated in terms of these letters. A well-labelled diagram will prevent sign errors and help you decide which formula applies. Always sketch the triangle first and mark every given side and angle.
这种标记非常重要，因为正弦定理和余弦定理都是基于这些字母表述的。一张标注清晰的图可以避免符号错误，并帮助你判断该用哪个公式。务必先画三角形草图，并标出所有已知的边和角。
2. The Sine Rule | 正弦定理
The sine rule states that the ratio of a side length to the sine of its opposite angle is the same for all three sides of the triangle. This relationship holds for any triangle, not just right-angled ones, which makes it an extremely powerful tool.
正弦定理指出：三角形中任意边的长度与其对角正弦之比，对三条边而言都相等。这一关系对任意三角形都成立，而不仅限于直角三角形，因此它是一个非常强大的工具。
a / sin A = b / sin B = c / sin C
When you need to find a side, use the form with the unknown side on the numerator. When you need to find an angle, use the reciprocal form:
当需要求边时，使用未知边在分子上的形式。当需要求角时，使用倒数形式：
sin A / a = sin B / b = sin C / c
The sine rule is most commonly used when you know two angles and one side, because once two angles are known the third angle is immediately found using the fact that the angles of a triangle sum to 180°.
正弦定理最常用于已知两角和一边的情况，因为一旦知道两个角，第三个角就可以利用三角形内角和为 180° 立即求出。
3. Worked Example: Applying the Sine Rule | 例题：应用正弦定理
Example: In triangle ABC, angle A = 40°, angle B = 70° and side a = 8 cm. Find side b and side c.
例题：在三角形 ABC 中，角 A = 40°，角 B = 70°，边 a = 8 cm。求边 b 和边 c。
First, find the third angle: angle C = 180° − 40° − 70° = 70°. Then apply the sine rule to find side b:
首先求第三个角：角 C = 180° − 40° − 70° = 70°。然后应用正弦定理求边 b：
b / sin 70° = 8 / sin 40° ⇒ b = 8 × sin 70° / sin 40° ≈ 11.7 cm
To find side c, use the same ratio: c is opposite angle C = 70°, so c = 8 × sin 70° / sin 40° ≈ 11.7 cm. In this case b and c are equal because angles B and C are equal.
求边 c 时使用同一比例：c 是角 C = 70° 的对边，所以 c = 8 × sin 70° / sin 40° ≈ 11.7 cm。本例中 b 和 c 相等，因为角 B 和角 C 相等。
Always check that your calculator is in degree mode (DEG) before evaluating sines. A common mistake is leaving it in radians mode, which produces completely different and incorrect values.
在计算正弦之前，务必检查计算器处于角度模式（DEG）。常见错误是让计算器停留在弧度模式，这会得出完全错误的结果。
4. The Cosine Rule | 余弦定理
The cosine rule connects the three sides of a triangle to one of its angles. It can be viewed as a generalisation of Pythagoras’ theorem: when the angle is 90°, cos 90° = 0, and the formula reduces to a² = b² + c², which is exactly Pythagoras’ theorem.
余弦定理将三角形的三条边与其中一个角联系起来。它可以看作勾股定理的推广：当角为 90° 时，cos 90° = 0，公式简化为 a² = b² + c²，这正是勾股定理。
a² = b² + c² − 2bc·cos A
To find an angle when all three sides are known, rearrange the formula to make cos A the subject:
当已知三边求角时，将公式变形，使 cos A 成为主项：
cos A = (b² + c² − a²) / 2bc
Notice the pattern: the side you are finding or the angle you are finding determines the placement of each term. The side a is always opposite angle A, and the two sides b and c are the sides adjacent to angle A.
注意其中的规律：你要求的边或角决定了各项的位置。边 a 始终是角 A 的对边，而 b 和 c 是角 A 的两条邻边。
5. Worked Example: Applying the Cosine Rule | 例题：应用余弦定理
Example 1: In triangle PQR, PQ = 9 cm, PR = 7 cm and angle P = 50°. Find QR.
例 1：在三角形 PQR 中，PQ = 9 cm，PR = 7 cm，角 P = 50°。求 QR。
Here QR is opposite angle P, so let QR = p. Apply the cosine rule with b = 9, c = 7 and A = 50°:
这里 QR 是角 P 的对边，设 QR = p。应用余弦定理，取 b = 9，c = 7，A = 50°：
p² = 9² + 7² − 2 × 9 × 7 × cos 50° = 81 + 49 − 126 × 0.6428 ≈ 49.0
p = √49.0 ≈ 7.0 cm
Example 2: A triangle has sides of length 5 cm, 6 cm and 7 cm. Find the largest angle.
例 2：一个三角形的三边长分别为 5 cm、6 cm、7 cm。求最大角。
The largest angle is opposite the longest side, 7 cm. Using the rearranged cosine rule with a = 7, b = 6, c = 5:
最大角对最长边 7 cm。使用变形后的余弦定理，取 a = 7，b = 6，c = 5：
cos A = (6² + 5² − 7²) / (2 × 6 × 5) = (36 + 25 − 49) / 60 = 12 / 60 = 0.2
A = cos⁻¹(0.2) ≈ 78.5°
Always take the square root when finding a side, and be careful to follow the correct order of operations when evaluating the expression.
求边时务必开平方根，并在计算表达式时注意运算顺序。
6. The Area Formula | 面积公式
You already know the basic area formula Area = ½ × base × height. However, in many triangle problems the perpendicular height is not given. When you know two sides and the included angle, you can use the trigonometric area formula:
你已经知道基本面积公式 Area = ½ × 底 × 高。然而，在许多三角形问题中并未给出垂直高度。当你知道两条边及其夹角时，可以使用三角面积公式：
Area = ½ab·sin C
Here a and b are any two sides of the triangle, and C is the angle between them. The formula works for any choice of two sides, provided the angle used is the one included between them.
其中 a 和 b 是三角形的任意两条边，C 是它们之间的夹角。只要所使用的角是所选两条边的夹角，公式对任意两边组合都适用。
This formula is especially valuable in multi-part IGCSE questions. A typical problem might ask you to use the cosine rule to find a missing side, then use the sine rule to find an angle, and finally use the area formula to find the area.
这个公式在多小问的 IGCSE 综合题中尤为宝贵。典型问题可能要求你先用余弦定理求一条缺失边，再用正弦定理求一个角，最后用面积公式求面积。
7. Choosing the Correct Rule | 选择正确的规则
Many students struggle not with the calculations, but with deciding which rule to use. The table below summarises the decision process based on the information given in the question.
许多学生的困难不在于计算本身，而在于判断该用哪条规则。下表根据题目给出的信息总结了决策流程。
| Given Information | Rule to Use |
| Two angles and one side (AAS / ASA) | Sine rule |
| Two sides and a non-included angle (SSA) | Sine rule — check ambiguous case |
| Two sides and the included angle (SAS) | Cosine rule to find the third side |
| Three sides (SSS) | Cosine rule to find an angle |
|
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