📚 Sine and Cosine Rules | 正弦定理与余弦定理
In IGCSE Mathematics, solving non-right-angled triangles is a core skill. The sine and cosine rules allow you to find unknown sides and angles in any triangle, not just right-angled ones. This article explains how to use these rules, when to apply them, and how to avoid common exam mistakes.
在 IGCSE 数学中，解非直角三角形是一项核心技能。正弦定理与余弦定理能帮助你在任意三角形中求未知边和未知角，而不仅限于直角三角形。本文将讲解如何使用这两条定理、何时应用它们，以及如何避免常见的考试错误。
1. Why We Need the Sine and Cosine Rules | 为什么需要正弦定理与余弦定理
Right-angled trigonometry using sine, cosine and tangent only works when one angle is 90°. Many real-life and exam problems involve triangles with no right angle. For these triangles, Pythagoras' theorem and basic SOH CAH TOA do not apply directly. The sine rule and cosine rule are the standard tools for non-right-angled triangles.
使用正弦、余弦和正切的直角三角形三角学只适用于有一个角为 90° 的情况。许多现实生活和考试问题涉及没有直角的三角形。对于这些三角形，勾股定理和基本的 SOH CAH TOA 不能直接使用。正弦定理与余弦定理是解非直角三角形的标准工具。
2. Labelling a Triangle | 三角形的标记
Before using the rules, label the triangle consistently. Use capital letters A, B, C for the angles, and lower-case letters a, b, c for the sides opposite those angles. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. This labelling is essential for applying the formulae correctly.
在使用定理之前，要统一标记三角形。用大写字母 A、B、C 表示角，用小写字母 a、b、c 表示这些角所对的边。边 a 对角 A，边 b 对角 B，边 c 对角 C。这种标记对于正确应用公式至关重要。
3. The Sine Rule: Formula and Proof Idea | 正弦定理：公式与证明思路
The sine rule states that in any triangle ABC, the ratio of a side length to the sine of its opposite angle is constant. The formula is written as:
正弦定理指出，在任意三角形 ABC 中，边长与其对角正弦的比值是常数。公式写作：
a / sin A = b / sin B = c / sin C
The proof uses the area formula ½ ab sin C. Since the area can be written in three equivalent ways, equating any two gives the sine rule. This proof shows why the rule works for any triangle, whether acute or obtuse.
证明使用了面积公式 ½ ab sin C。由于面积可以用三种等价方式表示，将任意两种相等即可得到正弦定理。该证明说明了为什么这一定理适用于任何三角形，无论是锐角三角形还是钝角三角形。
4. Using the Sine Rule to Find a Missing Side | 用正弦定理求未知边
angle A = 40°, angle B = 60°, and side a = 8 cm. To find side b, rearrange the sine rule:
如果你知道两个角和一个边，就可以求另一个边。例如，假设角 A = 40°，角 B = 60°，边 a = 8 cm。要求边 b，重新整理正弦定理：
b = a sin B / sin A
≈ 8 × 0.8660 / 0.6428 ≈ 10.78 cm. Always keep the units and round only at the final step.
代入数值：b = 8 × sin 60° / sin 40°。因为 sin 60° ≈ 0.8660，sin 40° ≈ 0.6428，得到 b ≈ 8 × 0.8660 / 0.6428 ≈ 10.78 cm。始终保留单位，并且只在最后一步取近似值。
5. Using the Sine Rule to Find a Missing Angle | 用正弦定理求未知角
if you know side a = 12 cm, side b = 15 cm, and angle A = 50°, you can find angle B using:
如果你知道两条边和一个非夹角，正弦定理也可以求未知角。例如，已知边 a = 12 cm，边 b = 15 cm，角 A = 50°，可以用以下式子求角 B：
sin B = b sin A / a
.9575) ≈ 73.2°. Be careful: the sine function has two possible angles between 0° and 180°.
代入：sin B = 15 × sin 50° / 12。因为 sin 50° ≈ 0.7660，得到 sin B ≈ 15 × 0.7660 / 12 ≈ 0.9575。然后 B = sin⁻¹(0.9575) ≈ 73.2°。注意：正弦函数在 0° 到 180° 之间有两个可能的角。
6. The Ambiguous Case of the Sine Rule | 正弦定理的歧义情况
θ₁. You must check whether both solutions are valid by checking the sum of angles in the triangle.
用正弦定理求角时，可能有两个答案。这称为歧义情况，发生在已知两条边和一个非夹角时。如果 sin θ = k，那么 θ 可以是 θ₁ = sin⁻¹(k) 或 θ₂ = 180° − θ₁。你必须检查两个解是否都有效，方法是检查三角形内角和。
oth triangles are possible. In an exam, state both possible values unless the question specifies an acute triangle.
例如，如果 sin B ≈ 0.9575，那么 B₁ ≈ 73.2°，B₂ ≈ 180° − 73.2° ≈ 106.8°。如果角 A = 50°，那么 A + B₂ ≈ 50° + 106.8° ≈ 156.8°，小于 180°，所以两种三角形都可能存在。在考试中，除非题目指明是锐角三角形，否则要写出所有可能的值。
7. The Cosine Rule: Formula and Proof Idea | 余弦定理：公式与证明思路
The cosine rule relates the three sides of a triangle and one angle. It has two common forms. To find a missing side, use:
余弦定理将三角形的三条边与一个角联系起来。它有两种常用形式。求未知边时使用：
a² = b² + c² − 2bc cos A
To find a missing angle, rearrange it as:
求未知角时，将其变形为：
cos A = (b² + c² − a²) / 2bc
The cosine rule is actually a generalisation of Pythagoras' m. theorem. When angle A = 90°, cos A = 0, so the formula becomes a² = b² + c², which is Pythagoras' theorem. This connection helps you remember the structure of the rule.
余弦定理实际上是勾股定理的推广。当角 A = 90° 时，cos A = 0，公式变为 a² = b² + c²，这就是勾股定理。这个联系有助于你记住余弦定理的结构。
8. Using the Cosine Rule to Find a Missing Side | 用余弦定理求未知边
r example, suppose side b = 10 cm, side c = 12 cm, and angle A = 60°. To find side a, use:
当你知道两条边及其夹角时，用余弦定理求未知边。例如，假设边 b = 10 cm，边 c = 12 cm，角 A = 60°。要求边 a，使用：
a² = b² + c² − 2bc cos A
20 = 124. Therefore a = √124 ≈ 11.14 cm. Do not forget to take the square root at the end.
代入：a² = 10² + 12² − 2 × 10 × 12 × cos 60°。因为 cos 60° = 0.5，得到 a² = 100 + 144 − 120 = 124。因此 a = √124 ≈ 11.14 cm。别忘了最后要开平方根。
9. Using the Cosine Rule to Find a Missing Angle | 用余弦定理求未知角
ind an angle. For example, suppose a = 7 cm, b = 8 cm, and c = 9 cm. To find angle A, use:
如果你知道三条边，用余弦定理的变形公式求角。例如，假设 a = 7 cm，b = 8 cm，c = 9 cm。要求角 A，使用：
cos A = (b² + c² − a²) / 2bc
cosine rule gives a unique angle between 0° and 180°, so there is no ambiguous case here.
代入：cos A = (8² + 9² − 7²) / (2 × 8 × 9) = (64 + 81 − 49) / 144 = 96 / 144 = 2/3。然后 A = cos⁻¹(2/3) ≈ 48.2°。余弦定理在 0° 到 180° 之间给出唯一的角度，因此这里没有歧义情况。
10. Area of a Triangle Using Sine | 用正弦求三角形面积
The area of any triangle can be found if you know two sides and the included angle. The formula is:
如果你知道两条边及其夹角，就可以求任意三角形的面积。公式为：
Area = ½ ab sin C
y useful when the height of the triangle is not known. It also appears in proof questions involving the sine rule.
例如，如果 a = 6 cm，b = 10 cm，C = 30°，那么面积 = ½ × 6 × 10 × sin 30° = 30 × 0.5 = 15 cm²。这个公式在不知道三角形高的时候特别有用。它也出现在涉及正弦定理的证明题中。
11. Choosing the Right Rule | 选择正确的定理
Choosing whether to use the sine rule or cosine rule depends on the information given. Use the sine rule when you know two angles and one side, or two sides and a non-included angle. Use the cosine rule when you know two sides and the included angle, or all three sides.
选择使用正弦定理还是余弦定理取决于已知的信息。当你知道两个角和一个边，或两条边和一个非夹角时，使用正弦定理。当你知道两条边及其夹角，或三条边时，使用余弦定理。
| Given Information | 已知信息 | Rule to Use | 使用的定理 |
|---|---|
| Two angles and one side | 两个角和一个边 | Sine rule | 正弦定理 |
| Two sides and a non-included angle | 两条边和一个非夹角 | Sine rule | 正弦定理 |
| Two sides and the included angle | 两条边及其夹角 | Cosine rule | 余弦定理 |
| Three sides | 三条边 | Cosine rule | 余弦定理 |
Before substituting values, always label the triangle and identify which information you have. This simple step prevents most errors.
在代入数值之前，一定要标记三角形并确定你有哪些信息。这个简单的步骤可以避免大多数错误。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
Many students lose marks by using the wrong rule or by rounding too early. Always keep full calculator accuracy until the final answer. If a question asks for an angle, check whether the ambiguous case applies to the sine rule. If it does, state both possible angles or explain why only one is valid.
许多学生因为用错定理或过早取近似值而失分。始终在最后一步才取近似值，之前保持计算器的完整精度。如果题目要求求角，检查正弦定理是否存在歧义情况。如果存在，写出两个可能的角，或解释为什么只有一个有效。
Another common mistake is forgetting to take the square root when using the cosine rule to find a side. After calculating a², make sure you find a itself. Also, do not use the sine rule when you only know three sides, as it cannot solve that case.
另一个常见错误是用余弦定理求边时忘记开平方根。算出 a² 之后，一定要再求 a 本身。此外，当你只知道三条边时，不要使用正弦定理，因为它无法解决这种情况。
Practise past paper questions under timed conditions. Write the formula first, then substitute the values clearly, and show all working. This will help you gain method marks even if the final answer is incorrect.
在计时条件下练习历年真题。先写出公式，再清楚地代入数值，并展示所有步骤。这样即使最终答案错误，也能得到方法分。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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