📚 Sine and Cosine Rules: Solving Non-Right-Angled Triangles | 正弦定理与余弦定理：解任意三角形
In IGCSE Mathematics, right-angled trigonometry using sine, cosine and tangent is not enough for every triangle problem. Many exam questions involve triangles with no right angle, especially in higher-tier papers. This revision guide explains the sine rule, the cosine rule and the area formula for any triangle, with worked examples, flowcharts and exam tips.
在 IGCSE 数学中，仅靠直角三角形中的 sin、cos 和 tan 无法解决所有三角形问题。许多考题涉及没有直角的三角形，尤其是在高阶试卷中。本复习指南讲解正弦定理、余弦定理和任意三角形的面积公式，并配有例题、选择流程和考试技巧。
1. Labelling a Triangle Correctly | 正确标记三角形
Before using any rule, label each side and angle correctly. In triangle ABC, angle A is at vertex A, angle B is at vertex B, and angle C is at vertex C. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.
使用任何定理之前，必须正确标记每条边和每个角。在三角形 ABC 中，角 A 位于顶点 A，角 B 位于顶点 B，角 C 位于顶点 C。边 a 对着角 A，边 b 对着角 B，边 c 对着角 C。
This convention is not optional: the formulas only work when each side is paired with its opposite angle. If a question labels sides differently, redraw the triangle or relabel it on your diagram before substituting values.
这一规定不是可有可无的：只有当每条边与它所对的角正确配对时，公式才成立。如果题目对边的标记不同，请重新画图或在图上重新标记，再代入数值。
2. The Sine Rule: Formula and When to Use It | 正弦定理：公式与适用条件
a / sin A = b / sin B = c / sin C
The sine rule links each side to the sine of its opposite angle. Use it when you know two angles and one side, or when you know two sides and a non-included angle. The version shown is ideal for finding a missing side.
正弦定理将每条边与其所对角的正弦联系起来。当你已知两个角与一条边，或已知两条边与一个非夹角时使用。上述形式最适合求缺失的边。
For finding a missing angle, the rearranged form is often easier:
求缺失的角时，通常使用变形后的形式更方便：
sin A / a = sin B / b = sin C / c
Whichever version you use, always match each side with the sine of the angle directly opposite it.
无论使用哪种形式，都要始终将每条边与其正对的角的正弦相匹配。
3. Sine Rule: Finding a Missing Side | 正弦定理：求缺失的边
n triangle ABC, A = 40°, B = 62°, and b = 10 cm. Find side a to three significant figures.
例题：在三角形 ABC 中，A = 40°，B = 62°，b = 10 cm。求边 a，答案保留三位有效数字。
Write the part of the sine rule involving a and b:
写出正弦定理中涉及 a 和 b 的部分：
a / sin 40° = 10 / sin 62°
Multiply both sides by sin 40° to isolate a:
两边同时乘以 sin 40°，求出 a：
a = (10 × sin 40°) / sin 62° ≈ 7.29 cm
Remember to set your calculator to degree mode. Also keep the full display value until the final step, then round to three significant figures.
请记住将计算器设置为角度模式。还要保留计算器显示的全部数值，直到最后一步再保留三位有效数字。
4. Sine Rule: Finding a Missing Angle and the Ambiguous Case | 正弦定理：求缺失的角与多解情况
Example: In triangle ABC, a = 6 cm, b = 9 cm, and A = 30°. Find angle B.
例题：在三角形 ABC 中，a = 6 cm，b = 9 cm，A = 30°。求角 B。
Use the sine rule with sin B and b:
使用含有 sin B 和 b 的正弦定理：
sin B / 9 = sin 30° / 6
Multiply by 9:
两边乘以 9：
sin B = (9 × sin 30°) / 6 = 0.75
A + B = 30° + 131.4° = 161.4° is less than 180°, both solutions can form a valid triangle.
因此 B = sin⁻¹(0.75) ≈ 48.6°。但由于 sin(180° − B) = sin B，可能存在第二个角：B = 180° − 48.6° = 131.4°。因为 A + B = 30° + 131.4° = 161.4° 小于 180°，所以两个解都能构成有效三角形。
In IGCSE questions, this ambiguous case is usually tested only when you are given two sides and a non-included angle. Always check whether the second possible angle still leaves a positive third angle.
在 IGCSE 考题中，这种多解情况通常只在已知两条边和一个非夹角时出现。始终要检查第二个可能的角是否仍能为第三个角留下正的角度。
5. The Cosine Rule: Formula for Finding a Side | 余弦定理：求边的公式
a² = b² + c² − 2bc cos A
The cosine rule is used when you know two sides and the included angle, or when you know all three sides. It is also called the law of cosines. The side on the left of the formula is always opposite the angle in the cosine term.
余弦定理用于已知两条边及其夹角，或已知三条边时。它也叫余弦定律。公式左边的边总是与余弦项中的角相对。
You can cycle the letters for all three sides:
你可以轮换字母得到所有三条边的公式：
b² = a² + c² − 2ac cos B
c² = a² + b² − 2ab cos C
Use this version when the given angle is between the two known sides and you need the side opposite that angle.
当已知角位于两条已知边之间，并且需要求该角对面的边时，使用这个版本。
6. Cosine Rule Example: Finding a Side | 余弦定理例题：求边
Example: In triangle ABC, b = 5 cm, c = 8 cm, and A = 60°. Find side a.
例题：在三角形 ABC 中，b = 5 cm，c = 8 cm，A = 60°。求边 a。
Substitute the known values into the cosine rule:
将已知值代入余弦定理：
a² = 5² + 8² − 2 × 5 × 8 × cos 60°
Since cos 60° = 0.5, this becomes:
因为 cos 60° = 0.5，所以得到：
a² = 25 + 64 − 80 × 0.5 = 89 − 40 = 49
al version of Pythagoras: when A = 90°, cos A = 0 and the formula reduces to a² = b² + c².
因此 a = 7 cm。注意余弦定理是毕达哥拉斯定理的一般形式：当 A = 90° 时，cos A = 0，公式化简为 a² = b² + c²。
7. The Cosine Rule: Finding an Angle | 余弦定理：求角
cos A = (b² + c² − a²) / (2bc)
To find an angle when all three sides are known, rearrange the cosine rule. The angle chosen must be opposite the side in the numerator on the right side.
已知三条边求角时，需将余弦定理变形。所选角必须对着右边分子中的那条边。
Example: In triangle ABC, a = 7 cm, b = 6 cm, c = 5 cm. Find angle A.
例题：在三角形 ABC 中，a = 7 cm，b = 6 cm，c = 5 cm。求角 A。
cos A = (6² + 5² − 7²) / (2 × 6 × 5)
cos A = (36 + 25 − 49) / 60 = 12 / 60 = 0.2
Then A = cos⁻¹(0.2) ≈ 78.5°. Always write cos⁻¹, not sin⁻¹, when using this rearrangement.
因此 A = cos⁻¹(0.2) ≈ 78.5°。使用这个变形公式时，一定要写 cos⁻¹，而不是 sin⁻¹。
8. Area of a Triangle Using Sine | 用正弦求三角形面积
Area = ½ ab sin C
If you know two sides and the included angle, the area is half the product of those sides multiplied by the sine of the included angle. This formula works for any triangle, not just right-angled triangles.
若已知两条边及其夹角，面积等于这两条边乘积的一半乘以夹角的正弦。该公式适用于任何三角形，而不仅是直角三角形。
Example: Given a = 6 cm, b = 7 cm, and C = 30°, find the area.
例题：已知 a = 6 cm，b = 7 cm，C = 30°，求面积。
Area = ½ × 6 × 7 × sin 30° = 10.5 cm²
Always check that the angle is between the two sides you are multiplying. If the angle is opposite one of them, you cannot use this formula directly.
始终检查这个角是否在你相乘的两条边之间。如果角对着其中一条边，就不能直接使用这个公式。
9. Choosing the Right Rule: Decision Framework | 选择正确的方法：判断框架
One of the most common IGCSE difficulties is deciding whether to use the sine rule, the cosine rule, or basic right-angled trigonometry. Use this table as a quick checkpoint
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