Sine Rule and Its Typical Applications | 正弦定理及其典型应用

📚 Sine Rule and Its Typical Applications | 正弦定理及其典型应用

The sine rule is one of the most powerful tools in A-Level trigonometry. It connects the sides of a triangle to the sines of their opposite angles, allowing us to solve non-right-angled triangles and model real-world situations.

正弦定理是 A-Level 三角学中最有力的工具之一。它将三角形的边与其对角的正弦值联系起来，使我们能够解非直角三角形，并对现实情境进行建模。

1. The Statement of the Sine Rule | 正弦定理的表述

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

对任意三角形 ABC，设边 a、b、c 分别对角 A、B、C，正弦定理表述为：

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

This single equality means that the ratio of a side to the sine of its opposite angle is constant for all three vertices.

这一等式意味着，对于三角形的三个顶点，边与对角正弦之比恒为常数。

Equivalently, the rule can be written in its reciprocal form:

等价地，该定理也可写成倒数形式：

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

The reciprocal form is especially useful when finding an angle, because it places the unknown angle in the numerator.

倒数形式在求角时特别有用，因为它使未知角位于分子位置。

2. Derivation of the Sine Rule | 正弦定理的推导

Consider triangle ABC. Drop a perpendicular from vertex C to side AB, meeting it at point D. Let the height be h.

考虑三角形 ABC。从顶点 C 向边 AB 作垂线，垂足为 D。设高为 h。

In right-angled triangle ACD, we have sin A = h / b, so h = b sin A.

在直角三角形 ACD 中，有 sin A = h / b，因此 h = b sin A。

In right-angled triangle BCD, we have sin B = h / a, so h = a sin B.

在直角三角形 BCD 中，有 sin B = h / a，因此 h = a sin B。

Equating the two expressions for h gives b sin A = a sin B, which rearranges to a / sin A = b / sin B. Repeating this process for another altitude gives the full sine rule.

将 h 的两个表达式相等，得 b sin A = a sin B，整理后得到 a / sin A = b / sin B。对其他高重复此过程，即可得到完整的正弦定理。

b sin A = a sin B → a / sin A = b / sin B

This derivation works for obtuse triangles as well because the sine of an obtuse angle is positive and the altitude may fall outside the base.

该推导对钝角三角形同样成立，因为钝角的正弦为正，且高可能落在底边延长线上。

3. The Ambiguous Case (SSA) | 正弦定理的模糊情形（已知两边一对角）

When we are given two sides and a non-included angle (SSA), the sine rule may produce two possible triangles. This is called the ambiguous case.

当已知两边及其中一边的对角（SSA）时，正弦定理可能会产生两个可能的三角形，这称为模糊情形。

Suppose we know sides a, b and angle A. We calculate sin B = b sin A / a.

假设已知边 a、b 和角 A。我们计算 sin B = b sin A / a。

If b sin A > a, then sin B > 1, and no triangle exists.

若 b sin A > a，则 sin B > 1，三角形不存在。
If b sin A = a, then sin B = 1, giving B = 90° and exactly one triangle.

若 b sin A = a，则 sin B = 1，得 B = 90°，只有唯一一个三角形。
If b sin A < a, there may be two possible values of B: one acute and one obtuse, provided the obtuse value also satisfies A + B < 180°.

若 b sin A < a，则 B 可能有两个值：一个锐角和一个钝角，前提是钝角值也满足 A + B < 180°。

For example, if A = 30°, a = 7, b = 10, then sin B = 10 sin 30° / 7 = 5/7 ≈ 0.714. This gives B ≈ 45.6° or B ≈ 134.4°. Since A + 134.4° = 164.4° < 180°, both triangles are valid.

例如，若 A = 30°，a = 7，b = 10，则 sin B = 10 sin 30° / 7 = 5/7 ≈ 0.714。由此得 B ≈ 45.6° 或 B ≈ 134.4°。因为 A + 134.4° = 164.4° < 180°，所以两个三角形都成立。

Always check whether both angles are mathematically possible; reject any angle that makes the sum of angles exceed 180°.

务必检查两个角是否在数学上可行；舍去任何使内角之和超过 180° 的角。

4. Finding Missing Sides | 求未知边

The sine rule is most straightforward when we know two angles and one side (AAS or ASA). In this case there is no ambiguity.

当已知两角和一边（AAS 或 ASA）时，正弦定理最为直接，此时不存在模糊性。

Example: In triangle ABC, A = 40°, B = 65°, and a = 12. Find side b.

例：在三角形 ABC 中，A = 40°，B = 65°，a = 12。求边 b。

Using the rule a / sin A = b / sin B:

利用规则 a / sin A = b / sin B：

12 / sin 40° = b / sin 65°

b = 12 sin 65° / sin 40° ≈ 12 × 0.9063 / 0.6428 ≈ 16.9.

b = 12 sin 65° / sin 40° ≈ 12 × 0.9063 / 0.6428 ≈ 16.9。

When using the sine rule to find a side, choose the ratio that contains the known side and the target side, then cross-multiply.

用正弦定理求边时，选择包含已知边和目标边的比例式，然后交叉相乘。

Remember to keep your calculator in degree or radian mode according to the question; mixing modes is a common source of error.

注意根据题目将计算器调至角度制或弧度制；混用模式是常见错误来源。

5. Finding Missing Angles | 求未知角

To find an angle, use the reciprocal form: sin A / a = sin B / b. Rearrange to isolate the sine of the unknown angle.

求角时，使用倒数形式：sin A / a = sin B / b。整理出未知角的正弦。

Example: In triangle ABC, a = 8, b = 11, and B = 50°. Find angle A.

例：在三角形 ABC 中，a = 8，b = 11，B = 50°。求角 A。

sin A / 8 = sin 50° / 11 → sin A = 8 sin 50° / 11

sin A ≈ 8 × 0.7660 / 11 ≈ 0.5571, so A ≈ 33.8° or A ≈ 146.2°.

sin A ≈ 8 × 0.7660 / 11 ≈ 0.5571，因此 A ≈ 33.8° 或 A ≈ 146.2°。

But B = 50°, and A + B = 146.2° + 50° = 196.2° > 180°, so reject the obtuse value. Thus A ≈ 33.8°.

但 B = 50°，且 A + B = 146.2° + 50° = 196.2° > 180°，所以舍去钝角值。因此 A ≈ 33.8°。

When finding an angle, always compare the lengths of the sides. The largest side is opposite the largest angle. This helps you decide between acute and obtuse possibilities.

求角时，始终比较边长。最大的边对应最大的角。这有助于判断锐角还是钝角。

6. Application: Area of a Triangle | 应用：三角形面积

The sine rule leads directly to a formula for the area of a triangle when two sides and the included angle are known.

正弦定理直接引出了已知两边及其夹角时的三角形面积公式。

Area = ½ ab sin C, where a and b are two sides and C is the included angle between them.

面积 = ½ ab sin C，其中 a、b 为两条边，C 为它们的夹角。

This formula works for any triangle, not only right-angled ones. It is derived from the height h = b sin A used in the sine rule derivation.

该公式适用于任意三角形，而不仅仅是直角三角形。它由正弦定理推导中使用的高 h = b sin A 得到。

Example: Find the area of a triangle with sides 6 cm and 9 cm and included angle 40°.

例：求两边分别为 6 cm 和 9 cm，夹角为 40° 的三角形面积。

Area = ½ × 6 × 9 × sin 40° ≈ 27 × 0.6428 ≈ 17.36 cm²

This formula is especially useful in mechanics and vectors when calculating the magnitude of a cross product.

该公式在力学和向量中计算叉积大小时尤其有用。

7. Application: Circumradius | 应用：外接圆半径

The sine rule can be extended to include the circumradius R of the triangle:

正弦定理可扩展为包含三角形外接圆半径 R：

a / sin A = b / sin B = c / sin C = 2R

Here R is the radius of the circle that passes through all three vertices of the triangle.

其中 R 是经过三角形三个顶点的圆的半径。

To find R, simply compute one side divided by twice the sine of its opposite angle:

求 R 只需计算一边除以其对角的二倍正弦：

R = a / (2 sin A)

Example: If a = 10 and A = 60°, then R = 10 / (2 sin 60°) = 10 / (2 × 0.8660) ≈ 5.77.

例：若 a = 10，A = 60°，则 R = 10 / (2 sin 60°) = 10 / (2 × 0.8660) ≈ 5.77。

This relationship is often tested in questions involving chords, circles, or advanced geometry.

这一关系经常在与弦、圆或高级几何相关的题目中出现。

8. Real-World Navigation Problems | 实际导航问题

Sine rule is widely used in navigation, surveying, and engineering to find distances that are not directly measurable.

正弦定理广泛用于导航、测量和工程中，以计算无法直接测量的距离。

Typical problem: A boat observes a lighthouse at a bearing of 30°. After sailing 5 km east, the bearing is 50°. Find the boat’s distance from the lighthouse at the second observation.

典型问题：一艘船以方位角 30° 观察到灯塔。向东航行 5 km 后，方位角变为 50°。求第二次观测时船与灯塔的距离。

Solution: The triangle is formed by the two boat positions and the lighthouse. The interior angles can be calculated from the bearings. Suppose the angles are 20° and 40° and the known side is 5 km. Using the sine rule:

解法：两个船位与灯塔构成三角形。内角可由方位角计算。设某两个内角为 20° 和 40°，已知边为 5 km。使用正弦定理：

d / sin 40° = 5 / sin 20° → d = 5 sin 40° / sin 20° ≈ 9.40 km

Always draw a clear diagram and label all known angles. Convert bearings to interior angles by subtracting from 90° or 180° as appropriate.

务必绘制清晰图表并标出所有已知角。将方位角转换为内角时，需要根据情况从 90° 或 180° 中减去。

Another common context is finding the height of a mountain or building when observing from two different points.

另一个常见情境是从两个不同观测点求山或建筑物的高度。

9. Common Mistakes and Tips | 常见错误与提示

Many students lose marks on sine rule questions due to small but critical mistakes. Recognising these pitfalls is essential for exam success.

许多学生在正弦定理题目中因为微小但关键的失误而失分。识别这些陷阱对考试成功至关重要。

Mistake 1: Forgetting the ambiguous case when finding an angle from two sides and one angle. Always check whether a second valid angle exists.

错误一：在已知两边一角求角时忘记模糊情形。始终检查是否还存在第二个有效角。
Mistake 2: Using degrees when the question expects radians, or vice versa. The sine rule itself is mode-neutral, but numerical values depend on the unit.

错误二：题目要求弧度却使用角度制，反之亦然。正弦定理本身与模式无关，但数值取决于单位。
Mistake 3: Reversing the ratio. Ensure the side is paired with the sine of its opposite angle, not an adjacent angle.

错误三：比例倒置。确保边与对角正弦对应，而非邻角。
Mistake 4: Not rejecting impossible angles. If A + B ≥ 180°, that triangle cannot exist.

错误四：未舍去不可能的角。若 A + B ≥ 180°，则该三角形不存在。

Tip: When using a calculator, verify the sine values are between −1 and 1. If your computed sine is greater than 1, your data is likely invalid.

小贴士：使用计算器时，验证正弦值在 −1 到 1 之间。如果计算出的正弦大于 1，你的数据很可能无效。

10. Practice Questions | 练习题

Apply the sine rule to the following problems. Try them before looking at the answers.

用正弦定理解下列题目。先尝试自己做再看答案。

Question	Answer
1. Triangle ABC: a = 7, b = 9, A = 35°. Find B.	sin B = 9 sin 35° / 7 ≈ 0.737, B ≈ 47.5° or 132.5°; both valid?
2. In triangle ABC, A = 50°, C = 70°, c = 10. Find a.	a = 10 sin 50° / sin 70° ≈ 8.15
3. Find the circumradius R of a triangle with side a = 8 and opposite angle A = 45°.	R = 8 / (2 sin 45°) ≈ 5.66
4. Calculate the area of a triangle with sides 5 and 7 and included angle 60°.	Area = ½ × 5 × 7 × sin 60° ≈ 15.16

For question 1, 132.5° + 35° = 167.5° < 180°, so the obtuse angle is also valid. The two possible triangles are different by the length of side b.

对于第 1 题，132.5° + 35° = 167.5° < 180°，因此钝角也成立。两个可能三角形的另一边长度不同。

Practising these problems will help you internalise the sine rule and avoid common errors in exams.

练习这些题目将帮助你把正弦定理内化，并在考试中避免常见错误。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Sine Rule and Its Typical Applications | 正弦定理及其典型应用