📚 The Sine and Cosine Rules: Solving Triangles | 正弦定理与余弦定理：解三角形
Trigonometry is a key topic in IGCSE Mathematics, and the sine rule and cosine rule are two powerful formulas used to solve non-right-angled triangles. Once you know when and how to apply them, many exam questions become straightforward.
三角学是 IGCSE 数学中的重点内容。正弦定理与余弦定理是解非直角三角形的两大有力工具。只要掌握它们的适用条件和正确用法，很多考试题都会变得简单明了。
1. What Are the Two Rules? | 两大定理是什么
The sine rule connects a side to the sine of its opposite angle. The cosine rule connects a side to the other two sides and the included angle. Both rules work for any triangle, not just a right-angled triangle.
正弦定理将一条边与其对角的正弦值联系起来；余弦定理则将一条边与另外两条边及其夹角联系起来。两大定理对任意三角形都适用，不限于直角三角形。
- Use the sine rule when you know two angles and one side, or two sides and a non-included angle.
- 当你已知两角一边，或两边及其中的一对角时，应使用正弦定理。
- Use the cosine rule when you know two sides and the included angle, or all three sides.
- 当你已知两边及其夹角，或已知三边时，应使用余弦定理。
2. The Sine Rule: Formula | 正弦定理：公式
For any triangle ABC, label sides a, b and c 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 the ratio of each side to the sine of its opposite angle is constant.
对任意三角形 ABC，设边 a 与角 A 相对，边 b 与角 B 相对，边 c 与角 C 相对。正弦定理说明：每条边与其对角正弦值的比是一个常数。
a / sin A = b / sin B = c / sin C
calculating an angle, it is often easier to use the reciprocal form sin A / a = sin B / b.
这个等式可以通过变形来求缺失的边或角。当需要求角时，使用倒数的形式 sin A / a = sin B / b 通常更为方便。
3. Using the Sine Rule to Find a Side | 用正弦定理求边
and side a = 12 cm. To find side b, write the sine rule in the form b / sin B = a / sin A.
假设在三角形 ABC 中，角 A = 50°，角 B = 72°，边 a = 12 cm。要求边 b，可将正弦定理写成 b / sin B = a / sin A 的形式。
b = 12 × sin 72° ÷ sin 50° = 14.9 cm
Always check that your calculator is in degree mode, not radian mode, before evaluating sine values.
在计算正弦值之前，务必检查计算器处于角度模式（Degree Mode），而不是弧度模式（Radian Mode）。
4. Using the Sine Rule to Find an Angle | 用正弦定理求角
. If b = 15, c = 10 and angle B = 80°, then sin C can be found from c / sin C = b / sin B.
求角时使用倒数形式。若 b = 15，c = 10，角 B = 80°，则由 c / sin C = b / sin B 可求出 sin C。
sin C = 10 × sin 80° ÷ 15 = 0.657
.p.). Remember that the inverse sine function gives a principal value, but another possible angle exists; this leads to the ambiguous case.
因此 C = arcsin(0.657) = 41.0°（精确到一位小数）。注意反三角函数给出一个主值，但还可能存在另一个符合条件的角，这就引出了“两解情况”。
5. The Ambiguous Case | 两解情况
ple, if sin C = 0.657, then C could be 41.0° or 139.0°, because sin(180° − 41°) = sin 41°.
当你已知两边及其中一边对角（SSA）时，正弦定理可能产生两个不同的三角形。例如，若 sin C = 0.657，则 C 可能是 41.0° 或 139.0°，因为 sin(180° − 41°) = sin 41°。
- If the obtuse option makes the angle sum exceed 180°, discard it.
- 如果钝角选项会使三角形内角和超过 180°，就应舍去。
- If both options are valid, give both answers clearly.
- 如果两个选项都满足条件，则需要明确写出两个答案。
- This situation only arises with SSA, never with ASA or AAS.
- 这种情况只会在 SSA 中出现，而不会出现在 ASA 或 AAS 中。
6. The Cosine Rule: Formula | 余弦定理：公式
The cosine rule is used when you have two sides and the included angle (SAS) or three sides (SSS). For a triangle with sides a, b and c, and angle A opposite side a, the rule is as follows.
余弦定理适用于已知两边及其夹角（SAS）或已知三边（SSS）的情形。对于边 a、b、c，角 A 与边 a 相对，规则如下。
a² = b² + c² − 2bc cos A
ng the triangle, you can also write b² = a² + c² − 2ac cos B and c² = a² + b² − 2ab cos C.
通过重新标注三角形，也可写为 b² = a² + c² − 2ac cos B 和 c² = a² + b² − 2ab cos C。
7. Using the Cosine Rule to Find a Side | 用余弦定理求边
se b = 8 cm, c = 11 cm and angle A = 35°. To find side a, substitute into the cosine rule.
设 b = 8 cm，c = 11 cm，角 A = 35°。要求边 a，代入余弦定理即可。
a² = 8² + 11² − 2 × 8 × 11 × cos 35°
a² = 64 + 121 − 144.1 = 40.9
a = √40.9 = 6.39 cm
This method is reliable because the cosine rule always gives a unique value for a side when the included angle is known.
这种方法非常可靠，因为当夹角已知时，余弦定理求边只会得到唯一值。
8. Using the Cosine Rule to Find an Angle | 用余弦定理求角
Rearrange the cosine rule to make cos A the subject. This form is useful when all three sides are known.
将余弦定理变形，用其他量表示 cos A。当三边都已知时，这个形式非常有用。
cos A = (b² + c² − a²) ÷ (2bc)
9, c = 12. Then cos A = (9² + 12² − 7²) ÷ (2 × 9 × 12) = 176 ÷ 216 = 0.8148, so A = 35.4°.
例：a = 7，b = 9，c = 12。则 cos A = (9² + 12² − 7²) ÷ (2 × 9 × 12) = 176 ÷ 216 = 0.8148，因此 A = 35.4°。
The inverse cosine function gives a unique angle between 0° and 180°, so there is no ambiguous case with the cosine rule.
反余弦函数在 0° 到 180° 之间给出唯一角，因此余弦定理不存在两解情况。
9. Area of a Triangle Using Trigonometry | 用三角函数求三角形面积
For any triangle, if you know two sides and the included angle, the area can be found using the formula below. This is especially useful in non-right-angled triangles where the perpendicular height is not given.
对任意三角形，若已知两边及其夹角，可用下面这个公式求面积。对于没有给出垂直高度的非直角三角形，这个公式尤其方便。
Area = ½ ab sin C
Here, a and b are two sides, and C is the angle between them. Make sure the angle is measured in degrees, and give your final answer in square units.
其中 a 和 b 是两条边，C 是它们的夹角。确保角度使用度数表示，最终答案要使用适当的面积单位。
10. Which Rule Should I Use? | 如何选择合适定理
Exam questions often require you to identify the most efficient method. The table below summarises the choice based on the information given.
考试题目经常要求你判断最有效的解题方法。下表总结了根据已知条件应如何选择。
| Given Information | Rule to Use |
| Two angles and one side (ASA or AAS) | Sine Rule |
| Two sides and a non-included angle (SSA) | Sine Rule, check ambiguous case |
| Two sides and the included angle (SAS) | Cosine Rule for the third side |
| All three sides (SSS) | Cosine Rule for an angle |
11. Worked Example: Mixed Problem | 综合例题
A triangular garden has sides 8 m, 11 m and 14 m. Find the largest angle and the area of the garden. The largest angle is opposite the longest side, which is 14 m.
一块三角形花园的三边长分别为 8 m、11 m 和 14 m。求最大角以及花园面积。最大角对着最长边 14 m。
cos A = (8² + 11² − 14²) ÷ (2 × 8 × 11) = (64 + 121 − 196) ÷ 176 = −11 ÷ 176 = −0.0625
A = cos⁻¹(−0.0625) = 93.6°
× 8 × 11 × sin 93.6° ≈ 43.9 m². The largest angle is 93.6°, so this is an obtuse triangle.
求面积时，用两边及其夹角：Area = ½ × 8 × 11 × sin 93.6° ≈ 43.9 m²。最大角为 93.6°，因此这是钝角三角形。
12. Common Mistakes and Checklist | 常见错误与检查清单
The most common errors in IGCSE trigonometry questions come from careless use of the calculator or mixing up the formulas. Follow the checklist below to stay accurate.
IGCSE 三角学题目中最常见的错误来自计算器使用不当或混淆公式。请按照下面的清单来确保准确性。
- Always use degree mode on your calculator.
- 始终将计算器设置为角度模式。
- Label the triangle clearly before substituting values.
- 代入数值前先清楚标注三角形的边和角。
- Check whether the sine rule has two possible answers.
- 检查正弦定理是否可能存在两个答案。
- Use the cosine rule for SAS and SSS, not the sine rule.
- 在处理 SAS 和 SSS 时使用余弦定理，而不是正弦定理。
- Round only at the final answer, not during intermediate steps.
- 只在最后答案处四舍五入，不要在中间步骤中过早取近似值。
Once you are comfortable with these two rules, solving triangle problems becomes a systematic process. Practice identifying the correct rule quickly, and always sketch the triangle before starting.
当你熟练掌握这两个定理后，解三角形问题将变成一个系统化的过程。练习快速判断应使用的定理，并在做题前先画出三角形草图。
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