📚 Cosine Rule | 余弦定理
The Cosine Rule, also known as the Law of Cosines, extends trigonometry beyond right‑angled triangles. It relates the lengths of the sides of any triangle to the cosine of one of its angles. For IGCSE Edexcel, you must be able to use it to find an unknown side given two sides and the included angle (SAS), or to find an angle when all three sides are known (SSS). Mastering this rule is essential for solving non‑right‑angled triangles and tackling real‑world problems.
余弦定理（也称为余弦定律）将三角学延伸至非直角三角形。它将任意三角形的边长与其中一个角的余弦联系起来。在 Edexcel IGCSE 考试中，你需要能够利用已知两边及其夹角求第三边（SAS），或已知三边求角度（SSS）。掌握这一定理对于求解非直角三角形和解决实际问题至关重要。
1. Introduction to the Cosine Rule | 余弦定理简介
The cosine rule is a generalisation of Pythagoras' theorem. In a triangle with side lengths a, b and c, where side a is opposite angle A, side b opposite angle B and side c opposite angle C, the rule is usually written in two main forms.
余弦定理是勾股定理的推广。在一个三角形中，边长分别为 a、b 和 c，且边 a 对角 A，边 b 对角 B，边 c 对角 C，定理通常有两种主要表示形式。
gle C = 90°, then cos 90° = 0 and the formula reduces to c² = a² + b², which is Pythagoras' theorem. This shows why the cosine rule works for any triangle.
当角 C = 90° 时，cos 90° = 0，公式简化为 c² = a² + b²，这正是勾股定理。这解释了为什么余弦定理适用于任意三角形。
2. The Formula for Finding a Side | 求边的公式
To find an unknown side opposite a known angle, use:
已知一个角及其两边，求这个角所对的边，使用：
a² = b² + c² − 2bc cos A
to match the given information. For example, to find side b, use b² = a² + c² − 2ac cos B.
你可以根据已知信息轮换字母。例如，要求边 b，使用 b² = a² + c² − 2ac cos B。
In this formula, the side on the left‑hand side matches the angle inside the cosine term, and the other two sides are the ones forming that angle.
在该公式中，左边的边与余弦项内的角对应，另外两条边则是夹角的两边。
3. The Formula for Finding an Angle | 求角的公式
When all three sides are known, you can find any angle by rearranging the cosine rule:
当三条边都已知时，可以通过变形余弦定理求任意角：
cos A = (b² + c² − a²) / (2bc)
Then use the inverse cosine function (cos⁻¹) on your calculator to obtain angle A. Always ensure your calculator is in degree mode.
然后利用计算器的反余弦函数（cos⁻¹）求出角 A。务必确保计算器设置为角度制。
Similarly, cos B = (a² + c² − b²) / (2ac) and cos C = (a² + b² − c²) / (2ab).
类似地，cos B = (a² + c² − b²) / (2ac)，cos C = (a² + b² − c²) / (2ab)。
4. Deriving the Cosine Rule (Overview) | 余弦定理的推导（概述）
A common proof uses the altitude from a vertex to split the triangle into two right‑angled triangles. By applying Pythagoras' theorem to both smaller triangles and using the definition of cosine in the right‑angled triangle, you can eliminate the height and obtain the cosine rule.
常见的证明方法是作一条高线，将三角形分成两个直角三角形。对两个小三角形分别应用勾股定理，并利用直角三角形中余弦的定义，可以消去高，从而导出余弦定理。
While you are not asked to reproduce the full derivation in the Edexcel IGCSE exam, understanding where the rule comes from helps you remember the formula and its connection to Pythagoras.
虽然在 Edexcel IGCSE 考试中不必完整重现推导过程，但了解定理的来源有助于记忆公式以及与勾股定理的联系。
5. Using the Cosine Rule to Find a Side (SAS) | 已知两边及夹角求第三边
When you are given two sides and the angle between them, the cosine rule gives the opposite side directly. Let's work through an example:
当已知两边和它们的夹角时，可以直接用余弦定理求出对边。我们来看一个例子：
Given b = 8 cm, c = 5 cm and angle A = 60°. Find side a.
已知 b = 8 cm，c = 5 cm，∠A = 60°，求边 a。
Substitute into a² = b² + c² − 2bc cos A:
代入公式 a² = b² + c² − 2bc cos A：
a² = 8² + 5² − 2 × 8 × 5 × cos 60°
a² = 64 + 25 − 80 × 0.5
a² = 89 − 40 = 49
Take the positive square root (side lengths are positive): a = √49 = 7 cm.
取正平方根（边长为正）：a = √49 = 7 cm。
Always check the order of operations: square the sides, compute the product, then combine. Using brackets on your calculator can prevent errors.
务必注意运算顺序：先平方边长，再计算乘积，然后合并。在计算器上使用括号可以避免错误。
6. Using the Cosine Rule to Find an Angle (SSS) | 已知三边求角度
If you know all three side lengths, you can determine any interior angle. Use the rearranged form:
如果已知三条边长，就可以求出任意内角。使用变形公式：
cos A = (b² + c² − a²) / (2bc)
ple: a = 7 cm, b = 9 cm, c = 12 cm. Find the largest angle (opposite the longest side, c).
例题：a = 7 cm，b = 9 cm，c = 12 cm。求最大角（对最长边 c）。
cos C = (7² + 9² − 12²) / (2 × 7 × 9)
cos C = (49 + 81 − 144) / 126 = (−14) / 126 ≈ −0.1111
ecimal place). Because cos C is negative, angle C is obtuse, which makes sense as it is the largest angle.
现在使用 cos⁻¹(−0.1111) ≈ 96.4°（保留一位小数）。因为 cos C 为负值，角 C 是钝角，这也符合它是最大角的预期。
Always round your final answer to an appropriate degree of accuracy as specified in the question, and state the angle with the degree symbol.
永远按照题目要求的精度对最终答案进行舍入，并用度数符号标注角度。
7. When to Use the Cosine Rule vs. the Sine Rule | 何时使用余弦定理与正弦定理
Choosing the correct rule depends on the information you have:
选择正确的定理取决于已知信息：
- SAS (two sides and the included angle) → Cosine Rule to find the third side.
- SSS (three sides) → Cosine Rule to find any angle.
- AAS or ASA (two angles and one side) → Sine Rule.
- SSA (two sides and a non‑included angle) → Sine Rule, but check for the ambiguous case.
- SAS（两边及其夹角）→ 使用余弦定理求第三边。
- SSS（三条边）→ 使用余弦定理求角度。
- AAS 或 ASA（两角及一边）→ 使用正弦定理。
- SSA（两边及一个非夹角）→ 使用正弦定理，但需注意多解情况。
The cosine rule does not suffer from ambiguity because the inverse cosine function gives a unique angle between 0° and 180° for any value from −1 to 1.
余弦定理不会产生多解问题，因为反余弦函数对 −1 到 1 之间的任何值都会给出 0° 至 180° 之间的唯一角度。
8. Step‑by‑Step Worked Example (Finding a Side) | 分步示例（求边）
ngle PQR, p = 10.2 cm, q = 6.8 cm and angle R = 47°. Find side r to 3 significant figures.
在三角形 PQR 中，p = 10.2 cm，q = 6.8 cm，∠R = 47°。求边 r，精确至三位有效数字。
- Write the formula with correct labelling: r² = p² + q² − 2pq cos R.
- Substitute: r² = 10.2² + 6.8² − 2 × 10.2 × 6.8 × cos 47°.
- Compute squares: 104.04 + 46.24 = 150.28.
- Compute 2pq cos R: 2 × 10.2 × 6.8 × cos 47° ≈ 138.72 × 0.6820 ≈ 94.59.
- Subtract: r² ≈ 150.28 − 94.59 = 55.69.
- Square root: r ≈ √55.69 ≈ 7.46 cm (to 3 s.f.).
- 用正确的字母标注写出公式：r² = p² + q² − 2pq cos R。
- 代入：r² = 10.2² + 6.8² − 2 × 10.2 × 6.8 × cos 47°。
- 计算平方：104.04 + 46.24 = 150.28。
- 计算 2pq cos R：2 × 10.2 × 6.8 × cos 47° ≈ 138.72 × 0.6820 ≈ 94.59。
- 相减：r² ≈ 150.28 − 94.59 = 55.69。
- 开平方：r ≈ √55.69 ≈ 7.46 cm（保留三位有效数字）。
9. Step‑by‑Step Worked Example (Finding an Angle) | 分步示例（求角）
5.4 m, y = 7.2 m and z = 8.1 m. Calculate angle X, giving your answer to 1 decimal place.
三角形 XYZ 的边长为 x = 5.4 m，y = 7.2 m，z = 8.1 m。计算角 X，答案保留一位小数。
- Use cos X = (y² + z² − x²) / (2yz).
- Compute y² + z² = 51.84 + 65.61 = 117.45.
- Subtract x²: 117.45 − 29.16 = 88.29.
- Compute denominator: 2 × 7.2 × 8.1 = 116.64.
- cos X = 88.29 / 116.64 ≈ 0.7569.
- X = cos⁻¹(0.7569) ≈ 40.8° (to 1 d.p.).
- 使用 cos X = (y² + z² − x²) / (2yz)。
- 计算 y² + z² = 51.84 + 65.61 = 117.45。
- 减去 x²：117.45 − 29.16 = 88.29。
- 计算分母：2 × 7.2 × 8.1 = 116.64。
- cos X = 88.29 / 116.64 ≈ 0.7569。
- X = cos⁻¹(0.7569) ≈ 40.8°（保留一位小数）。
10. Applications to Bearings and Navigation | 方位与导航的应用
Cosine rule problems often feature bearings. A ship sails from port A on a bearing of 065° for 12 km to point B, then changes bearing to 140° and sails 9 km to point C. To find the direct distance from A to C, you can sketch the triangle and find the included angle at B.
余弦定理问题常涉及方位角。一艘船从港口 A 沿 065° 的方位航行 12 km 到达 B 点，然后转向 140° 方位航行 9 km 到达 C 点。要求 A 到 C 的直线距离，可以画出三角形，并求出 B 处的夹角。
75°. This gives AC ≈ 13.2 km. A clear diagram is essential to identify the correct angle.
B 处的夹角等于方位角之差：140° − 065° = 75°。然后应用余弦定理：AC² = AB² + BC² − 2×AB×BC×cos 75°，求得 AC ≈ 13.2 km。清晰的图示对于确定正确的角至关重要。
Bearings are always measured clockwise from north. Convert the given bearings into interior angles of the triangle carefully, as the angle used in the formula must be the angle between the two known sides.
方位角始终从正北顺时针测量。要仔细将给出的方位角转化为三角形的内角，因为公式中使用的角必须是已知两边之间的夹角。
11. Common Pitfalls and How to Avoid Them | 常见错误及避免方法
Many errors come from mislabelling the triangle or using the formula with the wrong angle. Always ensure the angle in the cosine term lies between the two sides used in the product 2bc.
许多错误源于三角形标注错误或公式中使用了错误的角。务必保证余弦项中的角是乘积 2bc 中那两条边的夹角。
- Forgetting to take the square root when finding a side.
- Using the cosine rule for a side when you actually have AAS – the sine rule would be much faster.
- Entering an angle in radians when your calculator is in radian mode, or vice versa.
- Mishandling negative cosine values for obtuse angles – remember that cos θ is negative for 90° < θ < 180°, and your calculator will correctly give an obtuse angle.
- Rounding intermediate values too early; keep full precision in your calculator until the final answer.
- 求边时忘记开平方根。
- 在已知两角一边（AAS）的情况下误用余弦定理求边——此时用正弦定理会更快。
- 计算器处于弧度模式时却输入了角度值，或相反。
- 对钝角出现余弦负值处理不当——牢记当 90° < θ < 180° 时 cos θ 为负，计算器会正确给出钝角。
- 过早对中间值进行舍入；在得出最终答案之前，计算器中应保留全部精度。
12. Exam Tips and Summary | 考试提示与总结
In the exam, read the question carefully to identify what is given and what is required. Draw and label the triangle, noting which side is opposite which angle. If you are given SAS, reach for the cosine rule first. If you are given SSS, use the cosine rule to find the largest angle first to check for an obtuse possibility.
考试中，仔细审题，明确已知和所求。画出并标注三角形，注明各边所对的角。如果已知两边及其夹角（SAS），首选余弦定理。如果已知三边（SSS），先用余弦定理求最大角，以检查是否存在钝角。
. You can derive the angle version by rearranging the side version, so memorising one form can help you retrieve the other.
记住余弦定理的三种形式：求边 a² = b² + c² − 2bc cos A；求角 cos A = (b² + c² − a²) / (2bc)。你可以通过变形从求边公式得到求角公式，因此记住一种形式即可推得另一种。
Practice plenty of mixed questions combining the sine
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