📚 Sine and Cosine Rule for Non-Right-Angled Triangles | 非直角三角形解三角形的正弦与余弦定理
When a triangle is not right-angled, the familiar trigonometric ratios of SOH-CAH-TOA no longer apply directly. To solve such triangles — that is, to find unknown sides and angles — we need two powerful tools: the sine rule and the cosine rule. These rules form a core part of the IB Mathematics syllabus and appear regularly in both Paper 1 and Paper 2.
当一个三角形不是直角三角形时，我们熟知的 SOH-CAH-TOA 三角比便不再直接适用。为了解这类三角形——即求未知边和角——我们需要两个强大的工具：正弦定理和余弦定理。这两条定理是 IB 数学课程的核心内容，在 Paper 1 和 Paper 2 中均频繁出现。
1. From Right Triangles to General Triangles | 从直角三角形到一般三角形
Recall that in a right-angled triangle with angle θ, we define sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. These definitions rely on the presence of a 90° angle, which allows us to form right-angled ratios. However, most triangles encountered in real life — and in exam problems — are not right-angled.
回顾在含有角 θ 的直角三角形中，我们定义 sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。这些定义依赖于 90° 角的存在，从而可以构造直角比值。然而，现实生活中以及考试题目里遇到的大多数三角形并非直角三角形。
To extend trigonometry beyond right angles, we need formulas that relate all three sides and all three angles of any triangle. The sine rule and the cosine rule achieve exactly this. They allow us to solve a triangle when we know either two angles and one side, two sides and one angle, or all three sides.
为了将三角学扩展到直角三角形之外，我们需要能够联系任意三角形三条边和三个角的公式。正弦定理和余弦定理正是为此而生。当我们已知两角一边、两边一角或三边时，它们都能帮助我们求解三角形。
2. Notation for a General Triangle | 一般三角形的记号
Before stating the rules, we must agree on a standard notation. In triangle ABC, the side opposite vertex A is denoted by lowercase a, the side opposite vertex B by b, and the side opposite vertex C by c. Thus side a is between vertices B and C, side b is between A and C, and side c is between A and B.
在陈述定理之前，我们必须约定一种标准记号。在三角形 ABC 中，顶点 A 所对的边记为小写字母 a，顶点 B 所对的边记为 b，顶点 C 所对的边记为 c。也就是说，边 a 位于顶点 B 和 C 之间，边 b 位于 A 和 C 之间，边 c 位于 A 和 B 之间。
This consistent labelling makes the sine rule and cosine rule easy to state and apply. Always begin every problem by drawing a clear diagram and labelling the vertices with capital letters and the opposite sides with the corresponding lowercase letters.
这种一致的标记方法使正弦定理和余弦定理易于陈述和应用。每道题开始时都应先画出清晰的图形，用大写字母标记顶点，并用对应的小写字母标记对边。
3. The Sine Rule | 正弦定理
The sine rule states that in any triangle ABC, the ratio of each side to the sine of its opposite angle is constant:
正弦定理指出：在任意三角形 ABC 中，每条边与其对角正弦之比为常数：
a/sin A = b/sin B = c/sin C
Equivalently, we may write sin A/a = sin B/b = sin C/c. The rule can be used in two situations: (i) when we know two angles and any one side (ASA or AAS), and (ii) when we know two sides and a non-included angle (SSA), though this second case requires careful attention to the ambiguous case.
等价地，我们也可以写成 sin A/a = sin B/b = sin C/c。正弦定理可用于两种情况：(i) 已知两角和任意一边（ASA 或 AAS）；(ii) 已知两边和其中一边的对角（SSA），但第二种情形需要特别注意解的情况可能不唯一。
To prove the sine rule, drop a perpendicular from vertex C to side c (or its extension), and call its length h. Then in the two right-angled triangles formed, h = b sin A and h = a sin B. Equating these gives a/sin A = b/sin B. Repeating the argument with a different altitude gives the full equality.
为了证明正弦定理，从顶点 C 向边 c（或其延长线）作垂线，设其长度为 h。在由此形成的两个直角三角形中，h = b sin A 且 h = a sin B。令二者相等，得到 a/sin A = b/sin B。用另外一条高重复该论证，即可得到完整的等式。
4. Applying the Sine Rule (ASA/AAS) | 正弦定理的应用（两角一边）
When two angles and one side of a triangle are known, the sine rule provides a direct method to find the remaining sides. The third angle is immediately found from the fact that the angle sum of a triangle is 180°.
当已知一个三角形的两角和一边时，正弦定理提供了一种直接求其余两边的方法。第三个角可由三角形内角和为 180° 立即求出。
Example: In triangle ABC, A = 40°, B = 60°, and a = 8 cm. Find side b and side c.
例：在三角形 ABC 中，A = 40°，B = 60°，a = 8 cm。求边 b 和边 c。
First, C = 180° − 40° − 60° = 80°. Using the sine rule: b/sin 60° = 8/sin 40°, so b = 8 sin 60°/sin 40° ≈ 10.78 cm. Similarly, c = 8 sin 80°/sin 40° ≈ 12.26 cm.
首先，C = 180° − 40° − 60° = 80°。利用正弦定理：b/sin 60° = 8/sin 40°，因此 b = 8 sin 60°/sin 40° ≈ 10.78 cm。同理，c = 8 sin 80°/sin 40° ≈ 12.26 cm。
A common exam tip: always use the exact given values, not rounded intermediate results, when computing subsequent quantities. Round only at the final step to an appropriate degree of accuracy, typically 1 decimal place for lengths and 1 degree for angles.
考试常见提示：计算后续量时，应使用给定的精确值，而不是中间结果的四舍五入值。只在最后一步按适当精度取近似值，长度通常保留 1 位小数，角度保留 1 度。
5. The Ambiguous Case of the Sine Rule (SSA) | 正弦定理的含混情况（SSA）
When we are given two sides and a non-included angle (SSA), the sine rule may yield zero, one, or two possible triangles. This is called the ambiguous case. It arises because the sine function is positive in both the first and second quadrants, so an angle θ and 180° − θ have the same sine value.
当我们已知两边和其中一边的对角（SSA）时，正弦定理可能得出零个、一个或两个可能的三角形。这就是所谓的含混情况（ambiguous case）。它的产生是因为正弦函数在第一象限和第二象限都为正，因此角 θ 与 180° − θ 具有相同的正弦值。
Suppose we know sides a and b and angle A. Let h = b sin A, the height of the triangle. Then:
假设已知边 a、b 和角 A。令 h = b sin A，即三角形的高。则：
- If a < h, no triangle exists.
- 如果 a < h，则三角形不存在。
- If a = h, exactly one right-angled triangle exists.
- 如果 a = h，则恰有一个直角三角形存在。
- If h < a < b, two distinct triangles exist.
- 如果 h < a < b，则存在两个不同的三角形。
- If a ≥ b, exactly one triangle exists.
- 如果 a ≥ b，则恰有一个三角形存在。
Example: Given a = 12 cm, b = 15 cm, and A = 35°, determine how many triangles are possible.
例：已知 a = 12 cm，b = 15 cm，A = 35°，判断有几个可能的三角形。
Here h = 15 sin 35° ≈ 8.60 cm. Since 8.60 < 12 < 15, we have h < a < b, so two triangles are possible. The angle B can be either B₁ = sin⁻¹(15 sin 35°/12) ≈ 45.8° or B₂ = 180° − 45.8° ≈ 134.2°.
此处 h = 15 sin 35° ≈ 8.60 cm。因为 8.60 < 12 < 15，即 h < a < b，所以存在两个可能的三角形。角 B 可以是 B₁ = sin⁻¹(15 sin 35°/12) ≈ 45.8°，或 B₂ = 180° − 45.8° ≈ 134.2°。
In an exam, if the problem does not specify “the acute case” or “the obtuse case,” you must consider both possibilities and state clearly which one satisfies the given conditions.
在考试中，如果题目没有特别说明”锐角情况”或”钝角情况”，你必须同时考虑两种可能性，并明确指出哪一种满足给定条件。
6. The Cosine Rule | 余弦定理
The cosine rule generalises Pythagoras’ theorem to any triangle. It relates the square of one side to the other two sides and the cosine of the included angle:
余弦定理将勾股定理推广到任意三角形。它把一条边的平方与另外两条边及其夹角的余弦联系起来：
a² = b² + c² − 2bc cos A
By cyclic permutation, we also have b² = a² + c² − 2ac cos B and c² = a² + b² − 2ab cos C. Notice that if A = 90°, then cos A = 0 and the formula reduces to a² = b² + c², which is exactly Pythagoras’ theorem. Thus the cosine rule is a genuine extension of Pythagoras.
通过轮换，我们还有 b² = a² + c² − 2ac cos B 和 c² = a² + b² − 2ab cos C。注意若 A = 90°，则 cos A = 0，公式退化为 a² = b² + c²，正是勾股定理。因此余弦定理是勾股定理的真正推广。
To prove the cosine rule, place triangle ABC with A at the origin and side c along the x-axis. Then B = (c, 0), and C = (b cos A, b sin A). The distance formula between B and C gives a² = (c − b cos A)² + (b sin A)² = b² + c² − 2bc cos A, after expanding and using cos²A + sin²A = 1.
为证明余弦定理，将三角形 ABC 置于坐标平面，A 在原点，边 c 沿 x 轴方向。则 B = (c, 0)，C = (b cos A, b sin A)。利用两点间距离公式可得 a² = (c − b cos A)² + (b sin A)² = b² + c² − 2bc cos A，展开并使用 cos²A + sin²A = 1 即得。
7. Applying the Cosine Rule (SAS) | 余弦定理的应用（两边及夹角）
The cosine rule is used directly when two sides and the included angle are known (SAS). We substitute the known values into a² = b² + c² − 2bc cos A to find the third side opposite the given angle.
当已知两边及其夹角（SAS）时，可直接使用余弦定理。我们将已知值代入 a² = b² + c² − 2bc cos A，求出给定角所对的第三边。
Example: In triangle ABC, b = 7 cm, c = 9 cm, and A = 48°. Find side a.
例：在三角形 ABC 中，b = 7 cm，c = 9 cm，A = 48°。求边 a。
Using the formula: a² = 7² + 9² − 2 × 7 × 9 × cos 48° = 49 + 81 − 126 × 0.6691 ≈ 130 − 84.31 = 45.69. Therefore a ≈ √45.69 ≈ 6.76 cm.
代入公式：a² = 7² + 9² − 2 × 7 × 9 × cos 48° = 49 + 81 − 126 × 0.6691 ≈ 130 − 84.31 = 45.69。因此 a ≈ √45.69 ≈ 6.76 cm。
Once the third side is found, the remaining angles can be found either by the sine rule or by a rearranged cosine rule. The rearranged form is particularly useful because it avoids the ambiguous case:
求出第三边后，其余角可用正弦定理或重排后的余弦定理求得。重排形式尤为有用，因为可以避免含混情况：
cos A = (b² + c² − a²)/(2bc)
This form is convenient for finding any angle when the calculator gives a unique inverse cosine value between 0° and 180°.
这种形式便于在已知三边时求任意角，因为计算器给出的反余弦值在 0° 到 180° 之间是唯一的。
8. Applying the Cosine Rule (SSS) | 余弦定理的应用（三边）
When all three sides of a triangle are known (SSS), we use the rearranged cosine rule to find each angle. The largest angle is opposite the longest side, and it is wise to find the largest angle first: if the triangle is obtuse, the cosine of the largest angle will be negative, confirming the presence of an obtuse angle.
当已知三角形的三边（SSS）时，我们使用重排后的余弦定理求各个角。最大角对着最长边，明智的做法是先求最大角：若三角形为钝角三角形，则最大角的余弦为负值，从而确认钝角的存在。
Example: A triangle has sides a = 6 cm, b = 7 cm, c = 10 cm. Find the largest angle.
例：一个三角形的三边为 a = 6 cm，b = 7 cm，c = 10 cm。求最大角。
The largest angle is C, opposite side c = 10 cm. Using cos C = (a² + b² − c²)/(2ab) = (36 + 49 − 100)/(2 × 6 × 7) = (−15)/84 ≈ −0.1786. Hence C = cos⁻¹(−0.1786) ≈ 100.3°.
最大角为 C，对着边 c = 10 cm。利用 cos C = (a² + b² − c²)/(2ab) = (36 + 49 − 100)/(2 × 6 × 7) = (−15)/84 ≈ −0.1786。因此 C = cos⁻¹(−0.1786) ≈ 100.3°。
After finding the largest angle, the other two angles can safely be found using the sine rule, since they must both be acute. This strategy avoids the ambiguous case completely.
求出最大角后，其余两个角可放心使用正弦定理求得，因为它们必然都是锐角。这种策略可以完全避免含混情况。
9. The Area Formula for a Non-Right-Angled Triangle | 非直角三角形的面积公式
Closely connected to the sine rule is the area formula for any triangle:
与正弦定理紧密相关的是任意三角形的面积公式：
Area = ½ ab sin C
This formula states that the area of a triangle is half the product of two sides and the sine of the included angle. It follows directly from the standard formula Area = ½ × base × height: if we take side c as the base, then the height is h = b sin A = a sin B, and Area = ½ c × b sin A = ½ bc sin A.
该公式说明三角形的面积等于两边及其夹角正弦乘积的一半。它直接由标准公式”面积 = ½ × 底 × 高”推出：若以边 c 为底，则高为 h = b sin A = a sin B，故面积 = ½ × c × b sin A = ½ bc sin A。
Example: Find the area of a triangle with sides b = 8 cm and c = 11 cm, and included angle A = 55°.
例：求两边长 b = 8 cm、c = 11 cm 且夹角 A = 55° 的三角形面积。
Area = ½ × 8 × 11 × sin 55° = 44 × 0.8192 ≈ 36.05 cm².
面积 = ½ × 8 × 11 × sin 55° = 44 × 0.8192 ≈ 36.05 cm²。
This formula is often useful in multi-part questions where an area is required after solving part of a triangle. It appears in IB Paper 1 and can be combined with either the sine rule or the cosine rule in a single problem.
该公式在多步问题中常用，当需要在求出部分三角形后进一步求面积时尤为方便。它在 IB Paper 1 中出现，并可与正弦定理或余弦定理结合在一道题中使用。
10. Choosing Between the Sine Rule and the Cosine Rule | 正弦定理与余弦定理的选择
A frequent difficulty for students is deciding which rule to use. The choice depends entirely on the information given:
学生经常遇到的困难是如何判断该用哪条定理。选择完全取决于题目给出的条件：
| Given information | Suggested rule |
| Two angles and a side (ASA/AAS) | Sine rule |
| Two sides and a non-included angle (SSA) | Sine rule, with careful check of ambiguous case |
| Two sides and the included angle (SAS) | Cosine rule first, then sine rule |
| Three sides (SSS) | Cosine rule (rearranged) |
给定条件 | 建议使用的定理
两角一边（ASA/AAS） | 正弦定理
两边及一边的对角（SSA） | 正弦定理，注意含混情况
两边及夹角（SAS） | 先用余弦定理，再用正弦定理
三边（SSS） | 余弦定理（重排形式）
A good general strategy is: use the cosine rule whenever you have more sides than angles; use the sine rule whenever you have more angles than sides. If you are solving for an angle in an SSS triangle, the rearranged cosine rule is safer than the sine rule because it gives a unique answer.
一个好的通用策略是：当条件中边多于角时用余弦定理；当角多于边时用正弦定理。若在 SSS 三角形中求角，重排的余弦定理比正弦定理更安全，因为它给出唯一答案。
11. Worked Examples and Exam Tips | 例题与考试技巧
Example 1 (two bearings): A boat travels 12 km on a bearing of 030°, then turns and travels 9 km on a bearing of 110°. Find the boat’s distance from its starting point.
例 1（两个方位角）：一艘船沿方位角 030° 航行 12 km，然后转向沿方位角 110° 航行 9 km。求船离出发点的距离。
The angle between the two paths is 110° − 30° = 80°. The distance d from the start is the side opposite this angle, with the two known sides being 12 and 9. By the cosine rule: d² = 12² + 9² − 2 × 12 × 9 × cos 80° = 144 + 81 − 216 × 0.17365 ≈ 225 − 37.51 = 187.49. So d ≈ 13.69 km.
两条路径之间的夹角为 110° − 30° = 80°。距出发点的距离 d 是该角所对的边，两条已知边为 12 和 9。由余弦定理：d² = 12² + 9² − 2 × 12 × 9 × cos 80° = 144 + 81 − 216 × 0.17365 ≈ 225 − 37.51 = 187.49。因此 d ≈ 13.69 km。
Example 2 (triangle in a diagram): In triangle XYZ, XY = 8 cm, YZ = 10 cm, and angle YZX = 30°. Find angle XZY and then the area of the triangle.
例 2（图形中的三角形）：在三角形 XYZ 中，XY = 8 cm，YZ = 10 cm，角 YZX = 30°。求角 XZY 及三角形面积。
Here we know two sides (XY and YZ) and a non-included angle, so we use the sine rule: sin X/10 = sin 30°/8, giving sin X = 10 × 0.5/8 = 0.625. Thus X ≈ 38.7° or 141.3°. The second value is impossible because the sum of angles would exceed 180°. So X ≈ 38.7°, hence angle Y = 180° − 30° − 38.7° ≈ 111.3°. Then Area = ½ × 8 × 10 × sin 111.3° ≈ 37.2 cm².
此题已知两边（XY 和 YZ）及其中一边的对角，因此使用正弦定理：sin X/10 = sin 30°/8，得 sin X = 10 × 0.5/8 = 0.625。故 X ≈ 38.7° 或 141.3°。第二个值不可能，因为内角和将超过 180°。所以 X ≈ 38.7°，从而角 Y = 180° − 30° − 38.7° ≈ 111.3°。面积 = ½ × 8 × 10 × sin 111.3° ≈ 37.2 cm²。
Exam tips: (1) Always sketch the triangle first. (2) Check whether your answer is reasonable: the largest side must face the largest angle. (3) Use degrees mode on your calculator unless the question specifies radians. (4) In IB exams, give answers to 3 significant figures unless told otherwise.
考试提示：(1) 先画三角形草图。(2) 检查答案是否合理：最大边必须对着最大角。(3) 除非题目要求弧度，否则计算器使用角度模式。(4) 在 IB 考试中，除非另有说明，答案保留 3 位有效数字。
12. Summary | 总结
The sine rule and the cosine rule are the two fundamental tools for solving non-right-angled triangles. The sine rule, a/sin A = b/sin B = c/sin C, is best used with two angles and a side, but requires caution in the SSA ambiguous case. The cosine rule, a² = b² + c² − 2bc cos A, handles SAS and SSS situations and generalises Pythagoras’ theorem.
正弦定理和余弦定理是解非直角三角形的两大基本工具。正弦定理 a/sin A = b/sin B = c/sin C 最适合已知两角一边的情形，但在 SSA 含混情况下需要格外小心。余弦定理 a² = b² + c² − 2bc cos A 处理 SAS 和 SSS 情形，并且是勾股定理的推广。
The area formula, Area = ½ ab sin C, completes the toolkit for non-right-angled triangle problems. Together, these formulas allow us to find every side, every angle, and the area of any triangle. With consistent notation, careful diagram-drawing, and smart rule selection, these problems become routine.
面积公式”面积 = ½ ab sin C”完善了解非直角三角形问题的工具箱。这些公式加在一起，使我们能够求出任意三角形的每条边、每个角和面积。只要记号一致、认真画图并合理选择定理，这类问题就能迎刃而解。
Master these rules, and you will be well prepared for the trigonometry questions in the IB Mathematics exams.
掌握这些定理，你就能从容应对 IB 数学考试中的三角函数问题。
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