📚 IGCSE Mathematics: Sine Rule, Cosine Rule and Triangle Area | IGCSE 数学：正弦定理、余弦定理与三角形面积
In IGCSE Mathematics, especially on the Extended syllabus, you must be able to solve non-right-angled triangles. The sine rule, the cosine rule and the area formula using sine are essential tools for finding unknown sides, unknown angles and areas when you are not given a right angle.
在 IGCSE 数学（尤其是扩展课程）中，你必须会解非直角三角形。正弦定理、余弦定理以及含正弦的面积公式是必备工具，用于在题目没有给出直角时求未知边、未知角和面积。
1. When to Use the Sine and Cosine Rules | 何时使用正弦定理与余弦定理
For a non-right-angled triangle, you cannot use basic SOHCAHTOA or the Pythagorean theorem directly. Instead, you choose the sine rule or the cosine rule depending on the information you are given.
对于非直角三角形，你不能直接使用基本的 SOHCAHTOA 或勾股定理，而要根据已知信息选择正弦定理或余弦定理。
The sine rule is usually used when you know two angles and one side, or two sides and a non-included angle. The cosine rule is used when you know two sides and the included angle, or all three sides.
正弦定理通常用于已知两角一边，或已知两边及一个非夹角。余弦定理用于已知两边及夹角，或已知三条边。
Always label the vertices with capital letters A, B, C and the opposite sides with lowercase letters a, b, c. This labelling is the key to applying the formulas correctly.
始终用大写字母 A、B、C 标记顶点，并用小写字母 a、b、c 标记对应的对边。这种标记是正确应用公式的关键。
Use the following summary table to decide quickly:
可以使用下面的总结表快速判断：
| Information given | 已知信息 | Recommended rule | 推荐公式 |
| Two angles and one side | 两角一边 | Sine rule | 正弦定理 |
| Two sides and a non-included angle | 两边及非夹角 | Sine rule, but check for ambiguous case | 正弦定理，但需检查两解情况 |
| Two sides and the included angle | 两边及夹角 | Cosine rule | 余弦定理 |
| Three sides | 三条边 | Cosine rule, rearranged for an angle | 余弦定理变形求角 |
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 constant for all three sides of a triangle.
正弦定理指出，三角形中任意一条边的长度与其对角的正弦值之比对于三条边都是相等的。
a / sin A = b / sin B = c / sin C
sin A / a = sin B / b = sin C / c
Use the first form when you are finding a missing side, and use the second form when you are finding a missing angle. Both forms are equivalent, but choosing the right one makes the algebra easier.
求未知边时使用第一种形式，求未知角时使用第二种形式。两种形式是等价的，但选择合适的形式可以使代数运算更简单。
ow angle A, angle B and side a, you can find side b by rearranging: b = (a sin B) / sin A.
例如，如果你知道角 A、角 B 和边 a，可以通过移项求边 b：b = (a sin B) / sin A。
The sine rule is particularly efficient when two angles are given, because the third angle is found immediately from the angle sum of 180°.
当已知两个角时，正弦定理特别高效，因为第三个角可以立即由三角形内角和 180° 求出。
3. The Cosine Rule | 余弦定理
The cosine rule links the three sides of a triangle with the cosine of one of its angles. It is especially useful when you do not have a matching side-angle pair.
余弦定理将三角形的三条边与其中一个角的余弦联系起来。当你没有对应的边角对时，这个定理尤其有用。
a² = b² + c² − 2bc cos A
be rearranged for the other sides: b² = a² + c² − 2ac cos B and c² = a² + b² − 2ab cos C.
该公式可以变形为其他边的形式：b² = a² + c² − 2ac cos B 和 c² = a² + b² − 2ab cos C。
le when all three sides are known, rearrange the formula as: cos A = (b² + c² − a²) / 2bc.
当已知三条边求角度时，将公式变形为：cos A = (b² + c² − a²) / 2bc。
Always take care with the order of operations: multiply the two sides first, then multiply by 2 and by the cosine of the included angle, before subtracting from the sum of squares.
务必注意运算顺序：先平方相加，然后减去 2 乘以两条边的乘积再乘以夹角的余弦，注意括号的使用。
The cosine rule is also used in bearings, navigation and 3D problems when two lengths and the angle between them are known.
余弦定理也常用于方位角、航海和三维问题中，当已知两条长度及其夹角时非常实用。
4. Area of a Triangle Using Sine | 用正弦求三角形面积
For any triangle, the area can be found using two sides and the included angle. This is often called the sine area formula.
对于任意三角形，面积都可以用两条边和它们的夹角来求。这通常被称为正弦面积公式。
Area = ½ ab sin C = ½ bc sin A = ½ ca sin B
This formula is particularly useful when you know two sides and the angle between them, and it avoids the need to calculate a perpendicular height separately.
当你知道两条边及其夹角时，这个公式特别有用，它避免了单独计算垂线高度。
xample, if a = 6 cm, b = 8 cm and angle C = 30°, the area is ½ × 6 × 8 × sin 30° = 12 cm².
例如，如果 a = 6 cm，b =
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