📚 Sine Rule and Cosine Rule | 正弦定理与余弦定理
In IGCSE mathematics, you must be able to solve non-right-angled triangles. The sine rule and cosine rule are the two essential tools for finding unknown sides and angles when you cannot use Pythagoras’ theorem or basic trigonometric ratios.
在 IGCSE 数学中，你必须会解非直角三角形。当你无法使用勾股定理或基本三角比时，正弦定理和余弦定理是求未知边和角的两个核心工具。
1. Why We Need New Rules | 为什么需要新定理
Right-angled triangle trigonometry only works when one angle is exactly 90°. However, many IGCSE problems involve acute or obtuse triangles, where no right angle is given. In these cases, the sine rule and cosine rule allow us to find any unknown side or angle using limited information.
直角三角形的三角比只在一个角恰好为 90° 时才能使用。然而，许多 IGCSE 题目涉及锐角三角形或钝角三角形，没有给出直角。在这些情况下，正弦定理和余弦定理让我们仅凭有限信息就能求出任意未知边或角。
2. Labelling Conventions | 三角形的标记约定
In any triangle ABC, the common convention is to label the side opposite angle A as a, the side opposite angle B as b, and the side opposite angle C as c. This pairing between an angle and its opposite side is the foundation of both rules.
在任意三角形 ABC 中，常用约定是把角 A 的对边标为 a，角 B 的对边标为 b，角 C 的对边标为 c。角与其对边的这种一一对应关系是两条定理的基础。
3. The Sine Rule Formula | 正弦定理公式
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，三角形的任意一边长度与其对角正弦之比在整个三角形中保持不变。
a / sin A = b / sin B = c / sin C
Equivalently, you can invert the fractions: sin A / a = sin B / b = sin C / c. You should choose the version that makes the unknown the numerator.
等价地，你可以把分式倒过来：sin A / a = sin B / b = sin C / c。你应选择让未知量位于分子的那种形式。
4. Using the Sine Rule to Find a Side | 用正弦定理求边
Use the sine rule to find a missing side when you know two angles and one side (AAS or ASA). You need a complete angle-side pair and one additional angle or side.
当已知两个角和一个边（角角边或角边角）时，可以用正弦定理求缺失的边。你需要一对完整的角-边以及另一个角或边。
Example: In triangle ABC, A = 50°, B = 60°, and a = 8 cm. Find b.
例题：在三角形 ABC 中，A = 50°，B = 60°，a = 8 cm，求 b。
Write the ratio: a / sin A = b / sin B. Substitute: 8 / sin 50° = b / sin 60°.
写出比例式：a / sin A = b / sin B。代入：8 / sin 50° = b / sin 60°。
Rearrange: b = 8 sin 60° / sin 50° ≈ 8 × 0.8660 / 0.7660 ≈ 9.04 cm.
移项：b = 8 sin 60° / sin 50° ≈ 8 × 0.8660 / 0.7660 ≈ 9.04 cm。
5. Using the Sine Rule to Find an Angle | 用正弦定理求角
You can also rearrange the sine rule to find an unknown angle when you know two sides and one corresponding opposite angle. This situation is often abbreviated as SSA.
当你已知两边和一个对应的对角时，也可以把正弦定理变形来求未知角。这种情况通常简写为边边角（SSA）。
Example: In triangle ABC, a = 7 cm, b = 9 cm, and A = 40°. Find angle B.
例题：在三角形 ABC 中，a = 7 cm，b = 9 cm，A = 40°，求角 B。
Use sin B / b = sin A / a, so sin B = b sin A / a = 9 sin 40° / 7 ≈ 0.8265.
使用 sin B / b = sin A / a，所以 sin B = b sin A / a = 9 sin 40° / 7 ≈ 0.8265。
Then B ≈ 55.7°. However, since sin(180° − θ) = sin θ, another possible answer is 124.3°. This is the ambiguous case.
然后 B ≈ 55.7°。但是，由于 sin(180° − θ) = sin θ，另一个可能的答案是 124.3°。这就是模糊情形。
6. The Ambiguous Case of the Sine Rule | 正弦定理的模糊情形
When using the sine rule to find an angle from SSA data, two different triangles may satisfy the given information: one with an acute angle and one with an obtuse angle. You must decide whether the obtuse answer is valid using the angle sum of 180°.
当使用正弦定理根据 SSA 数据求角时，可能会有两个不同的三角形满足给定条件：一个是锐角，另一个是钝角。你必须利用三角形内角和为 180° 来判断钝角答案是否成立。
For example, with B ≈ 55.7° or 124.3° and A = 40°, both sums A + B are less than 180°, so both triangles are possible. If the obtuse candidate made the sum exceed 180°, it would be rejected.
例如，当 B ≈ 55.7° 或 124.3°，且 A = 40° 时，两组 A + B 的和都小于 180°，因此两个三角形都可能成立。如果钝角候选值使角度和超过 180°，则它会被排除。
7. The Cosine Rule Formula | 余弦定理公式
The cosine rule links the three sides of a triangle to the cosine of one angle. It is especially useful when the sine rule cannot be applied because no complete angle-side pair is known.
余弦定理把三角形的三条边与其中一个角的余弦联系起来。当没有完整的角-边对导致正弦定理无法使用时，它特别有用。
a² = b² + c² − 2bc cos A
Similar versions exist for the other angles: b² = a² + c² − 2ac cos B and c² = a² + b² − 2ab cos C.
对其他角也有类似的版本：b² = a² + c² − 2ac cos B 和 c² = a² + b² − 2ab cos C。
8. Using the Cosine Rule to Find a Side | 用余弦定理求边
Use the cosine rule to find a missing side when you know two sides and the included angle (SAS). The included angle is the angle between the two known sides.
当已知两边及其夹角（SAS）时，可以用余弦定理求缺失的边。夹角是两条已知边之间的角。
Example: Given a = 8 cm, b = 5 cm, and C = 60°, find side c.
例题：已知 a = 8 cm，b = 5 cm，C = 60°，求边 c。
Using c² = a² + b² − 2ab cos C: c² = 8² + 5² − 2 × 8 × 5 × cos 60° = 64 + 25 − 80 × 0.5 = 49.
使用 c² = a² + b² − 2ab cos C：c² = 8² + 5² − 2 × 8 × 5 × cos 60° = 64 + 25 − 80 × 0.5 = 49。
Therefore c = 7 cm.
因此 c = 7 cm。
9. Using the Cosine Rule to Find an Angle | 用余弦定理求角
If you know all three sides but none of the angles (SSS), rearrange the cosine rule to solve for an angle.
如果你已知三条边但不知道任何角（SSS），可以重新整理余弦定理来求角。
cos A = (b² + c² − a²) / (2bc)
Example: In triangle ABC, a = 7 cm, b = 8 cm, c = 9 cm. Find angle A.
例题：在三角形 ABC 中，a = 7 cm，b = 8 cm，c = 9 cm，求角 A。
cos A = (8² + 9² − 7²) / (2 × 8 × 9) = (64 + 81 − 49
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