Edexcel IGCSE Mathematics: The Sine Rule and Its Applications | 正弦定理及其应用

📚 Edexcel IGCSE Mathematics: The Sine Rule and Its Applications | 正弦定理及其应用

In any triangle, the relationship between sides and angles is not always captured by right-angled trigonometry. The sine rule extends trigonometric reasoning to any triangle, making it an essential tool for solving problems involving non-right-angled triangles.

在任意三角形中，边与角之间的关系并不总能通过直角三角形三角比来描述。正弦定理将三角推理推广到任意三角形，是解决非直角三角形问题的核心工具。

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1. The Formula | 公式

For any triangle ABC, with sides a, b and c opposite angles A, B and C respectively, the sine rule states:

对于任意三角形 ABC，设边 a、b、c 分别对应角 A、B、C，则正弦定理表述为：

a / sin A = b / sin B = c / sin C

This single relationship connects a side to the sine of its opposite angle. It can be used in two main ways: finding an unknown side when two angles and one side are known, or finding an unknown angle when two sides and one opposite angle are known.

这一关系将边与其对角的正弦值联系起来。它有两种主要用法：已知两角和一边求未知边，或者已知两边和其中一边的对角求未知角。

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2. When to Use the Sine Rule | 何时使用正弦定理

The sine rule is applicable in two classic situations, often abbreviated as ASA (two angles and one side) or SSA (two sides and a non-included angle).

正弦定理适用于两种经典情形，通常简称为 ASA（两角一边）或 SSA（两边及其中一边对角）。

• ASA: Two angles and any side are known. The third angle can be found using the angle sum, then the sine rule gives the remaining sides.

• ASA（两角一边）：已知两个角及任意一边。利用内角和求出第三个角，再用正弦定理求出其余两边。

• SSA: Two sides and an angle opposite one of them are known. The sine rule can be used to find a missing angle, but care is needed because this case may be ambiguous.

• SSA（两边及对角）：已知两边及其中一边的对角。可用正弦定理求未知角，但需要注意该情形可能产生“两解”问题。

Always check whether the triangle is right-angled first. If it is, basic right-angled trigonometry is faster. The sine rule is for non-right-angled triangles, although it also works for right-angled ones.

解题前先判断是否为直角三角形。若是，使用直角三角形三角比更快捷。正弦定理主要针对非直角三角形，但在直角三角形中同样适用。

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3. Finding a Missing Side | 求未知边

When two angles and one side are known, the sine rule gives the unknown side directly by rearranging the formula.

当已知两个角和一个边时，通过变形正弦定理公式可直接求出未知边。

a = b × (sin A / sin B)

For example, in triangle ABC, angle A = 40°, angle B = 60°, and side b = 10 cm. Then:

例如，在三角形 ABC 中，角 A = 40°，角 B = 60°，边 b = 10 cm。则：

a = 10 × (sin 40° / sin 60°) = 10 × (0.6428 / 0.8660) ≈ 7.42 cm

Always store intermediate values carefully, and round only at the final step unless instructed otherwise.

中间值应尽量保留足够精度，仅在最后一步按要求四舍五入。

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4. Finding a Missing Angle | 求未知角

To find an angle, substitute the known sides and the sine of the known opposite angle, then solve for sin A.

求角时，将已知边和已知对角的 sin 值代入，然后解出 sin A。

sin A = a × (sin B / b)

Take the inverse sine to find A. For instance, if a = 7, b = 9, and angle B = 52°, then:

通过反正弦求出角 A。例如，若 a = 7，b = 9，角 B = 52°，则：

sin A = 7 × (sin 52° / 9) = 7 × (0.7880 / 9) ≈ 0.6129

A = sin⁻¹(0.6129) ≈ 37.8°

Your calculator should be in degree mode. If working in radians, adjust accordingly.

计算器应处于角度模式（DEG）。若使用弧度，需相应切换。

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5. The Ambiguous Case | 两解（含混）情形

When two sides and a non-included angle are given (SSA), there may be zero, one, or two possible triangles. This is called the ambiguous case.

当已知两边及其中一边的对角（SSA）时，可能没有解、有一个解或两个解。这被称为两解（含混）情形。

For a triangle with known sides a and b, and known angle A (opposite side a), the possibilities are:

对于已知边 a、b 和角 A（A 的对边为 a）的三角形，可能性如下：

Condition | 条件	Number of solutions | 解的个数
a < b × sin A	0 (no triangle) | 0（无三角形）
a = b × sin A	1 (right triangle) | 1（直角三角形）
b × sin A < a < b	2 (two possible angles) | 2（两个可能角）
a ≥ b	1 (only one triangle) | 1（唯一三角形）

For example, if b = 8, a = 6, and A = 35°, then b×sin A = 8 × 0.5736 = 4.59, which is less than a, so two triangles satisfy the data. Always check whether the problem requires one or two answers.

例如，若 b = 8，a = 6，A = 35°，则 b×sin A = 8 × 0.5736 = 4.59，小于 a，因此满足条件的有两个三角形。解题时务必判断题目要求一个解还是两个解。

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6. The Sine Rule for Area | 正弦定理与面积

Although not strictly part of the sine rule, the area formula derived from two sides and the included angle is often taught together with it.

尽管面积公式并非正弦定理本身，但与正弦定理密切相关，常一同讲解。

Area = ½ × a × b × sin C

Here, C is the angle between sides a and b. This formula provides an alternative to using base and height, especially in non-right-angled triangles.

其中 C 为边 a 与 b 之间的夹角。该公式避免了寻找高，特别适用于非直角三角形。

For example, if two sides are 12 cm and 15 cm with an included angle of 30°, the area is:

例如，两条边分别为 12 cm 和 15 cm，夹角为 30°，则面积为：

Area = ½ × 12 × 15 × sin 30° = ½ × 180 × ½ = 45 cm²

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7. Sine Rule vs Cosine Rule | 正弦定理与余弦定理的对比

Both rules handle non-right-angled triangles. The key is knowing which to apply. The cosine rule is used when you have two sides and the included angle (SAS) or three sides (SSS).

两个定理都处理非直角三角形。关键在于判断使用哪个：已知两边及夹角（SAS）或三边（SSS）时用余弦定理。

Situation | 情形	Rule | 适用定理
Two angles and a side (ASA) | 两角一边	Sine rule | 正弦定理
Two sides and a non-included angle (SSA) | 两边及对角	Sine rule (check ambiguity) | 正弦定理（注意两解）
Two sides and the included angle (SAS) | 两边及夹角	Cosine rule | 余弦定理
Three sides (SSS) | 三边	Cosine rule | 余弦定理

The sine rule is simpler but requires an angle opposite a known side. The cosine rule is more general but involves more calculation.

正弦定理形式简单，但需要一对已知的边角对；余弦定理更通用，但计算量稍大。

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8. Real-World Applications: Surveying | 实际应用：测量

Surveyors use the sine rule to measure distances that are physically inaccessible. By measuring angles from two known points, they can calculate the distance to a distant object.

测量员使用正弦定理计算无法直接到达的距离。通过从两个已知位置测量角度，可算出远处物体的距离。

Imagine two observation points A and B are 500 m apart. A landmark C is seen from A at an angle of 70° from AB, and from B at an angle of 60° from BA. Then angle C = 180° – 70° – 60° = 50°.

假设两个观测点 A 和 B 相距 500 m。从 A 看目标 C 与 AB 方向夹角为 70°，从 B 看与 BA 方向夹角为 60°。则角 C = 180° – 70° – 60° = 50°。

Using the sine rule:

使用正弦定理：

AC / sin 60° = 500 / sin 50°

AC = 500 × sin 60° / sin 50° ≈ 500 × 0.8660 / 0.7660 ≈ 565.4 m

This technique, known as triangulation, is widely used in mapmaking and navigation.

这种技术称为三角测量法，广泛用于制图和导航。

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9. Navigation and Bearings | 航向与方位角

Bearings describe directions in degrees clockwise from north. Sine rule problems involving bearings often ask for the distance travelled or the bearing of a final position.

方位角表示从正北方向顺时针旋转的角度。涉及方位角的正弦定理题目通常要求计算航行距离或最终位置的方位。

A boat sails 40 km on a bearing of 050°, then changes course to travel 60 km on a bearing of 120°. The angle between the two paths is 120° – 50° = 70°. This is an included angle, so the distance back to the start is found using the cosine rule; however, if an angle needs to be found, the sine rule may follow.

一艘船先按方位角 050° 航行 40 km，然后改变航向按方位角 120° 航行 60 km。两条路径的夹角为 120° – 50° = 70°。这是夹角，回程距离用余弦定理求；若要求某个角，则可能用到正弦定理。

In mixed problems, draw a clear diagram first. Label all known angles and sides, then decide which rule fits.

在混合问题中，先画清晰示意图，标出所有已知角和边，再判断使用哪个定理。

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10. Three-Dimensional Problems | 三维问题

In 3D problems, the sine rule is often applied to a triangular face within a solid. For example, a roof truss or a pyramid face may form a non-right triangle.

在三维问题中，正弦定理常被用于立体图形中的某个三角形截面。例如，屋顶桁架或棱锥的侧面可能形成非直角三角形。

When working in 3D, first identify the relevant plane. Draw the triangle separately, and then use the sine rule as usual.

处理三维问题时，先确定相关平面，将三角形单独画出，再照常应用正弦定理。

For instance, in a pyramid with a triangular face having sides 10 m and 14 m and an angle of 45° opposite the 14 m side, you may need to find another angle. Use the sine rule carefully, considering the ambiguous case.

例如，棱锥的一个三角形侧面两边长 10 m 和 14 m，其中 14 m 所对角为 45°。求其他角时需小心使用正弦定理，并考虑两解情形。

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11. Common Mistakes and Tips | 常见错误与技巧

Students often confuse the sine rule with the cosine rule, misuse the formula, or forget to check the ambiguous case. Avoiding these errors improves accuracy in exams.

学生常混淆正弦定理与余弦定理、用错公式，或忘记检查两解情形。避免这些错误能显著提高考试准确率。

• Match sides with opposite angles: The sine rule only works when a side is paired with its opposite angle, never with an adjacent angle.

• 边与对角对应：正弦定理仅适用于边与其对角配对，切勿将边与邻角配对。

• Check calculator mode: A wrong mode gives completely incorrect sine values. Always set to DEG for degree problems.

• 检查计算器模式：模式错误会导致正弦值完全错误。角度问题一定要使用 DEG 模式。

• Draw a diagram: A labelled sketch prevents incorrect substitutions and helps you see which case applies.

• 画图标注：标好示意图可避免代入错误，并帮助判断属于哪种情形。

• Round only at the end: Keep full precision in intermediate calculations to avoid rounding errors propagating.

• 最后才取近似：中间计算保留完整精度，防止舍入误差累积。

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12. Exam-Style Worked Example | 考试典型例题

Let’s work through a complete example that resembles an Edexcel IGCSE question.

下面通过一道类似 Edexcel IGCSE 考法的完整例题进行说明。

Question: In triangle PQR, angle P = 48°, angle Q = 65°, and side PR = 9 cm. Find the length of side PQ.

题目：在三角形 PQR 中，角 P = 48°，角 Q = 65°，边 PR = 9 cm。求边 PQ 的长度。

First, find angle R: R = 180° – 48° – 65° = 67°. Side PR is opposite angle Q, and side PQ is opposite angle R. Therefore:

首先求角 R：R = 180° – 48° – 65° = 67°。边 PR 对角 Q，边 PQ 对角 R。因此：

PQ / sin 67° = 9 / sin 65°

PQ = 9 × sin 67° / sin 65° ≈ 9 × 0.9205 / 0.9063 ≈ 9.14 cm

Always state the final answer with the correct units, and check whether the answer is reasonable by comparing it with known sides.

最终答案需写上正确单位，并通过与已知边比较来检查结果是否合理。

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