📚 The Sine Rule and Its Applications | A-Level数学：正弦定理及应用
The sine rule is one of the most powerful tools in A-Level mathematics for solving problems involving non-right-angled triangles. It connects the lengths of sides to the sines of their opposite angles, allowing us to find unknown sides, unknown angles, and even the area of a triangle in a wide range of contexts.
正弦定理是A-Level数学中解决非直角三角形问题最有力的工具之一。它把三角形的边长与其对角的正弦值联系起来，使我们可以广泛地求解未知边长、未知角度，甚至三角形的面积。
1. The Statement of the Sine Rule | 正弦定理的表述
For any triangle ABC, with side a opposite angle A, side b opposite angle B, and side c opposite angle C, the sine rule states:
对于任意三角形ABC，设边a为角A的对边，边b为角B的对边，边c为角C的对边，正弦定理表述为：
a ⁄ sin A = b ⁄ sin B = c ⁄ sin C
Equivalently, the rule can be written in its reciprocal form, which is often more convenient when finding an angle:
等价地，该定理也可以写成倒数形式，这在求角度时通常更为方便：
sin A ⁄ a = sin B ⁄ b = sin C ⁄ c
The sine rule applies to any triangle, whether acute, obtuse, or right-angled. However, for right-angled triangles, the basic trigonometric ratios are usually simpler.
正弦定理适用于任何三角形，无论是锐角三角形、钝角三角形还是直角三角形。不过对于直角三角形，基本三角比通常更为简便。
2. Derivation: Where Does It Come From? | 推导：它从何而来？
Consider triangle ABC. Draw a perpendicular from vertex A to side BC, meeting BC at point D. Let the height of this perpendicular be h.
考虑三角形ABC。从顶点A向边BC作垂线，垂足为D。设这条高的长度为h。
In right-angled triangle ABD, we have sin B = h ⁄ c, so h = c sin B.
在直角三角形ABD中，我们有 sin B = h ⁄ c，因此 h = c sin B。
In right-angled triangle ACD, we have sin C = h ⁄ b, so h = b sin C.
在直角三角形ACD中，我们有 sin C = h ⁄ b，因此 h = b sin C。
B = c ⁄ sin C. Repeating this process with a different altitude gives the full sine rule.
将两个h的表达式相等，得到 c sin B = b sin C，整理后即 b ⁄ sin B = c ⁄ sin C。用另一条高重复此过程，即可得到完整的正弦定理。
3. Using the Sine Rule to Find a Side | 用正弦定理求边
When we know two angles and one side of a triangle, we can use the sine rule to find the length of another side. This situation is often called ASA or AAS.
当我们知道三角形的两个角和一个边时，可以用正弦定理求另一个边的长度。这种情况通常称为ASA或AAS。
Example: In triangle ABC, angle A = 40°, angle B = 65°, and side b = 12 cm. Find side a.
示例：在三角形ABC中，角A = 40°，角B = 65°，边b = 12 cm。求边a。
Using the sine rule:
使用正弦定理：
a ⁄ sin 40° = 12 ⁄ sin 65°
Therefore:
因此：
a = 12 sin 40° ⁄ sin 65° ≈ 12 × 0.6428 ⁄ 0.9063 ≈ 8.51 cm
Always check whether the problem requires the answer to a sensible number of significant figures. In A-Level exams, 3 significant figures is the standard unless stated otherwise.
始终检查题目要求答案保留到合理的有效数字。在A-Level考试中，除非另有说明，通常保留3位有效数字。
4. Using the Sine Rule to Find an Angle | 用正弦定理求角
When we know two sides and one angle that is not included between them, we can use the sine rule to find another angle. This is often called SSA or the ambiguous case, which we will examine in detail later.
当我们知道两条边和一个非夹角的角时，可以用正弦定理求另一个角。这通常称为SSA或模糊情形，我们稍后会详细讨论。
Example: In triangle PQR, side p = 7 cm, side q = 9 cm, and angle P = 35°. Find angle Q.
示例：在三角形PQR中，边p = 7 cm，边q = 9 cm，角P = 35°。求角Q。
Using the sine rule in reciprocal form:
使用倒数形式的正弦定理：
sin Q ⁄ q = sin P ⁄ p
So:
所以：
sin Q = 9 sin 35° ⁄ 7 ≈ 9 × 0.5736 ⁄ 7 ≈ 0.7375
could also be 180° − 47.5° = 132.5°. We need to decide which angle is geometrically valid.
因此Q ≈ 47.5°。然而，我们必须记住sin θ = sin(180° − θ)，所以Q也可能是180° − 47.5° = 132.5°。我们需要判断哪个角度在几何上有效。
5. The Ambiguous Case | 模糊情形
When using the sine rule to find an angle from two sides and a non-included angle, there may be two possible triangles that satisfy the given conditions. This is known as the ambiguous case.
当用正弦定理从两边和一非夹角求角时，可能存在两个满足条件的三角形。这被称为模糊情形。
a and b, and angle A, the possibilities depend on the height of the triangle, h = b sin A:
已知边a和b以及角A，可能性取决于三角形的高h = b sin A：
- If a < h, no triangle exists.
- If a = h, exactly one right-angled triangle exists.
- If h < a < b, two possible triangles exist.
- If a ≥ b, only one triangle exists.
- 如果a < h，则不存在三角形。
- 如果a = h，则恰好存在一个直角三角形。
- 如果h < a < b，则存在两个可能的三角形。
- 如果a ≥ b，则只存在一个三角形。
In exams, always check whether the given angles sum to less than 180° for each candidate solution. If both candidates give valid angle sums, then both triangles are possible unless the question specifies otherwise, such as "the triangle is acute" or "obtuse".
在考试中，务必检查每个候选解的角度总和是否小于180°。如果两个候选解都给出有效的角度和，那么两个三角形都有可能，除非题目另有说明，例如“三角形是锐角三角形”或“钝角三角形”。
6. Proof of the Ambiguous Case | 模糊情形的证明分析
To understand the ambiguous case algebraically, suppose we are given angle A, side b, and side a. The sine rule gives:
为了从代数上理解模糊情形，假设已知角A、边b和边a。正弦定理给出：
sin B = b sin A ⁄ a
gle. If it is less than 1, then there are two angles B in (0°, 180°) that satisfy the equation: an acute angle and an obtuse angle.
如果右侧大于1，则不存在三角形，因为正弦值不能超过1。如果恰好等于1，则B = 90°，恰好存在一个三角形。如果小于1，则存在两个满足方程的角度B在(0°, 180°)内：一个锐角和一个钝角。
valid. Because there are potentially two valid values of B, we may obtain two different valid values of C.
对于每个候选B，我们接着计算C = 180° − A − B。如果C > 0，则该候选解有效。由于可能有两个有效的B值，我们可能得到两个不同的有效C值。
Worked example: In triangle ABC, angle A = 30°, side a = 5, side b = 7. Find angle B.
完整示例：在三角形ABC中，角A = 30°，边a = 5，边b = 7。求角B。
sin B = 7 sin 30° ⁄ 5 = 7 × 0.5 ⁄ 5 = 0.7
= 180° − 30° − 135.6° = 14.4°, which is also valid. Therefore two triangles are possible.
因此B ≈ 44.4°或B ≈ 135.6°。当B = 44.4°时，C = 180° − 30° − 44.4° = 105.6°，有效。当B = 135.6°时，C = 180° − 30° − 135.6° = 14.4°，同样有效。因此可能存在两个三角形。
7. Area of a Triangle Using Sine | 用正弦求三角形面积
Another important application of the sine rule is the area formula. For any triangle, the area is given by:
正弦定理的另一个重要应用是面积公式。对于任意三角形，面积为：
Area = ½ ab sin C
Here, a and b are any two sides, and C is the included angle between them. This formula is especially useful when the perpendicular height is not given.
其中a和b是任意两边，C是它们之间的夹角。当垂线高未知时，这个公式尤其有用。
For example, in triangle ABC, AB = 8 cm, AC = 6 cm, and angle A = 50°. The area is:
例如，在三角形ABC中，AB = 8 cm，AC = 6 cm，角A = 50°。面积为：
Area = ½ × 8 × 6 × sin 50° = 24 × sin 50° ≈ 18.4 cm²
× height, where the height is expressed as one side times the sine of the included angle.
这个公式直接从标准面积公式Area = ½ × 底 × 高推导而来，其中高表示为一条边乘以夹角的正弦。
8. Finding the Circumradius | 求外接圆半径
The sine rule is connected to the circumcircle of the triangle. The extended sine rule states that for any triangle ABC with circumradius R:
正弦定理与三角形的外接圆有关。扩展正弦定理指出，对于任意外接圆半径为R的三角形ABC：
a ⁄ sin A = b ⁄ sin B = c ⁄ sin C = 2R
This means that if we know a side and its opposite angle, we can immediately find the circumradius of the triangle. Conversely, if we know the circumradius and one angle, we can find the opposite side.
这意味着如果我们知道一条边及其对角，就能立即求出三角形外接圆的半径。反之，如果我们知道外接圆半径和一个角，就能求出对边。
Example: In triangle ABC, side a = 10 cm and angle A = 40°. Then:
示例：在三角形ABC中，边a = 10 cm，角A = 40°。则：
2R = 10 ⁄ sin 40° ≈ 10 ⁄ 0.6428 ≈ 15.56
R ≈ 7.78 cm. This result is elegant and often appears in exam questions involving circles.
所以R ≈ 7.78 cm。这个结果很优雅，而且常常出现在涉及圆的考试题中。
9. Common Exam Problems and Strategies | 常见考试题型与策略
Sine rule problems in Edexcel A-Level exams typically fall into several categories:
Edexcel A-Level考试中的正弦定理问题通常分为几类：
- Finding a missing side given two angles and one side.
- Finding a missing angle given two sides and a non-included angle.
- Determining the number of possible triangles from given data.
- Solving problems in context, such as bearings, elevations, and distances.
- Combining the sine rule with the area formula or the cosine rule.
- 已知两角一边，求缺失的边。
- 已知两边和一非夹角，求缺失的角。
- 根据已知数据确定可能的三角形数量。
- 解决实际情境问题，如方位角、仰角和距离。
- 将正弦定理与面积公式或余弦定理结合使用。
A reliable strategy is to first draw a clear diagram and label given information. Then decide whether the sine rule or cosine rule is appropriate. Use the sine rule when you have an opposite pair of side and angle. Check the validity of any angle you find, especially when the ambiguous case may apply.
一个可靠的策略是首先画一个清晰的图并标出已知信息。然后决定使用正弦定理还是余弦定理。当你有一组对边和对角时使用正弦定理。检查你求出的任何角度的有效性，特别是可能适用模糊情形时。
10. Worked Example: Full Problem | 综合例题：完整解答
Let us solve a complete problem that combines several topics.
让我们解决一个综合了多个知识点的完整问题。
Problem: angle B = 25°. Find the possible values of angle C, side BC, and the area of the triangle.
题目：三角形ABC中，边AB = 15 cm，边AC = 10 cm，角B = 25°。求角C的可能值、边BC和三角形的面积。
Step 1: 15 (opposite angle C) and side b = AC = 10 (opposite angle B). We are given angle B = 25°.
步骤1：找出已知的对应边角对。我们已知边c = AB = 15（对角C）和边b = AC = 10（对角B）。已知角B = 25°。
Using the sine rule:
使用正弦定理：
sin C ⁄ c = sin B ⁄ b
sin C = 15 sin 25° ⁄ 10 = 15 × 0.4226 ⁄ 10 ≈ 0.6339
Step 2: Therefore C ≈ 39.3° or C ≈ 140.7°. We must check both:
步骤2：因此C ≈ 39.3°或C ≈ 140.7°。我们必须检查两者：
If C = 39.3°, then A = 180° − 25° − 39.3° = 115.7°, valid.
如果C = 39.3°，则A = 180° − 25° − 39.3° = 115.7°，有效。
If C = 140.7°, then A = 180° − 25° − 140.7° = 14.3°, also valid.
如果C = 140.7°，则A = 180° − 25° − 140.7° = 14.3°，同样有效。
Step 3: Find side a = BC in each case using a ⁄ sin A = b ⁄ sin B.
步骤3：在每种情况下用a ⁄ sin A = b ⁄ sin B求边a = BC。
Case 1:
情形1：
a = 10 sin 115.7° ⁄ sin 25° ≈ 10 × 0.9003 ⁄ 0.4226 ≈ 21.3 cm
Case 2:
情形2：
a = 10 sin 14.3° ⁄ sin 25° ≈ 10 × 0.2470 ⁄ 0.4226 ≈ 5.85 cm
Step 4: Calculate the area using Area = ½ b c sin A.
步骤4：使用Area = ½ b c sin A计算面积。
Case 1 area:
情形1面积：
Area = ½ × 10 × 15 × sin 115.7° ≈ 75 × 0.9003 ≈ 67.5 cm²
Case 2 area:
情形2面积：
Area = ½ × 10 × 15 × sin 14.3° ≈ 75 × 0.2470 ≈ 18.5 cm²
Thus the problem has two valid solutions, demonstrating the importance of considering the ambiguous case.
因此该问题有两个有效解，说明考虑模糊情形的重要性。
11. Applications in Bearings and 3D Problems | 在方位角与三维问题中的应用
The sine rule is widely used in navigation and surveying. Bearings are measured clockwise from north, and triangles formed by ships, lighthouses, or landmarks can often be solved using the sine rule.
正弦定理广泛用于导航和测量。方位角从正北方向顺时针测量，由船只、灯塔或地标形成的三角形通常可以用正弦定理求解。
For example, a ship observes a lighthouse at a bearing of 060°. After travelling 8 km on a bearing of 120°, the lighthouse is now on a bearing of 350°. The triangle formed by the ship's two positions and the lighthouse can be solved using the sine rule.
例如，一艘船在方位角060°处观察到一座灯塔。沿方位角120°方向行驶8 km后，灯塔现在位于方位角350°方向。由船只的两个位置和灯塔形成的三角形可以用正弦定理求解。
In 3D problems, the sine rule is often applied within a specific triangular plane. Once the relevant triangle is identified, the same techniques apply as in 2D.
在三维问题中，正弦定理通常应用于特定三角形平面内。一旦确定了相关三角形，方法与二维情况相同。
When solving bearing problems, draw the diagram carefully and mark all known angles. Remember that bearings are always three-figure bearings, and angles in a triangle sum to 180°.
解决方位角问题时，仔细画图并标出所有已知角。记住方位角总是用三位数表示，且三角形内角和为180°。
12. Common Mistakes and Exam Tips | 常见错误与考试建议
Students often make errors in sine rule problems. Here are the most frequent pitfalls and how to avoid them.
学生在正弦定理问题中经常犯错。以下是最常见的陷阱以及如何避免它们。
- Using the sine rule when the known angle is not opposite a known side. Always match a side with its opposite angle.
- Forgetting to check the ambiguous case when finding an angle from two sides and a non-included angle.
- Switching between degrees and radians incorrectly. Check the angle mode on your calculator.
- Rounding too early in multi-step calculations. Keep full precision until the final answer.
- Using the included angle incorrectly in the area formula. The area formula requires the angle between the two chosen sides.
- 当已知角不是已知边的对角时使用正弦定理。始终将边与其对角匹配。
- 由两边和一非夹角求角时忘记检查模糊情形。
- 在度数与弧度之间错误切换。检查计算器的角度模式。
- 多步计算中过早四舍五入。在最终答案前保留完整精度。
- 在面积公式中错误使用夹角。面积公式要求两个选定边之间的夹角。
In Edexcel exams, show your working clearly and state which form of the sine rule you are using. If there are two possible triangles, state both and justify your reasoning. This demonstrates full understanding and earns method marks.
在Edexcel考试中，清晰展示你的解题过程，并说明你使用的是正弦定理的哪种形式。如果存在两个可能的三角形，写出两者并说明理由。这展示了完整的理解并能获得方法分。
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