📚 Trigonometry: Sine Rule, Cosine Rule and Area of a Triangle | 三角函数：正弦定理、余弦定理与三角形面积
In IGCSE Mathematics, trigonometry is not limited to right-angled triangles. When a triangle has no 90° angle, the sine rule, cosine rule and the sine-based area formula allow you to find unknown sides, angles and areas. This revision guide explains when and how to use each tool, with clear examples and exam tips.
在 IGCSE 数学中，三角学并不局限于直角三角形。当三角形没有 90° 角时，正弦定理、余弦定理以及基于正弦的面积公式可以帮助你求出未知边、未知角和面积。本复习指南将说明何时以及如何使用每一种方法，并提供清晰的示例和考试提示。
1. When to Use Trigonometry on Non-Right-Angled Triangles | 何时对非直角三角形使用三角学
Triangles without a right angle are called oblique triangles. You cannot apply SOHCAHTOA directly, so you need rules that connect all three sides and angles. The sine rule works when you have a matching pair of a side and its opposite angle. The cosine rule works when you have either three sides or two sides and the included angle. The area formula uses two sides and the included angle.
没有直角的三角形叫作斜三角形。你不能直接使用 SOHCAHTOA，因此需要使用能够联系三条边和三个角的规则。正弦定理适用于已知一对边与对角的情况。余弦定理适用于已知三边或两边及夹角的情况。面积公式则使用两边及其夹角。
For right-angled triangles, use Pythagoras’ theorem or the basic trigonometric ratios sin, cos and tan. For oblique triangles, choose the rule according to the information you are given.
对于直角三角形，使用毕达哥拉斯定理或基本三角比 sin、cos 和 tan。对于斜三角形，要根据题目给出的信息选择相应的规则。
2. Labelling Triangles and Notation | 三角形标注与符号
Use capital letters A, B, C for angles at the vertices, and lowercase letters a, b, c for the sides opposite those angles. This convention is essential because the sine and cosine rules only work when sides and angles are paired correctly.
用大写字母 A、B、C 表示顶点处的角，用小写字母 a、b、c 表示这些角所对的边。这个约定非常重要，因为只有正确配对角和边，正弦定理和余弦定理才能成立。
In triangle ABC, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. If a question does not label the triangle, draw your own diagram and mark the given values clearly.
在三角形 ABC 中，边 a 对角 A，边 b 对角 B，边 c 对角 C。如果题目没有标注三角形，请自己画出图形并清楚地标出已知值。
3. The Sine Rule | 正弦定理
The sine rule states that the ratio of a side to the sine of its opposite angle is constant for all three sides of a triangle.
正弦定理指出，在任意三角形中，一条边与其对角正弦的比值对于三条边都相等。
a / sin A = b / sin B = c / sin C
sin A / a = sin B / b = sin C / c
Use the sine rule when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). The first version is useful for finding a side; the inverted version is useful for finding an angle.
当你已知两角和一边（AAS 或 ASA）或两边及一个非夹角（SSA）时，使用正弦定理。第一种形式用于求边；倒数形式用于求角。
4. Using the Sine Rule: Finding a Side | 使用正弦定理求边
Example: In triangle ABC, A = 42°, B = 63° and b = 8 cm. Find side a.
示例：在三角形 ABC 中，A = 42°，B = 63°，b = 8 cm。求边 a。
Write the sine rule using the two given angles and their opposite sides.
使用两个已知角及其对边写出正弦定理。
a / sin 42° = 8 / sin 63°
Rearrange and calculate:
移项并计算：
a = (8 sin 42°) / sin 63° ≈ 6.01 cm
Always check that your calculator is in degree mode. If your answer looks unreasonable, re-read the question and check the pairing of sides and angles.
始终检查计算器是否处于角度制模式。如果答案看起来不合理，请重新审题并检查边与角的配对是否正确。
5. Using the Sine Rule: Finding an Angle | 使用正弦定理求角
Example: In triangle ABC, a = 9 cm, b = 12 cm and A = 36°. Find angle B.
示例：在三角形 ABC 中，a = 9 cm，b = 12 cm，A = 36°。求角 B。
Use the inverted sine rule:
使用正弦定理的倒数形式：
sin B / 12 = sin 36° / 9
Rearrange and solve:
移项并求解：
sin B = (12 sin 36°) / 9 ≈ 0.784
B = sin⁻¹(0.784) ≈ 51.6°
However, because sin(180° – θ) = sin θ, angle B could also be 180° – 51.6° = 128.4° if the triangle can exist. This is called the ambiguous case and is discussed later.
但是，由于 sin(180° – θ) = sin θ，如果三角形可以存在，角 B 也可能是 180° – 51.6° = 128.4°。这就是后面要讨论的歧义情形。
6. The Cosine Rule | 余弦定理
The cosine rule links the three sides of a triangle with the cosine of one angle. It is useful when you have two sides and the included angle (SAS) or all three sides (SSS).
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。当你已知两边及夹角（SAS）或三边（SSS）时，它非常有用。
a² = b² + c² – 2bc cos A
cos A = (b² + c² – a²) / (2bc)
The first form is used to find a side when you know the other two sides and the included angle. The second form is used to find an angle when you know all three sides.
第一种形式用于已知另外两边及其夹角时求边。第二种形式用于已知三边时求角。
7. Using the Cosine Rule: Finding a Side | 使用余弦定理求边
Example: Given b = 10 cm, c = 14 cm and A = 60°, find side a.
示例：已知 b = 10 cm，c = 14 cm，A = 60°，求边 a。
a² = 10² + 14² – 2(10)(14) cos 60°
a² = 100 + 196 – 280 × 0.5
a² = 296 – 140 = 156
a = √156 ≈ 12.5 cm
Remember to take the positive square root because lengths are positive. Do not round until the final answer unless the question says otherwise.
请记住取正的平方根，因为长度为正。除非题目另有说明，否则在得到最终答案之前不要提前取近似值。
8. Using the Cosine Rule: Finding an Angle | 使用余弦定理求角
Example: Given a = 7 cm, b = 10 cm and c = 12 cm, find angle A.
示例：已知 a = 7 cm，b = 10 cm，c = 12 cm，求角 A。
cos A = (10² + 12² – 7²) / (2 × 10 × 12)
cos A = (100 + 144 – 49) / 240
cos A = 195 / 240 = 0.8125
A = cos⁻¹(0.8125) ≈ 35.7°
The cosine rule gives a unique angle between 0° and 180°, so there is no ambiguous case when you use SSS or SAS.
余弦定理会给出 0° 到 180° 之间的唯一角度，因此使用 SSS 或 SAS 时不会出现歧义情形。
9. Area of a Triangle Using Sine | 用正弦求三角形面积
When you know two sides and the included angle, the area of a triangle can be found using:
当你已知两边及其夹角时，可以用以下公式求三角形面积：
Area = 1/2 ab sin C
The letters can be changed according to the sides and angle you are using: 1/2 bc sin A or 1/2 ac sin B.
字母可以根据你使用的边和角进行替换：1/2 bc sin A 或 1/2 ac sin B。
Example: Given a = 8 cm, b = 11 cm and C = 40°, calculate the area.
示例：已知 a = 8 cm，b = 11 cm，C = 40°，计算面积。
Area = 1/2 × 8 × 11 × sin 40° ≈ 28.3 cm²
Include the correct square units in your final answer. This formula is particularly useful when the perpendicular height is not given.
在最终答案中要写出正确的平方单位。当题目没有给出垂直高度时，这个公式特别有用。
10. The Ambiguous Case of the Sine Rule | 正弦定理的歧义情形
When you are given two sides and a non-included angle (SSA), there may be two possible triangles, one triangle or none. Because sin(180° – θ) = sin θ, an acute calculated angle could have an obtuse supplement.
当你已知两边和一个非夹角（SSA）时，可能存在两个三角形、一个三角形或不存在三角形。由于 sin(180° – θ) = sin θ，一个锐角计算结果可能对应一个钝角补角。
Example: Suppose a = 6 cm, b = 7 cm and A = 40°. Using the sine rule gives B ≈ 48.6° or B ≈ 131.4°. Since 131.4° + 40° = 171.4° < 180°, both solutions are valid.
示例：假设 a = 6 cm，b = 7 cm，A = 40°。使用正弦定理可得 B ≈ 48.6° 或 B ≈ 131.4°。因为 131.4° + 40° = 171.4° < 180°，两个解都成立。
If the obtuse option plus the given angle is greater than or equal to 180°, it must be rejected. Also check that the remaining angle is positive.
如果钝角选项加上已知角大于或等于 180°，则必须舍去。同时还要检查剩余角是否为正。
11. Mixed Problem-Solving Strategy | 混合问题求解策略
Use a step-by-step decision process when a trigonometry problem involves a non-right-angled triangle.
当三角学问题涉及非直角三角形时，请使用分步决策法。
- Right-angled triangle: use SOHCAHTOA or Pythagoras’ theorem. 直角三角形：使用 SOHCAHTOA 或毕达哥拉斯定理。
- Two angles and one side: use the sine rule. 两角和一边：使用正弦定理。
- Two sides and a non-included angle: use the sine rule and check the ambiguous case. 两边及非夹角：使用正弦定理并检查歧义情形。
- Two sides and the included angle: use the cosine rule. 两边及夹角：使用余弦定理。
- Three sides: use the cosine rule to find an angle. 三边：使用余弦定理求角。
- Area with two sides and included angle: use Area = 1/2 ab sin C. 已知两边及夹角求面积：使用 Area = 1/2 ab sin C。
Drawing a clear labelled diagram is the most reliable way to avoid choosing the wrong rule.
画出清晰且标注准确的图形是避免选错规则的最可靠方法。
12. Exam Tips and Common Mistakes | 考试提示与常见错误
Before solving, check your calculator is in degree mode, not radian mode. Label the triangle correctly and write down the rule you are using.
解题前，检查计算器是否处于角度制模式，而不是弧度制模式。正确标注三角形，并写出你使用的规则。
Common mistakes include:
常见错误包括：
- Using the cosine rule when a clear opposite pair is given. 有明确对角关系时误用余弦定理。
- Forgetting to consider the ambiguous case in SSA problems. 在 SSA 问题中忘记考虑歧义情形。
- Rounding too early and losing accuracy. 过早取近似值导致精度下降。
- Giving a length without units or using incorrect square units for area. 长度不写单位，或面积使用错误的平方单位。
- Mislabelling sides and angles so the formula does not match the diagram. 边和角标错，导致公式与图形不符。
If your calculated angle does not fit the triangle angle sum, re-check every step before finalising your answer
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